Optimal. Leaf size=99 \[ \frac {15 b^4 x}{a^7}-\frac {5 b^3 x^2}{a^6}+\frac {2 b^2 x^3}{a^5}-\frac {3 b x^4}{4 a^4}+\frac {x^5}{5 a^3}+\frac {b^7}{2 a^8 (b+a x)^2}-\frac {7 b^6}{a^8 (b+a x)}-\frac {21 b^5 \log (b+a x)}{a^8} \]
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Rubi [A]
time = 0.06, antiderivative size = 99, 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^7}{2 a^8 (a x+b)^2}-\frac {7 b^6}{a^8 (a x+b)}-\frac {21 b^5 \log (a x+b)}{a^8}+\frac {15 b^4 x}{a^7}-\frac {5 b^3 x^2}{a^6}+\frac {2 b^2 x^3}{a^5}-\frac {3 b x^4}{4 a^4}+\frac {x^5}{5 a^3} \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 )^3} \, dx &=\int \frac {x^7}{(b+a x)^3} \, dx\\ &=\int \left (\frac {15 b^4}{a^7}-\frac {10 b^3 x}{a^6}+\frac {6 b^2 x^2}{a^5}-\frac {3 b x^3}{a^4}+\frac {x^4}{a^3}-\frac {b^7}{a^7 (b+a x)^3}+\frac {7 b^6}{a^7 (b+a x)^2}-\frac {21 b^5}{a^7 (b+a x)}\right ) \, dx\\ &=\frac {15 b^4 x}{a^7}-\frac {5 b^3 x^2}{a^6}+\frac {2 b^2 x^3}{a^5}-\frac {3 b x^4}{4 a^4}+\frac {x^5}{5 a^3}+\frac {b^7}{2 a^8 (b+a x)^2}-\frac {7 b^6}{a^8 (b+a x)}-\frac {21 b^5 \log (b+a x)}{a^8}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 85, normalized size = 0.86 \begin {gather*} \frac {300 a b^4 x-100 a^2 b^3 x^2+40 a^3 b^2 x^3-15 a^4 b x^4+4 a^5 x^5-\frac {10 b^6 (13 b+14 a x)}{(b+a x)^2}-420 b^5 \log (b+a x)}{20 a^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 94, normalized size = 0.95
method | result | size |
risch | \(\frac {x^{5}}{5 a^{3}}-\frac {3 b \,x^{4}}{4 a^{4}}+\frac {2 b^{2} x^{3}}{a^{5}}-\frac {5 b^{3} x^{2}}{a^{6}}+\frac {15 b^{4} x}{a^{7}}+\frac {-7 b^{6} x -\frac {13 b^{7}}{2 a}}{a^{7} \left (a x +b \right )^{2}}-\frac {21 b^{5} \ln \left (a x +b \right )}{a^{8}}\) | \(90\) |
norman | \(\frac {\frac {x^{7}}{5 a}-\frac {7 b \,x^{6}}{20 a^{2}}+\frac {7 b^{2} x^{5}}{10 a^{3}}-\frac {63 b^{7}}{2 a^{8}}-\frac {7 b^{3} x^{4}}{4 a^{4}}+\frac {7 b^{4} x^{3}}{a^{5}}-\frac {42 b^{6} x}{a^{7}}}{\left (a x +b \right )^{2}}-\frac {21 b^{5} \ln \left (a x +b \right )}{a^{8}}\) | \(92\) |
default | \(\frac {\frac {1}{5} a^{4} x^{5}-\frac {3}{4} a^{3} b \,x^{4}+2 a^{2} b^{2} x^{3}-5 a \,b^{3} x^{2}+15 b^{4} x}{a^{7}}+\frac {b^{7}}{2 a^{8} \left (a x +b \right )^{2}}-\frac {7 b^{6}}{a^{8} \left (a x +b \right )}-\frac {21 b^{5} \ln \left (a x +b \right )}{a^{8}}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 103, normalized size = 1.04 \begin {gather*} -\frac {14 \, a b^{6} x + 13 \, b^{7}}{2 \, {\left (a^{10} x^{2} + 2 \, a^{9} b x + a^{8} b^{2}\right )}} - \frac {21 \, b^{5} \log \left (a x + b\right )}{a^{8}} + \frac {4 \, a^{4} x^{5} - 15 \, a^{3} b x^{4} + 40 \, a^{2} b^{2} x^{3} - 100 \, a b^{3} x^{2} + 300 \, b^{4} x}{20 \, a^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 129, normalized size = 1.30 \begin {gather*} \frac {4 \, a^{7} x^{7} - 7 \, a^{6} b x^{6} + 14 \, a^{5} b^{2} x^{5} - 35 \, a^{4} b^{3} x^{4} + 140 \, a^{3} b^{4} x^{3} + 500 \, a^{2} b^{5} x^{2} + 160 \, a b^{6} x - 130 \, b^{7} - 420 \, {\left (a^{2} b^{5} x^{2} + 2 \, a b^{6} x + b^{7}\right )} \log \left (a x + b\right )}{20 \, {\left (a^{10} x^{2} + 2 \, a^{9} b x + a^{8} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 109, normalized size = 1.10 \begin {gather*} \frac {- 14 a b^{6} x - 13 b^{7}}{2 a^{10} x^{2} + 4 a^{9} b x + 2 a^{8} b^{2}} + \frac {x^{5}}{5 a^{3}} - \frac {3 b x^{4}}{4 a^{4}} + \frac {2 b^{2} x^{3}}{a^{5}} - \frac {5 b^{3} x^{2}}{a^{6}} + \frac {15 b^{4} x}{a^{7}} - \frac {21 b^{5} \log {\left (a x + b \right )}}{a^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 95, normalized size = 0.96 \begin {gather*} -\frac {21 \, b^{5} \log \left ({\left | a x + b \right |}\right )}{a^{8}} - \frac {14 \, a b^{6} x + 13 \, b^{7}}{2 \, {\left (a x + b\right )}^{2} a^{8}} + \frac {4 \, a^{12} x^{5} - 15 \, a^{11} b x^{4} + 40 \, a^{10} b^{2} x^{3} - 100 \, a^{9} b^{3} x^{2} + 300 \, a^{8} b^{4} x}{20 \, a^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.09, size = 104, normalized size = 1.05 \begin {gather*} \frac {x^5}{5\,a^3}-\frac {7\,b^6\,x+\frac {13\,b^7}{2\,a}}{a^9\,x^2+2\,a^8\,b\,x+a^7\,b^2}-\frac {21\,b^5\,\ln \left (b+a\,x\right )}{a^8}-\frac {3\,b\,x^4}{4\,a^4}+\frac {15\,b^4\,x}{a^7}+\frac {2\,b^2\,x^3}{a^5}-\frac {5\,b^3\,x^2}{a^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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