Final answer to the problem
$\frac{a^{\left(3+2x\right)}b^{2x}}{3+2x}+\frac{2a^{\left(2+x\right)}b^xc^{2x}}{2+x}-\frac{1}{2}c^{4x}a^2\ln\left(c\right)+ac^{4x}+C_0$
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Step-by-step Solution
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Find the integral
$\int\left(a^{\left(x+1\right)}b^x+c^{2x}\right)^2da$
Learn how to solve integrals of rational functions problems step by step online.
$\int\left(a^{\left(x+1\right)}b^x+c^{2x}\right)^2da$
Learn how to solve integrals of rational functions problems step by step online. Find the integral of (a^(x+1)b^x+c^(2x))^2. Find the integral. Rewrite the integrand \left(a^{\left(x+1\right)}b^x+c^{2x}\right)^2 in expanded form. Expand the integral \int\left(a^{\left(2x+2\right)}b^{2x}+2a^{\left(x+1\right)}b^xc^{2x}+c^{4x}\right)da into 3 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int2a^{\left(x+1\right)}b^xc^{2x}da results in: 2\int a^{\left(x+1\right)}b^xc^{2x}da.
Final answer to the problem
$\frac{a^{\left(3+2x\right)}b^{2x}}{3+2x}+\frac{2a^{\left(2+x\right)}b^xc^{2x}}{2+x}-\frac{1}{2}c^{4x}a^2\ln\left(c\right)+ac^{4x}+C_0$