Final answer to the problem
$4a^{4}x+4a^{3}x^2+\frac{8x^{3}a^{2}}{3}+8a^{3}x+8a^{2}x+ax^{4}+4a^2x^2+4ax^2+\frac{x^{5}}{5}+\frac{4x^{3}a}{3}+\frac{4}{3}x^{3}+C_0$
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Step-by-step Solution
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Find the integral
$\int\left(a^2+\left(a+x\right)^2\right)\left(\left(a+x\right)^2+\left(a+2\right)^2\right)dx$
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$\int\left(a^2+\left(a+x\right)^2\right)\left(\left(a+x\right)^2+\left(a+2\right)^2\right)dx$
Learn how to solve problems step by step online. Integrate the function (a^2+(a+x)^2)((a+x)^2+(a+2)^2). Find the integral. Rewrite the integrand \left(a^2+\left(a+x\right)^2\right)\left(\left(a+x\right)^2+\left(a+2\right)^2\right) in expanded form. Expand the integral \int\left(4a^{4}+8a^{3}x+8a^{2}x^{2}+8a^{3}+8a^{2}+4ax^{3}+8a^2x+8ax+x^{4}+4ax^{2}+4x^{2}\right)dx into 11 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int4a^{4}dx results in: 4a^{4}x.
Final answer to the problem
$4a^{4}x+4a^{3}x^2+\frac{8x^{3}a^{2}}{3}+8a^{3}x+8a^{2}x+ax^{4}+4a^2x^2+4ax^2+\frac{x^{5}}{5}+\frac{4x^{3}a}{3}+\frac{4}{3}x^{3}+C_0$