Optimal. Leaf size=81 \[ \frac {5 b^4 x}{a^6}-\frac {2 b^3 x^2}{a^5}+\frac {b^2 x^3}{a^4}-\frac {b x^4}{2 a^3}+\frac {x^5}{5 a^2}-\frac {b^6}{a^7 (b+a x)}-\frac {6 b^5 \log (b+a x)}{a^7} \]
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Rubi [A]
time = 0.04, antiderivative size = 81, 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 = {269, 45}
\begin {gather*} -\frac {b^6}{a^7 (a x+b)}-\frac {6 b^5 \log (a x+b)}{a^7}+\frac {5 b^4 x}{a^6}-\frac {2 b^3 x^2}{a^5}+\frac {b^2 x^3}{a^4}-\frac {b x^4}{2 a^3}+\frac {x^5}{5 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a+\frac {b}{x}\right )^2} \, dx &=\int \frac {x^6}{(b+a x)^2} \, dx\\ &=\int \left (\frac {5 b^4}{a^6}-\frac {4 b^3 x}{a^5}+\frac {3 b^2 x^2}{a^4}-\frac {2 b x^3}{a^3}+\frac {x^4}{a^2}+\frac {b^6}{a^6 (b+a x)^2}-\frac {6 b^5}{a^6 (b+a x)}\right ) \, dx\\ &=\frac {5 b^4 x}{a^6}-\frac {2 b^3 x^2}{a^5}+\frac {b^2 x^3}{a^4}-\frac {b x^4}{2 a^3}+\frac {x^5}{5 a^2}-\frac {b^6}{a^7 (b+a x)}-\frac {6 b^5 \log (b+a x)}{a^7}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 77, normalized size = 0.95 \begin {gather*} \frac {50 a b^4 x-20 a^2 b^3 x^2+10 a^3 b^2 x^3-5 a^4 b x^4+2 a^5 x^5-\frac {10 b^6}{b+a x}-60 b^5 \log (b+a x)}{10 a^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 78, normalized size = 0.96
method | result | size |
default | \(\frac {\frac {1}{5} a^{4} x^{5}-\frac {1}{2} a^{3} b \,x^{4}+a^{2} b^{2} x^{3}-2 a \,b^{3} x^{2}+5 b^{4} x}{a^{6}}-\frac {b^{6}}{a^{7} \left (a x +b \right )}-\frac {6 b^{5} \ln \left (a x +b \right )}{a^{7}}\) | \(78\) |
risch | \(\frac {5 b^{4} x}{a^{6}}-\frac {2 b^{3} x^{2}}{a^{5}}+\frac {b^{2} x^{3}}{a^{4}}-\frac {b \,x^{4}}{2 a^{3}}+\frac {x^{5}}{5 a^{2}}-\frac {b^{6}}{a^{7} \left (a x +b \right )}-\frac {6 b^{5} \ln \left (a x +b \right )}{a^{7}}\) | \(78\) |
norman | \(\frac {-\frac {6 b^{6}}{a^{7}}+\frac {x^{6}}{5 a}-\frac {3 b \,x^{5}}{10 a^{2}}+\frac {b^{2} x^{4}}{2 a^{3}}-\frac {b^{3} x^{3}}{a^{4}}+\frac {3 b^{4} x^{2}}{a^{5}}}{a x +b}-\frac {6 b^{5} \ln \left (a x +b \right )}{a^{7}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 82, normalized size = 1.01 \begin {gather*} -\frac {b^{6}}{a^{8} x + a^{7} b} - \frac {6 \, b^{5} \log \left (a x + b\right )}{a^{7}} + \frac {2 \, a^{4} x^{5} - 5 \, a^{3} b x^{4} + 10 \, a^{2} b^{2} x^{3} - 20 \, a b^{3} x^{2} + 50 \, b^{4} x}{10 \, a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 96, normalized size = 1.19 \begin {gather*} \frac {2 \, a^{6} x^{6} - 3 \, a^{5} b x^{5} + 5 \, a^{4} b^{2} x^{4} - 10 \, a^{3} b^{3} x^{3} + 30 \, a^{2} b^{4} x^{2} + 50 \, a b^{5} x - 10 \, b^{6} - 60 \, {\left (a b^{5} x + b^{6}\right )} \log \left (a x + b\right )}{10 \, {\left (a^{8} x + a^{7} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 78, normalized size = 0.96 \begin {gather*} - \frac {b^{6}}{a^{8} x + a^{7} b} + \frac {x^{5}}{5 a^{2}} - \frac {b x^{4}}{2 a^{3}} + \frac {b^{2} x^{3}}{a^{4}} - \frac {2 b^{3} x^{2}}{a^{5}} + \frac {5 b^{4} x}{a^{6}} - \frac {6 b^{5} \log {\left (a x + b \right )}}{a^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 85, normalized size = 1.05 \begin {gather*} -\frac {6 \, b^{5} \log \left ({\left | a x + b \right |}\right )}{a^{7}} - \frac {b^{6}}{{\left (a x + b\right )} a^{7}} + \frac {2 \, a^{8} x^{5} - 5 \, a^{7} b x^{4} + 10 \, a^{6} b^{2} x^{3} - 20 \, a^{5} b^{3} x^{2} + 50 \, a^{4} b^{4} x}{10 \, a^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 83, normalized size = 1.02 \begin {gather*} \frac {x^5}{5\,a^2}-\frac {6\,b^5\,\ln \left (b+a\,x\right )}{a^7}-\frac {b\,x^4}{2\,a^3}+\frac {5\,b^4\,x}{a^6}+\frac {b^2\,x^3}{a^4}-\frac {2\,b^3\,x^2}{a^5}-\frac {b^6}{a\,\left (x\,a^7+b\,a^6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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