Optimal. Leaf size=70 \[ -\frac{3 b}{a^4 (a+b x)}-\frac{b}{a^3 (a+b x)^2}-\frac{b}{3 a^2 (a+b x)^3}-\frac{4 b \log (x)}{a^5}+\frac{4 b \log (a+b x)}{a^5}-\frac{1}{a^4 x} \]
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Rubi [A] time = 0.0388295, antiderivative size = 70, 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} \[ -\frac{3 b}{a^4 (a+b x)}-\frac{b}{a^3 (a+b x)^2}-\frac{b}{3 a^2 (a+b x)^3}-\frac{4 b \log (x)}{a^5}+\frac{4 b \log (a+b x)}{a^5}-\frac{1}{a^4 x} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^2 (a+b x)^4} \, dx &=\int \left (\frac{1}{a^4 x^2}-\frac{4 b}{a^5 x}+\frac{b^2}{a^2 (a+b x)^4}+\frac{2 b^2}{a^3 (a+b x)^3}+\frac{3 b^2}{a^4 (a+b x)^2}+\frac{4 b^2}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac{1}{a^4 x}-\frac{b}{3 a^2 (a+b x)^3}-\frac{b}{a^3 (a+b x)^2}-\frac{3 b}{a^4 (a+b x)}-\frac{4 b \log (x)}{a^5}+\frac{4 b \log (a+b x)}{a^5}\\ \end{align*}
Mathematica [A] time = 0.102326, size = 64, normalized size = 0.91 \[ -\frac{\frac{a \left (22 a^2 b x+3 a^3+30 a b^2 x^2+12 b^3 x^3\right )}{x (a+b x)^3}-12 b \log (a+b x)+12 b \log (x)}{3 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 69, normalized size = 1. \begin{align*} -{\frac{1}{{a}^{4}x}}-{\frac{b}{3\,{a}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{b}{{a}^{3} \left ( bx+a \right ) ^{2}}}-3\,{\frac{b}{{a}^{4} \left ( bx+a \right ) }}-4\,{\frac{b\ln \left ( x \right ) }{{a}^{5}}}+4\,{\frac{b\ln \left ( bx+a \right ) }{{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04235, size = 123, normalized size = 1.76 \begin{align*} -\frac{12 \, b^{3} x^{3} + 30 \, a b^{2} x^{2} + 22 \, a^{2} b x + 3 \, a^{3}}{3 \,{\left (a^{4} b^{3} x^{4} + 3 \, a^{5} b^{2} x^{3} + 3 \, a^{6} b x^{2} + a^{7} x\right )}} + \frac{4 \, b \log \left (b x + a\right )}{a^{5}} - \frac{4 \, b \log \left (x\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.39027, size = 325, normalized size = 4.64 \begin{align*} -\frac{12 \, a b^{3} x^{3} + 30 \, a^{2} b^{2} x^{2} + 22 \, a^{3} b x + 3 \, a^{4} - 12 \,{\left (b^{4} x^{4} + 3 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + a^{3} b x\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} x^{4} + 3 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + a^{3} b x\right )} \log \left (x\right )}{3 \,{\left (a^{5} b^{3} x^{4} + 3 \, a^{6} b^{2} x^{3} + 3 \, a^{7} b x^{2} + a^{8} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.793333, size = 88, normalized size = 1.26 \begin{align*} - \frac{3 a^{3} + 22 a^{2} b x + 30 a b^{2} x^{2} + 12 b^{3} x^{3}}{3 a^{7} x + 9 a^{6} b x^{2} + 9 a^{5} b^{2} x^{3} + 3 a^{4} b^{3} x^{4}} + \frac{4 b \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1801, size = 96, normalized size = 1.37 \begin{align*} \frac{4 \, b \log \left ({\left | b x + a \right |}\right )}{a^{5}} - \frac{4 \, b \log \left ({\left | x \right |}\right )}{a^{5}} - \frac{12 \, a b^{3} x^{3} + 30 \, a^{2} b^{2} x^{2} + 22 \, a^{3} b x + 3 \, a^{4}}{3 \,{\left (b x + a\right )}^{3} a^{5} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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