Optimal. Leaf size=56 \[ -\frac{b^2}{a^3 x}-\frac{b^3 \log (x)}{a^4}+\frac{b^3 \log (a+b x)}{a^4}+\frac{b}{2 a^2 x^2}-\frac{1}{3 a x^3} \]
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Rubi [A] time = 0.0214565, antiderivative size = 56, 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} \[ -\frac{b^2}{a^3 x}-\frac{b^3 \log (x)}{a^4}+\frac{b^3 \log (a+b x)}{a^4}+\frac{b}{2 a^2 x^2}-\frac{1}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^4 (a+b x)} \, dx &=\int \left (\frac{1}{a x^4}-\frac{b}{a^2 x^3}+\frac{b^2}{a^3 x^2}-\frac{b^3}{a^4 x}+\frac{b^4}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac{1}{3 a x^3}+\frac{b}{2 a^2 x^2}-\frac{b^2}{a^3 x}-\frac{b^3 \log (x)}{a^4}+\frac{b^3 \log (a+b x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.0061387, size = 56, normalized size = 1. \[ -\frac{b^2}{a^3 x}-\frac{b^3 \log (x)}{a^4}+\frac{b^3 \log (a+b x)}{a^4}+\frac{b}{2 a^2 x^2}-\frac{1}{3 a x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 53, normalized size = 1. \begin{align*} -{\frac{1}{3\,a{x}^{3}}}+{\frac{b}{2\,{a}^{2}{x}^{2}}}-{\frac{{b}^{2}}{{a}^{3}x}}-{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{4}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) }{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0718, size = 69, normalized size = 1.23 \begin{align*} \frac{b^{3} \log \left (b x + a\right )}{a^{4}} - \frac{b^{3} \log \left (x\right )}{a^{4}} - \frac{6 \, b^{2} x^{2} - 3 \, a b x + 2 \, a^{2}}{6 \, a^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51962, size = 126, normalized size = 2.25 \begin{align*} \frac{6 \, b^{3} x^{3} \log \left (b x + a\right ) - 6 \, b^{3} x^{3} \log \left (x\right ) - 6 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}}{6 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.529138, size = 44, normalized size = 0.79 \begin{align*} - \frac{2 a^{2} - 3 a b x + 6 b^{2} x^{2}}{6 a^{3} x^{3}} + \frac{b^{3} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21334, size = 76, normalized size = 1.36 \begin{align*} \frac{b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{4}} - \frac{b^{3} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{6 \, a b^{2} x^{2} - 3 \, a^{2} b x + 2 \, a^{3}}{6 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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