Optimal. Leaf size=90 \[ -\frac{a^6}{3 b^7 (a+b x)^3}+\frac{3 a^5}{b^7 (a+b x)^2}-\frac{15 a^4}{b^7 (a+b x)}+\frac{10 a^2 x}{b^6}-\frac{20 a^3 \log (a+b x)}{b^7}-\frac{2 a x^2}{b^5}+\frac{x^3}{3 b^4} \]
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Rubi [A] time = 0.0565142, antiderivative size = 90, 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} \[ -\frac{a^6}{3 b^7 (a+b x)^3}+\frac{3 a^5}{b^7 (a+b x)^2}-\frac{15 a^4}{b^7 (a+b x)}+\frac{10 a^2 x}{b^6}-\frac{20 a^3 \log (a+b x)}{b^7}-\frac{2 a x^2}{b^5}+\frac{x^3}{3 b^4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^6}{(a+b x)^4} \, dx &=\int \left (\frac{10 a^2}{b^6}-\frac{4 a x}{b^5}+\frac{x^2}{b^4}+\frac{a^6}{b^6 (a+b x)^4}-\frac{6 a^5}{b^6 (a+b x)^3}+\frac{15 a^4}{b^6 (a+b x)^2}-\frac{20 a^3}{b^6 (a+b x)}\right ) \, dx\\ &=\frac{10 a^2 x}{b^6}-\frac{2 a x^2}{b^5}+\frac{x^3}{3 b^4}-\frac{a^6}{3 b^7 (a+b x)^3}+\frac{3 a^5}{b^7 (a+b x)^2}-\frac{15 a^4}{b^7 (a+b x)}-\frac{20 a^3 \log (a+b x)}{b^7}\\ \end{align*}
Mathematica [A] time = 0.0281285, size = 90, normalized size = 1. \[ -\frac{a^6}{3 b^7 (a+b x)^3}+\frac{3 a^5}{b^7 (a+b x)^2}-\frac{15 a^4}{b^7 (a+b x)}+\frac{10 a^2 x}{b^6}-\frac{20 a^3 \log (a+b x)}{b^7}-\frac{2 a x^2}{b^5}+\frac{x^3}{3 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 87, normalized size = 1. \begin{align*} 10\,{\frac{{a}^{2}x}{{b}^{6}}}-2\,{\frac{a{x}^{2}}{{b}^{5}}}+{\frac{{x}^{3}}{3\,{b}^{4}}}-{\frac{{a}^{6}}{3\,{b}^{7} \left ( bx+a \right ) ^{3}}}+3\,{\frac{{a}^{5}}{{b}^{7} \left ( bx+a \right ) ^{2}}}-15\,{\frac{{a}^{4}}{{b}^{7} \left ( bx+a \right ) }}-20\,{\frac{{a}^{3}\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 = 1.11261, size = 138, normalized size = 1.53 \begin{align*} -\frac{45 \, a^{4} b^{2} x^{2} + 81 \, a^{5} b x + 37 \, a^{6}}{3 \,{\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} - \frac{20 \, a^{3} \log \left (b x + a\right )}{b^{7}} + \frac{b^{2} x^{3} - 6 \, a b x^{2} + 30 \, a^{2} x}{3 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60368, size = 293, normalized size = 3.26 \begin{align*} \frac{b^{6} x^{6} - 3 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 73 \, a^{3} b^{3} x^{3} + 39 \, a^{4} b^{2} x^{2} - 51 \, a^{5} b x - 37 \, a^{6} - 60 \,{\left (a^{3} b^{3} x^{3} + 3 \, a^{4} b^{2} x^{2} + 3 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{3 \,{\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.758466, size = 105, normalized size = 1.17 \begin{align*} - \frac{20 a^{3} \log{\left (a + b x \right )}}{b^{7}} + \frac{10 a^{2} x}{b^{6}} - \frac{2 a x^{2}}{b^{5}} - \frac{37 a^{6} + 81 a^{5} b x + 45 a^{4} b^{2} x^{2}}{3 a^{3} b^{7} + 9 a^{2} b^{8} x + 9 a b^{9} x^{2} + 3 b^{10} x^{3}} + \frac{x^{3}}{3 b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20424, size = 112, normalized size = 1.24 \begin{align*} -\frac{20 \, a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac{45 \, a^{4} b^{2} x^{2} + 81 \, a^{5} b x + 37 \, a^{6}}{3 \,{\left (b x + a\right )}^{3} b^{7}} + \frac{b^{8} x^{3} - 6 \, a b^{7} x^{2} + 30 \, a^{2} b^{6} x}{3 \, b^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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