Optimal. Leaf size=83 \[ -\frac{4 a^3 (a+b x)^2}{b}-\frac{4 a^2 (a+b x)^3}{3 b}-\frac{32 a^5 \log (a-b x)}{b}-16 a^4 x-\frac{a (a+b x)^4}{2 b}-\frac{(a+b x)^5}{5 b} \]
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Rubi [A] time = 0.0319296, antiderivative size = 83, 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} \[ -\frac{4 a^3 (a+b x)^2}{b}-\frac{4 a^2 (a+b x)^3}{3 b}-\frac{32 a^5 \log (a-b x)}{b}-16 a^4 x-\frac{a (a+b x)^4}{2 b}-\frac{(a+b x)^5}{5 b} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^6}{a^2-b^2 x^2} \, dx &=\int \frac{(a+b x)^5}{a-b x} \, dx\\ &=\int \left (-16 a^4+\frac{32 a^5}{a-b x}-8 a^3 (a+b x)-4 a^2 (a+b x)^2-2 a (a+b x)^3-(a+b x)^4\right ) \, dx\\ &=-16 a^4 x-\frac{4 a^3 (a+b x)^2}{b}-\frac{4 a^2 (a+b x)^3}{3 b}-\frac{a (a+b x)^4}{2 b}-\frac{(a+b x)^5}{5 b}-\frac{32 a^5 \log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0074237, size = 65, normalized size = 0.78 \[ -\frac{16}{3} a^2 b^2 x^3-13 a^3 b x^2-\frac{32 a^5 \log (a-b x)}{b}-31 a^4 x-\frac{3}{2} a b^3 x^4-\frac{b^4 x^5}{5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 61, normalized size = 0.7 \begin{align*} -{\frac{{b}^{4}{x}^{5}}{5}}-{\frac{3\,a{b}^{3}{x}^{4}}{2}}-{\frac{16\,{b}^{2}{a}^{2}{x}^{3}}{3}}-13\,{a}^{3}b{x}^{2}-31\,{a}^{4}x-32\,{\frac{{a}^{5}\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 = 1.2065, size = 81, normalized size = 0.98 \begin{align*} -\frac{1}{5} \, b^{4} x^{5} - \frac{3}{2} \, a b^{3} x^{4} - \frac{16}{3} \, a^{2} b^{2} x^{3} - 13 \, a^{3} b x^{2} - 31 \, a^{4} x - \frac{32 \, a^{5} \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 = 1.94812, size = 147, normalized size = 1.77 \begin{align*} -\frac{6 \, b^{5} x^{5} + 45 \, a b^{4} x^{4} + 160 \, a^{2} b^{3} x^{3} + 390 \, a^{3} b^{2} x^{2} + 930 \, a^{4} b x + 960 \, a^{5} \log \left (b x - a\right )}{30 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.350551, size = 65, normalized size = 0.78 \begin{align*} - \frac{32 a^{5} \log{\left (- a + b x \right )}}{b} - 31 a^{4} x - 13 a^{3} b x^{2} - \frac{16 a^{2} b^{2} x^{3}}{3} - \frac{3 a b^{3} x^{4}}{2} - \frac{b^{4} x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17242, size = 97, normalized size = 1.17 \begin{align*} -\frac{32 \, a^{5} \log \left ({\left | b x - a \right |}\right )}{b} - \frac{6 \, b^{9} x^{5} + 45 \, a b^{8} x^{4} + 160 \, a^{2} b^{7} x^{3} + 390 \, a^{3} b^{6} x^{2} + 930 \, a^{4} b^{5} x}{30 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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