Optimal. Leaf size=72 \[ \frac {b^5}{a^6 (a x+b)}+\frac {5 b^4 \log (a x+b)}{a^6}-\frac {4 b^3 x}{a^5}+\frac {3 b^2 x^2}{2 a^4}-\frac {2 b x^3}{3 a^3}+\frac {x^4}{4 a^2} \]
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Rubi [A] time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {263, 43} \[ \frac {3 b^2 x^2}{2 a^4}+\frac {b^5}{a^6 (a x+b)}-\frac {4 b^3 x}{a^5}+\frac {5 b^4 \log (a x+b)}{a^6}-\frac {2 b x^3}{3 a^3}+\frac {x^4}{4 a^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+\frac {b}{x}\right )^2} \, dx &=\int \frac {x^5}{(b+a x)^2} \, dx\\ &=\int \left (-\frac {4 b^3}{a^5}+\frac {3 b^2 x}{a^4}-\frac {2 b x^2}{a^3}+\frac {x^3}{a^2}-\frac {b^5}{a^5 (b+a x)^2}+\frac {5 b^4}{a^5 (b+a x)}\right ) \, dx\\ &=-\frac {4 b^3 x}{a^5}+\frac {3 b^2 x^2}{2 a^4}-\frac {2 b x^3}{3 a^3}+\frac {x^4}{4 a^2}+\frac {b^5}{a^6 (b+a x)}+\frac {5 b^4 \log (b+a x)}{a^6}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 66, normalized size = 0.92 \[ \frac {3 a^4 x^4-8 a^3 b x^3+18 a^2 b^2 x^2+\frac {12 b^5}{a x+b}+60 b^4 \log (a x+b)-48 a b^3 x}{12 a^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 85, normalized size = 1.18 \[ \frac {3 \, a^{5} x^{5} - 5 \, a^{4} b x^{4} + 10 \, a^{3} b^{2} x^{3} - 30 \, a^{2} b^{3} x^{2} - 48 \, a b^{4} x + 12 \, b^{5} + 60 \, {\left (a b^{4} x + b^{5}\right )} \log \left (a x + b\right )}{12 \, {\left (a^{7} x + a^{6} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 73, normalized size = 1.01 \[ \frac {5 \, b^{4} \log \left ({\left | a x + b \right |}\right )}{a^{6}} + \frac {b^{5}}{{\left (a x + b\right )} a^{6}} + \frac {3 \, a^{6} x^{4} - 8 \, a^{5} b x^{3} + 18 \, a^{4} b^{2} x^{2} - 48 \, a^{3} b^{3} x}{12 \, a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 67, normalized size = 0.93 \[ \frac {x^{4}}{4 a^{2}}-\frac {2 b \,x^{3}}{3 a^{3}}+\frac {3 b^{2} x^{2}}{2 a^{4}}-\frac {4 b^{3} x}{a^{5}}+\frac {b^{5}}{\left (a x +b \right ) a^{6}}+\frac {5 b^{4} \ln \left (a x +b \right )}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 70, normalized size = 0.97 \[ \frac {b^{5}}{a^{7} x + a^{6} b} + \frac {5 \, b^{4} \log \left (a x + b\right )}{a^{6}} + \frac {3 \, a^{3} x^{4} - 8 \, a^{2} b x^{3} + 18 \, a b^{2} x^{2} - 48 \, b^{3} x}{12 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 72, normalized size = 1.00 \[ \frac {x^4}{4\,a^2}+\frac {5\,b^4\,\ln \left (b+a\,x\right )}{a^6}-\frac {2\,b\,x^3}{3\,a^3}-\frac {4\,b^3\,x}{a^5}+\frac {3\,b^2\,x^2}{2\,a^4}+\frac {b^5}{a\,\left (x\,a^6+b\,a^5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 71, normalized size = 0.99 \[ \frac {b^{5}}{a^{7} x + a^{6} b} + \frac {x^{4}}{4 a^{2}} - \frac {2 b x^{3}}{3 a^{3}} + \frac {3 b^{2} x^{2}}{2 a^{4}} - \frac {4 b^{3} x}{a^{5}} + \frac {5 b^{4} \log {\left (a x + b \right )}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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