Optimal. Leaf size=58 \[ \frac{a^3}{3 b^4 (a+b x)^3}-\frac{3 a^2}{2 b^4 (a+b x)^2}+\frac{3 a}{b^4 (a+b x)}+\frac{\log (a+b x)}{b^4} \]
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Rubi [A] time = 0.0303948, antiderivative size = 58, 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} \[ \frac{a^3}{3 b^4 (a+b x)^3}-\frac{3 a^2}{2 b^4 (a+b x)^2}+\frac{3 a}{b^4 (a+b x)}+\frac{\log (a+b x)}{b^4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{(a+b x)^4} \, dx &=\int \left (-\frac{a^3}{b^3 (a+b x)^4}+\frac{3 a^2}{b^3 (a+b x)^3}-\frac{3 a}{b^3 (a+b x)^2}+\frac{1}{b^3 (a+b x)}\right ) \, dx\\ &=\frac{a^3}{3 b^4 (a+b x)^3}-\frac{3 a^2}{2 b^4 (a+b x)^2}+\frac{3 a}{b^4 (a+b x)}+\frac{\log (a+b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0189906, size = 44, normalized size = 0.76 \[ \frac{\frac{a \left (11 a^2+27 a b x+18 b^2 x^2\right )}{(a+b x)^3}+6 \log (a+b x)}{6 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 55, normalized size = 1. \begin{align*}{\frac{{a}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{3\,{a}^{2}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}+3\,{\frac{a}{{b}^{4} \left ( bx+a \right ) }}+{\frac{\ln \left ( bx+a \right ) }{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06126, size = 95, normalized size = 1.64 \begin{align*} \frac{18 \, a b^{2} x^{2} + 27 \, a^{2} b x + 11 \, a^{3}}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac{\log \left (b x + a\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51631, size = 201, normalized size = 3.47 \begin{align*} \frac{18 \, a b^{2} x^{2} + 27 \, a^{2} b x + 11 \, a^{3} + 6 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.552948, size = 70, normalized size = 1.21 \begin{align*} \frac{11 a^{3} + 27 a^{2} b x + 18 a b^{2} x^{2}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac{\log{\left (a + b x \right )}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19585, size = 62, normalized size = 1.07 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{b^{4}} + \frac{18 \, a b x^{2} + 27 \, a^{2} x + \frac{11 \, a^{3}}{b}}{6 \,{\left (b x + a\right )}^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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