Optimal. Leaf size=66 \[ -\frac{2 a^2 (a+b x)^2}{b}-\frac{16 a^4 \log (a-b x)}{b}-8 a^3 x-\frac{2 a (a+b x)^3}{3 b}-\frac{(a+b x)^4}{4 b} \]
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Rubi [A] time = 0.0253941, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {627, 43} \[ -\frac{2 a^2 (a+b x)^2}{b}-\frac{16 a^4 \log (a-b x)}{b}-8 a^3 x-\frac{2 a (a+b x)^3}{3 b}-\frac{(a+b x)^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^5}{a^2-b^2 x^2} \, dx &=\int \frac{(a+b x)^4}{a-b x} \, dx\\ &=\int \left (-8 a^3+\frac{16 a^4}{a-b x}-4 a^2 (a+b x)-2 a (a+b x)^2-(a+b x)^3\right ) \, dx\\ &=-8 a^3 x-\frac{2 a^2 (a+b x)^2}{b}-\frac{2 a (a+b x)^3}{3 b}-\frac{(a+b x)^4}{4 b}-\frac{16 a^4 \log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0059495, size = 54, normalized size = 0.82 \[ -\frac{11}{2} a^2 b x^2-\frac{16 a^4 \log (a-b x)}{b}-15 a^3 x-\frac{5}{3} a b^2 x^3-\frac{b^3 x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 50, normalized size = 0.8 \begin{align*} -{\frac{{b}^{3}{x}^{4}}{4}}-{\frac{5\,a{b}^{2}{x}^{3}}{3}}-{\frac{11\,{a}^{2}b{x}^{2}}{2}}-15\,x{a}^{3}-16\,{\frac{{a}^{4}\ln \left ( bx-a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20587, size = 66, normalized size = 1. \begin{align*} -\frac{1}{4} \, b^{3} x^{4} - \frac{5}{3} \, a b^{2} x^{3} - \frac{11}{2} \, a^{2} b x^{2} - 15 \, a^{3} x - \frac{16 \, a^{4} \log \left (b x - a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82077, size = 122, normalized size = 1.85 \begin{align*} -\frac{3 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 66 \, a^{2} b^{2} x^{2} + 180 \, a^{3} b x + 192 \, a^{4} \log \left (b x - a\right )}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.327504, size = 53, normalized size = 0.8 \begin{align*} - \frac{16 a^{4} \log{\left (- a + b x \right )}}{b} - 15 a^{3} x - \frac{11 a^{2} b x^{2}}{2} - \frac{5 a b^{2} x^{3}}{3} - \frac{b^{3} x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17887, size = 82, normalized size = 1.24 \begin{align*} -\frac{16 \, a^{4} \log \left ({\left | b x - a \right |}\right )}{b} - \frac{3 \, b^{7} x^{4} + 20 \, a b^{6} x^{3} + 66 \, a^{2} b^{5} x^{2} + 180 \, a^{3} b^{4} x}{12 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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