Optimal. Leaf size=86 \[ \frac{3 a^2 x^2}{b^5}-\frac{a^6}{2 b^7 (a+b x)^2}+\frac{6 a^5}{b^7 (a+b x)}-\frac{10 a^3 x}{b^6}+\frac{15 a^4 \log (a+b x)}{b^7}-\frac{a x^3}{b^4}+\frac{x^4}{4 b^3} \]
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Rubi [A] time = 0.0525685, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{3 a^2 x^2}{b^5}-\frac{a^6}{2 b^7 (a+b x)^2}+\frac{6 a^5}{b^7 (a+b x)}-\frac{10 a^3 x}{b^6}+\frac{15 a^4 \log (a+b x)}{b^7}-\frac{a x^3}{b^4}+\frac{x^4}{4 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^6}{(a+b x)^3} \, dx &=\int \left (-\frac{10 a^3}{b^6}+\frac{6 a^2 x}{b^5}-\frac{3 a x^2}{b^4}+\frac{x^3}{b^3}+\frac{a^6}{b^6 (a+b x)^3}-\frac{6 a^5}{b^6 (a+b x)^2}+\frac{15 a^4}{b^6 (a+b x)}\right ) \, dx\\ &=-\frac{10 a^3 x}{b^6}+\frac{3 a^2 x^2}{b^5}-\frac{a x^3}{b^4}+\frac{x^4}{4 b^3}-\frac{a^6}{2 b^7 (a+b x)^2}+\frac{6 a^5}{b^7 (a+b x)}+\frac{15 a^4 \log (a+b x)}{b^7}\\ \end{align*}
Mathematica [A] time = 0.0398734, size = 77, normalized size = 0.9 \[ \frac{12 a^2 b^2 x^2-\frac{2 a^6}{(a+b x)^2}+\frac{24 a^5}{a+b x}-40 a^3 b x+60 a^4 \log (a+b x)-4 a b^3 x^3+b^4 x^4}{4 b^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 83, normalized size = 1. \begin{align*} -10\,{\frac{{a}^{3}x}{{b}^{6}}}+3\,{\frac{{a}^{2}{x}^{2}}{{b}^{5}}}-{\frac{a{x}^{3}}{{b}^{4}}}+{\frac{{x}^{4}}{4\,{b}^{3}}}-{\frac{{a}^{6}}{2\,{b}^{7} \left ( bx+a \right ) ^{2}}}+6\,{\frac{{a}^{5}}{{b}^{7} \left ( bx+a \right ) }}+15\,{\frac{{a}^{4}\ln \left ( bx+a \right ) }{{b}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994151, size = 123, normalized size = 1.43 \begin{align*} \frac{12 \, a^{5} b x + 11 \, a^{6}}{2 \,{\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} + \frac{15 \, a^{4} \log \left (b x + a\right )}{b^{7}} + \frac{b^{3} x^{4} - 4 \, a b^{2} x^{3} + 12 \, a^{2} b x^{2} - 40 \, a^{3} x}{4 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51429, size = 247, normalized size = 2.87 \begin{align*} \frac{b^{6} x^{6} - 2 \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{4} - 20 \, a^{3} b^{3} x^{3} - 68 \, a^{4} b^{2} x^{2} - 16 \, a^{5} b x + 22 \, a^{6} + 60 \,{\left (a^{4} b^{2} x^{2} + 2 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{4 \,{\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.651687, size = 92, normalized size = 1.07 \begin{align*} \frac{15 a^{4} \log{\left (a + b x \right )}}{b^{7}} - \frac{10 a^{3} x}{b^{6}} + \frac{3 a^{2} x^{2}}{b^{5}} - \frac{a x^{3}}{b^{4}} + \frac{11 a^{6} + 12 a^{5} b x}{2 a^{2} b^{7} + 4 a b^{8} x + 2 b^{9} x^{2}} + \frac{x^{4}}{4 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20584, size = 112, normalized size = 1.3 \begin{align*} \frac{15 \, a^{4} \log \left ({\left | b x + a \right |}\right )}{b^{7}} + \frac{12 \, a^{5} b x + 11 \, a^{6}}{2 \,{\left (b x + a\right )}^{2} b^{7}} + \frac{b^{9} x^{4} - 4 \, a b^{8} x^{3} + 12 \, a^{2} b^{7} x^{2} - 40 \, a^{3} b^{6} x}{4 \, b^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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