Optimal. Leaf size=93 \[ \frac{6 b^2}{a^5 (a+b x)}+\frac{3 b^2}{2 a^4 (a+b x)^2}+\frac{b^2}{3 a^3 (a+b x)^3}+\frac{10 b^2 \log (x)}{a^6}-\frac{10 b^2 \log (a+b x)}{a^6}+\frac{4 b}{a^5 x}-\frac{1}{2 a^4 x^2} \]
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Rubi [A] time = 0.0481338, antiderivative size = 93, 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 = {44} \[ \frac{6 b^2}{a^5 (a+b x)}+\frac{3 b^2}{2 a^4 (a+b x)^2}+\frac{b^2}{3 a^3 (a+b x)^3}+\frac{10 b^2 \log (x)}{a^6}-\frac{10 b^2 \log (a+b x)}{a^6}+\frac{4 b}{a^5 x}-\frac{1}{2 a^4 x^2} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 (a+b x)^4} \, dx &=\int \left (\frac{1}{a^4 x^3}-\frac{4 b}{a^5 x^2}+\frac{10 b^2}{a^6 x}-\frac{b^3}{a^3 (a+b x)^4}-\frac{3 b^3}{a^4 (a+b x)^3}-\frac{6 b^3}{a^5 (a+b x)^2}-\frac{10 b^3}{a^6 (a+b x)}\right ) \, dx\\ &=-\frac{1}{2 a^4 x^2}+\frac{4 b}{a^5 x}+\frac{b^2}{3 a^3 (a+b x)^3}+\frac{3 b^2}{2 a^4 (a+b x)^2}+\frac{6 b^2}{a^5 (a+b x)}+\frac{10 b^2 \log (x)}{a^6}-\frac{10 b^2 \log (a+b x)}{a^6}\\ \end{align*}
Mathematica [A] time = 0.10632, size = 79, normalized size = 0.85 \[ \frac{\frac{a \left (110 a^2 b^2 x^2+15 a^3 b x-3 a^4+150 a b^3 x^3+60 b^4 x^4\right )}{x^2 (a+b x)^3}-60 b^2 \log (a+b x)+60 b^2 \log (x)}{6 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 88, normalized size = 1. \begin{align*} -{\frac{1}{2\,{a}^{4}{x}^{2}}}+4\,{\frac{b}{{a}^{5}x}}+{\frac{{b}^{2}}{3\,{a}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{3\,{b}^{2}}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}}+6\,{\frac{{b}^{2}}{{a}^{5} \left ( bx+a \right ) }}+10\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{6}}}-10\,{\frac{{b}^{2}\ln \left ( bx+a \right ) }{{a}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04163, size = 146, normalized size = 1.57 \begin{align*} \frac{60 \, b^{4} x^{4} + 150 \, a b^{3} x^{3} + 110 \, a^{2} b^{2} x^{2} + 15 \, a^{3} b x - 3 \, a^{4}}{6 \,{\left (a^{5} b^{3} x^{5} + 3 \, a^{6} b^{2} x^{4} + 3 \, a^{7} b x^{3} + a^{8} x^{2}\right )}} - \frac{10 \, b^{2} \log \left (b x + a\right )}{a^{6}} + \frac{10 \, b^{2} \log \left (x\right )}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.61244, size = 363, normalized size = 3.9 \begin{align*} \frac{60 \, a b^{4} x^{4} + 150 \, a^{2} b^{3} x^{3} + 110 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x - 3 \, a^{5} - 60 \,{\left (b^{5} x^{5} + 3 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{3} + a^{3} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{5} x^{5} + 3 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{3} + a^{3} b^{2} x^{2}\right )} \log \left (x\right )}{6 \,{\left (a^{6} b^{3} x^{5} + 3 \, a^{7} b^{2} x^{4} + 3 \, a^{8} b x^{3} + a^{9} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.8636, size = 104, normalized size = 1.12 \begin{align*} \frac{- 3 a^{4} + 15 a^{3} b x + 110 a^{2} b^{2} x^{2} + 150 a b^{3} x^{3} + 60 b^{4} x^{4}}{6 a^{8} x^{2} + 18 a^{7} b x^{3} + 18 a^{6} b^{2} x^{4} + 6 a^{5} b^{3} x^{5}} + \frac{10 b^{2} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19504, size = 116, normalized size = 1.25 \begin{align*} -\frac{10 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{6}} + \frac{10 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac{60 \, a b^{4} x^{4} + 150 \, a^{2} b^{3} x^{3} + 110 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x - 3 \, a^{5}}{6 \,{\left (b x + a\right )}^{3} a^{6} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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