Optimal. Leaf size=89 \[ -\frac {10 b^3 \log (x)}{a^6}+\frac {10 b^3 \log (a+b x)}{a^6}-\frac {4 b^3}{a^5 (a+b x)}-\frac {6 b^2}{a^5 x}-\frac {b^3}{2 a^4 (a+b x)^2}+\frac {3 b}{2 a^4 x^2}-\frac {1}{3 a^3 x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, 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} \[ -\frac {4 b^3}{a^5 (a+b x)}-\frac {b^3}{2 a^4 (a+b x)^2}-\frac {6 b^2}{a^5 x}-\frac {10 b^3 \log (x)}{a^6}+\frac {10 b^3 \log (a+b x)}{a^6}+\frac {3 b}{2 a^4 x^2}-\frac {1}{3 a^3 x^3} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x^4 (a+b x)^3} \, dx &=\int \left (\frac {1}{a^3 x^4}-\frac {3 b}{a^4 x^3}+\frac {6 b^2}{a^5 x^2}-\frac {10 b^3}{a^6 x}+\frac {b^4}{a^4 (a+b x)^3}+\frac {4 b^4}{a^5 (a+b x)^2}+\frac {10 b^4}{a^6 (a+b x)}\right ) \, dx\\ &=-\frac {1}{3 a^3 x^3}+\frac {3 b}{2 a^4 x^2}-\frac {6 b^2}{a^5 x}-\frac {b^3}{2 a^4 (a+b x)^2}-\frac {4 b^3}{a^5 (a+b x)}-\frac {10 b^3 \log (x)}{a^6}+\frac {10 b^3 \log (a+b x)}{a^6}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 79, normalized size = 0.89 \[ -\frac {\frac {a \left (2 a^4-5 a^3 b x+20 a^2 b^2 x^2+90 a b^3 x^3+60 b^4 x^4\right )}{x^3 (a+b x)^2}-60 b^3 \log (a+b x)+60 b^3 \log (x)}{6 a^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 141, normalized size = 1.58 \[ -\frac {60 \, a b^{4} x^{4} + 90 \, a^{2} b^{3} x^{3} + 20 \, a^{3} b^{2} x^{2} - 5 \, a^{4} b x + 2 \, a^{5} - 60 \, {\left (b^{5} x^{5} + 2 \, a b^{4} x^{4} + a^{2} b^{3} x^{3}\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} x^{5} + 2 \, a b^{4} x^{4} + a^{2} b^{3} x^{3}\right )} \log \relax (x)}{6 \, {\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 86, normalized size = 0.97 \[ \frac {10 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{6}} - \frac {10 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{6}} - \frac {60 \, a b^{4} x^{4} + 90 \, a^{2} b^{3} x^{3} + 20 \, a^{3} b^{2} x^{2} - 5 \, a^{4} b x + 2 \, a^{5}}{6 \, {\left (b x + a\right )}^{2} a^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.94 \[ -\frac {b^{3}}{2 \left (b x +a \right )^{2} a^{4}}-\frac {4 b^{3}}{\left (b x +a \right ) a^{5}}-\frac {10 b^{3} \ln \relax (x )}{a^{6}}+\frac {10 b^{3} \ln \left (b x +a \right )}{a^{6}}-\frac {6 b^{2}}{a^{5} x}+\frac {3 b}{2 a^{4} x^{2}}-\frac {1}{3 a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 97, normalized size = 1.09 \[ -\frac {60 \, b^{4} x^{4} + 90 \, a b^{3} x^{3} + 20 \, a^{2} b^{2} x^{2} - 5 \, a^{3} b x + 2 \, a^{4}}{6 \, {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} + \frac {10 \, b^{3} \log \left (b x + a\right )}{a^{6}} - \frac {10 \, b^{3} \log \relax (x)}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 91, normalized size = 1.02 \[ \frac {20\,b^3\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^6}-\frac {\frac {1}{3\,a}+\frac {10\,b^2\,x^2}{3\,a^3}+\frac {15\,b^3\,x^3}{a^4}+\frac {10\,b^4\,x^4}{a^5}-\frac {5\,b\,x}{6\,a^2}}{a^2\,x^3+2\,a\,b\,x^4+b^2\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 92, normalized size = 1.03 \[ \frac {- 2 a^{4} + 5 a^{3} b x - 20 a^{2} b^{2} x^{2} - 90 a b^{3} x^{3} - 60 b^{4} x^{4}}{6 a^{7} x^{3} + 12 a^{6} b x^{4} + 6 a^{5} b^{2} x^{5}} + \frac {10 b^{3} \left (- \log {\relax (x )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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