Optimal. Leaf size=122 \[ \frac{5}{64 a^6 b (a-b x)}-\frac{5}{32 a^6 b (a+b x)}+\frac{1}{64 a^5 b (a-b x)^2}-\frac{3}{32 a^5 b (a+b x)^2}-\frac{1}{16 a^4 b (a+b x)^3}-\frac{1}{32 a^3 b (a+b x)^4}+\frac{15 \tanh ^{-1}\left (\frac{b x}{a}\right )}{64 a^7 b} \]
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Rubi [A] time = 0.0788462, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {627, 44, 208} \[ \frac{5}{64 a^6 b (a-b x)}-\frac{5}{32 a^6 b (a+b x)}+\frac{1}{64 a^5 b (a-b x)^2}-\frac{3}{32 a^5 b (a+b x)^2}-\frac{1}{16 a^4 b (a+b x)^3}-\frac{1}{32 a^3 b (a+b x)^4}+\frac{15 \tanh ^{-1}\left (\frac{b x}{a}\right )}{64 a^7 b} \]
Antiderivative was successfully verified.
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Rule 627
Rule 44
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^2 \left (a^2-b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a-b x)^3 (a+b x)^5} \, dx\\ &=\int \left (\frac{1}{32 a^5 (a-b x)^3}+\frac{5}{64 a^6 (a-b x)^2}+\frac{1}{8 a^3 (a+b x)^5}+\frac{3}{16 a^4 (a+b x)^4}+\frac{3}{16 a^5 (a+b x)^3}+\frac{5}{32 a^6 (a+b x)^2}+\frac{15}{64 a^6 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac{1}{64 a^5 b (a-b x)^2}+\frac{5}{64 a^6 b (a-b x)}-\frac{1}{32 a^3 b (a+b x)^4}-\frac{1}{16 a^4 b (a+b x)^3}-\frac{3}{32 a^5 b (a+b x)^2}-\frac{5}{32 a^6 b (a+b x)}+\frac{15 \int \frac{1}{a^2-b^2 x^2} \, dx}{64 a^6}\\ &=\frac{1}{64 a^5 b (a-b x)^2}+\frac{5}{64 a^6 b (a-b x)}-\frac{1}{32 a^3 b (a+b x)^4}-\frac{1}{16 a^4 b (a+b x)^3}-\frac{3}{32 a^5 b (a+b x)^2}-\frac{5}{32 a^6 b (a+b x)}+\frac{15 \tanh ^{-1}\left (\frac{b x}{a}\right )}{64 a^7 b}\\ \end{align*}
Mathematica [A] time = 0.0531594, size = 98, normalized size = 0.8 \[ \frac{\frac{2 a \left (50 a^3 b^2 x^2+10 a^2 b^3 x^3+17 a^4 b x-16 a^5-30 a b^4 x^4-15 b^5 x^5\right )}{(a-b x)^2 (a+b x)^4}-15 \log (a-b x)+15 \log (a+b x)}{128 a^7 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 126, normalized size = 1. \begin{align*}{\frac{15\,\ln \left ( bx+a \right ) }{128\,b{a}^{7}}}-{\frac{5}{32\,{a}^{6}b \left ( bx+a \right ) }}-{\frac{3}{32\,{a}^{5}b \left ( bx+a \right ) ^{2}}}-{\frac{1}{16\,{a}^{4}b \left ( bx+a \right ) ^{3}}}-{\frac{1}{32\,{a}^{3}b \left ( bx+a \right ) ^{4}}}-{\frac{15\,\ln \left ( bx-a \right ) }{128\,b{a}^{7}}}-{\frac{5}{64\,{a}^{6}b \left ( bx-a \right ) }}+{\frac{1}{64\,{a}^{5}b \left ( bx-a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.31346, size = 211, normalized size = 1.73 \begin{align*} -\frac{15 \, b^{5} x^{5} + 30 \, a b^{4} x^{4} - 10 \, a^{2} b^{3} x^{3} - 50 \, a^{3} b^{2} x^{2} - 17 \, a^{4} b x + 16 \, a^{5}}{64 \,{\left (a^{6} b^{7} x^{6} + 2 \, a^{7} b^{6} x^{5} - a^{8} b^{5} x^{4} - 4 \, a^{9} b^{4} x^{3} - a^{10} b^{3} x^{2} + 2 \, a^{11} b^{2} x + a^{12} b\right )}} + \frac{15 \, \log \left (b x + a\right )}{128 \, a^{7} b} - \frac{15 \, \log \left (b x - a\right )}{128 \, a^{7} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.70633, size = 547, normalized size = 4.48 \begin{align*} -\frac{30 \, a b^{5} x^{5} + 60 \, a^{2} b^{4} x^{4} - 20 \, a^{3} b^{3} x^{3} - 100 \, a^{4} b^{2} x^{2} - 34 \, a^{5} b x + 32 \, a^{6} - 15 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} - a^{2} b^{4} x^{4} - 4 \, a^{3} b^{3} x^{3} - a^{4} b^{2} x^{2} + 2 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right ) + 15 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} - a^{2} b^{4} x^{4} - 4 \, a^{3} b^{3} x^{3} - a^{4} b^{2} x^{2} + 2 \, a^{5} b x + a^{6}\right )} \log \left (b x - a\right )}{128 \,{\left (a^{7} b^{7} x^{6} + 2 \, a^{8} b^{6} x^{5} - a^{9} b^{5} x^{4} - 4 \, a^{10} b^{4} x^{3} - a^{11} b^{3} x^{2} + 2 \, a^{12} b^{2} x + a^{13} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.04247, size = 158, normalized size = 1.3 \begin{align*} - \frac{16 a^{5} - 17 a^{4} b x - 50 a^{3} b^{2} x^{2} - 10 a^{2} b^{3} x^{3} + 30 a b^{4} x^{4} + 15 b^{5} x^{5}}{64 a^{12} b + 128 a^{11} b^{2} x - 64 a^{10} b^{3} x^{2} - 256 a^{9} b^{4} x^{3} - 64 a^{8} b^{5} x^{4} + 128 a^{7} b^{6} x^{5} + 64 a^{6} b^{7} x^{6}} - \frac{\frac{15 \log{\left (- \frac{a}{b} + x \right )}}{128} - \frac{15 \log{\left (\frac{a}{b} + x \right )}}{128}}{a^{7} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20814, size = 169, normalized size = 1.39 \begin{align*} -\frac{15 \, \log \left ({\left | -\frac{2 \, a}{b x + a} + 1 \right |}\right )}{128 \, a^{7} b} + \frac{\frac{24 \, a}{b x + a} - 11}{256 \, a^{7} b{\left (\frac{2 \, a}{b x + a} - 1\right )}^{2}} - \frac{\frac{5 \, a^{6} b^{11}}{b x + a} + \frac{3 \, a^{7} b^{11}}{{\left (b x + a\right )}^{2}} + \frac{2 \, a^{8} b^{11}}{{\left (b x + a\right )}^{3}} + \frac{a^{9} b^{11}}{{\left (b x + a\right )}^{4}}}{32 \, a^{12} b^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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