Optimal. Leaf size=86 \[ \frac{3 b^2 x^2}{a^5}-\frac{b^6}{2 a^7 (a x+b)^2}+\frac{6 b^5}{a^7 (a x+b)}-\frac{10 b^3 x}{a^6}+\frac{15 b^4 \log (a x+b)}{a^7}-\frac{b x^3}{a^4}+\frac{x^4}{4 a^3} \]
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Rubi [A] time = 0.0562362, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 43} \[ \frac{3 b^2 x^2}{a^5}-\frac{b^6}{2 a^7 (a x+b)^2}+\frac{6 b^5}{a^7 (a x+b)}-\frac{10 b^3 x}{a^6}+\frac{15 b^4 \log (a x+b)}{a^7}-\frac{b x^3}{a^4}+\frac{x^4}{4 a^3} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+\frac{b}{x}\right )^3} \, dx &=\int \frac{x^6}{(b+a x)^3} \, dx\\ &=\int \left (-\frac{10 b^3}{a^6}+\frac{6 b^2 x}{a^5}-\frac{3 b x^2}{a^4}+\frac{x^3}{a^3}+\frac{b^6}{a^6 (b+a x)^3}-\frac{6 b^5}{a^6 (b+a x)^2}+\frac{15 b^4}{a^6 (b+a x)}\right ) \, dx\\ &=-\frac{10 b^3 x}{a^6}+\frac{3 b^2 x^2}{a^5}-\frac{b x^3}{a^4}+\frac{x^4}{4 a^3}-\frac{b^6}{2 a^7 (b+a x)^2}+\frac{6 b^5}{a^7 (b+a x)}+\frac{15 b^4 \log (b+a x)}{a^7}\\ \end{align*}
Mathematica [A] time = 0.0471842, size = 73, normalized size = 0.85 \[ \frac{12 a^2 b^2 x^2-4 a^3 b x^3+a^4 x^4+\frac{2 b^5 (12 a x+11 b)}{(a x+b)^2}-40 a b^3 x+60 b^4 \log (a x+b)}{4 a^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 83, normalized size = 1. \begin{align*} -10\,{\frac{{b}^{3}x}{{a}^{6}}}+3\,{\frac{{b}^{2}{x}^{2}}{{a}^{5}}}-{\frac{b{x}^{3}}{{a}^{4}}}+{\frac{{x}^{4}}{4\,{a}^{3}}}-{\frac{{b}^{6}}{2\,{a}^{7} \left ( ax+b \right ) ^{2}}}+6\,{\frac{{b}^{5}}{{a}^{7} \left ( ax+b \right ) }}+15\,{\frac{{b}^{4}\ln \left ( ax+b \right ) }{{a}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17405, size = 123, normalized size = 1.43 \begin{align*} \frac{12 \, a b^{5} x + 11 \, b^{6}}{2 \,{\left (a^{9} x^{2} + 2 \, a^{8} b x + a^{7} b^{2}\right )}} + \frac{15 \, b^{4} \log \left (a x + b\right )}{a^{7}} + \frac{a^{3} x^{4} - 4 \, a^{2} b x^{3} + 12 \, a b^{2} x^{2} - 40 \, b^{3} x}{4 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42911, size = 247, normalized size = 2.87 \begin{align*} \frac{a^{6} x^{6} - 2 \, a^{5} b x^{5} + 5 \, a^{4} b^{2} x^{4} - 20 \, a^{3} b^{3} x^{3} - 68 \, a^{2} b^{4} x^{2} - 16 \, a b^{5} x + 22 \, b^{6} + 60 \,{\left (a^{2} b^{4} x^{2} + 2 \, a b^{5} x + b^{6}\right )} \log \left (a x + b\right )}{4 \,{\left (a^{9} x^{2} + 2 \, a^{8} b x + a^{7} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.455783, size = 92, normalized size = 1.07 \begin{align*} \frac{12 a b^{5} x + 11 b^{6}}{2 a^{9} x^{2} + 4 a^{8} b x + 2 a^{7} b^{2}} + \frac{x^{4}}{4 a^{3}} - \frac{b x^{3}}{a^{4}} + \frac{3 b^{2} x^{2}}{a^{5}} - \frac{10 b^{3} x}{a^{6}} + \frac{15 b^{4} \log{\left (a x + b \right )}}{a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08812, size = 112, normalized size = 1.3 \begin{align*} \frac{15 \, b^{4} \log \left ({\left | a x + b \right |}\right )}{a^{7}} + \frac{12 \, a b^{5} x + 11 \, b^{6}}{2 \,{\left (a x + b\right )}^{2} a^{7}} + \frac{a^{9} x^{4} - 4 \, a^{8} b x^{3} + 12 \, a^{7} b^{2} x^{2} - 40 \, a^{6} b^{3} x}{4 \, a^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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