Optimal. Leaf size=86 \[ -\frac {10 b^3 x}{a^6}+\frac {3 b^2 x^2}{a^5}-\frac {b x^3}{a^4}+\frac {x^4}{4 a^3}-\frac {b^6}{2 a^7 (b+a x)^2}+\frac {6 b^5}{a^7 (b+a x)}+\frac {15 b^4 \log (b+a x)}{a^7} \]
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Rubi [A]
time = 0.04, antiderivative size = 86, 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}{2 a^7 (a x+b)^2}+\frac {6 b^5}{a^7 (a x+b)}+\frac {15 b^4 \log (a x+b)}{a^7}-\frac {10 b^3 x}{a^6}+\frac {3 b^2 x^2}{a^5}-\frac {b x^3}{a^4}+\frac {x^4}{4 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+\frac {b}{x}\right )^3} \, dx &=\int \frac {x^6}{(b+a x)^3} \, dx\\ &=\int \left (-\frac {10 b^3}{a^6}+\frac {6 b^2 x}{a^5}-\frac {3 b x^2}{a^4}+\frac {x^3}{a^3}+\frac {b^6}{a^6 (b+a x)^3}-\frac {6 b^5}{a^6 (b+a x)^2}+\frac {15 b^4}{a^6 (b+a x)}\right ) \, dx\\ &=-\frac {10 b^3 x}{a^6}+\frac {3 b^2 x^2}{a^5}-\frac {b x^3}{a^4}+\frac {x^4}{4 a^3}-\frac {b^6}{2 a^7 (b+a x)^2}+\frac {6 b^5}{a^7 (b+a x)}+\frac {15 b^4 \log (b+a x)}{a^7}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 73, normalized size = 0.85 \begin {gather*} \frac {-40 a b^3 x+12 a^2 b^2 x^2-4 a^3 b x^3+a^4 x^4+\frac {2 b^5 (11 b+12 a x)}{(b+a x)^2}+60 b^4 \log (b+a x)}{4 a^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 83, normalized size = 0.97
method | result | size |
risch | \(\frac {x^{4}}{4 a^{3}}-\frac {b \,x^{3}}{a^{4}}+\frac {3 b^{2} x^{2}}{a^{5}}-\frac {10 b^{3} x}{a^{6}}+\frac {6 b^{5} x +\frac {11 b^{6}}{2 a}}{a^{6} \left (a x +b \right )^{2}}+\frac {15 b^{4} \ln \left (a x +b \right )}{a^{7}}\) | \(79\) |
norman | \(\frac {\frac {x^{6}}{4 a}-\frac {b \,x^{5}}{2 a^{2}}+\frac {5 b^{2} x^{4}}{4 a^{3}}+\frac {45 b^{6}}{2 a^{7}}-\frac {5 b^{3} x^{3}}{a^{4}}+\frac {30 b^{5} x}{a^{6}}}{\left (a x +b \right )^{2}}+\frac {15 b^{4} \ln \left (a x +b \right )}{a^{7}}\) | \(81\) |
default | \(\frac {\frac {1}{4} a^{3} x^{4}-a^{2} b \,x^{3}+3 a \,b^{2} x^{2}-10 b^{3} x}{a^{6}}-\frac {b^{6}}{2 a^{7} \left (a x +b \right )^{2}}+\frac {6 b^{5}}{a^{7} \left (a x +b \right )}+\frac {15 b^{4} \ln \left (a x +b \right )}{a^{7}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 91, normalized size = 1.06 \begin {gather*} \frac {12 \, a b^{5} x + 11 \, b^{6}}{2 \, {\left (a^{9} x^{2} + 2 \, a^{8} b x + a^{7} b^{2}\right )}} + \frac {15 \, b^{4} \log \left (a x + b\right )}{a^{7}} + \frac {a^{3} x^{4} - 4 \, a^{2} b x^{3} + 12 \, a b^{2} x^{2} - 40 \, b^{3} x}{4 \, a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 117, normalized size = 1.36 \begin {gather*} \frac {a^{6} x^{6} - 2 \, a^{5} b x^{5} + 5 \, a^{4} b^{2} x^{4} - 20 \, a^{3} b^{3} x^{3} - 68 \, a^{2} b^{4} x^{2} - 16 \, a b^{5} x + 22 \, b^{6} + 60 \, {\left (a^{2} b^{4} x^{2} + 2 \, a b^{5} x + b^{6}\right )} \log \left (a x + b\right )}{4 \, {\left (a^{9} x^{2} + 2 \, a^{8} b x + a^{7} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 92, normalized size = 1.07 \begin {gather*} \frac {12 a b^{5} x + 11 b^{6}}{2 a^{9} x^{2} + 4 a^{8} b x + 2 a^{7} b^{2}} + \frac {x^{4}}{4 a^{3}} - \frac {b x^{3}}{a^{4}} + \frac {3 b^{2} x^{2}}{a^{5}} - \frac {10 b^{3} x}{a^{6}} + \frac {15 b^{4} \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 = 0.52, size = 83, normalized size = 0.97 \begin {gather*} \frac {15 \, b^{4} \log \left ({\left | a x + b \right |}\right )}{a^{7}} + \frac {12 \, a b^{5} x + 11 \, b^{6}}{2 \, {\left (a x + b\right )}^{2} a^{7}} + \frac {a^{9} x^{4} - 4 \, a^{8} b x^{3} + 12 \, a^{7} b^{2} x^{2} - 40 \, a^{6} b^{3} x}{4 \, a^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 92, normalized size = 1.07 \begin {gather*} \frac {6\,b^5\,x+\frac {11\,b^6}{2\,a}}{a^8\,x^2+2\,a^7\,b\,x+a^6\,b^2}+\frac {x^4}{4\,a^3}+\frac {15\,b^4\,\ln \left (b+a\,x\right )}{a^7}-\frac {b\,x^3}{a^4}-\frac {10\,b^3\,x}{a^6}+\frac {3\,b^2\,x^2}{a^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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