Optimal. Leaf size=77 \[ \frac {6 a^2 x}{b^5}-\frac {3 a x^2}{2 b^4}+\frac {x^3}{3 b^3}+\frac {a^5}{2 b^6 (a+b x)^2}-\frac {5 a^4}{b^6 (a+b x)}-\frac {10 a^3 \log (a+b x)}{b^6} \]
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Rubi [A]
time = 0.03, antiderivative size = 77, 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 = {45}
\begin {gather*} \frac {a^5}{2 b^6 (a+b x)^2}-\frac {5 a^4}{b^6 (a+b x)}-\frac {10 a^3 \log (a+b x)}{b^6}+\frac {6 a^2 x}{b^5}-\frac {3 a x^2}{2 b^4}+\frac {x^3}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^5}{(a+b x)^3} \, dx &=\int \left (\frac {6 a^2}{b^5}-\frac {3 a x}{b^4}+\frac {x^2}{b^3}-\frac {a^5}{b^5 (a+b x)^3}+\frac {5 a^4}{b^5 (a+b x)^2}-\frac {10 a^3}{b^5 (a+b x)}\right ) \, dx\\ &=\frac {6 a^2 x}{b^5}-\frac {3 a x^2}{2 b^4}+\frac {x^3}{3 b^3}+\frac {a^5}{2 b^6 (a+b x)^2}-\frac {5 a^4}{b^6 (a+b x)}-\frac {10 a^3 \log (a+b x)}{b^6}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 67, normalized size = 0.87 \begin {gather*} \frac {36 a^2 b x-9 a b^2 x^2+2 b^3 x^3+\frac {3 a^5}{(a+b x)^2}-\frac {30 a^4}{a+b x}-60 a^3 \log (a+b x)}{6 b^6} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.33, size = 116, normalized size = 1.51 \begin {gather*} \frac {-60 a^5 \text {Log}\left [a+b x\right ]-27 a^5-120 a^4 b x \text {Log}\left [a+b x\right ]+6 a^4 b x-60 a^3 b^2 x^2 \text {Log}\left [a+b x\right ]+63 a^3 b^2 x^2+20 a^2 b^3 x^3-5 a b^4 x^4+2 b^5 x^5}{6 b^6 \left (a^2+2 a b x+b^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 72, normalized size = 0.94
method | result | size |
risch | \(\frac {x^{3}}{3 b^{3}}-\frac {3 a \,x^{2}}{2 b^{4}}+\frac {6 a^{2} x}{b^{5}}+\frac {-5 a^{4} x -\frac {9 a^{5}}{2 b}}{b^{5} \left (b x +a \right )^{2}}-\frac {10 a^{3} \ln \left (b x +a \right )}{b^{6}}\) | \(68\) |
norman | \(\frac {\frac {x^{5}}{3 b}-\frac {5 a \,x^{4}}{6 b^{2}}+\frac {10 a^{2} x^{3}}{3 b^{3}}-\frac {15 a^{5}}{b^{6}}-\frac {20 a^{4} x}{b^{5}}}{\left (b x +a \right )^{2}}-\frac {10 a^{3} \ln \left (b x +a \right )}{b^{6}}\) | \(70\) |
default | \(\frac {\frac {1}{3} b^{2} x^{3}-\frac {3}{2} a b \,x^{2}+6 a^{2} x}{b^{5}}-\frac {5 a^{4}}{b^{6} \left (b x +a \right )}+\frac {a^{5}}{2 b^{6} \left (b x +a \right )^{2}}-\frac {10 a^{3} \ln \left (b x +a \right )}{b^{6}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 81, normalized size = 1.05 \begin {gather*} -\frac {10 \, a^{4} b x + 9 \, a^{5}}{2 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} - \frac {10 \, a^{3} \log \left (b x + a\right )}{b^{6}} + \frac {2 \, b^{2} x^{3} - 9 \, a b x^{2} + 36 \, a^{2} x}{6 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 107, normalized size = 1.39 \begin {gather*} \frac {2 \, b^{5} x^{5} - 5 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 63 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x - 27 \, a^{5} - 60 \, {\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 85, normalized size = 1.10 \begin {gather*} - \frac {10 a^{3} \log {\left (a + b x \right )}}{b^{6}} + \frac {6 a^{2} x}{b^{5}} - \frac {3 a x^{2}}{2 b^{4}} + \frac {- 9 a^{5} - 10 a^{4} b x}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} + \frac {x^{3}}{3 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 86, normalized size = 1.12 \begin {gather*} \frac {\frac {1}{3} x^{3} b^{6}-\frac {3}{2} x^{2} b^{5} a+6 x b^{4} a^{2}}{b^{9}}+\frac {\frac {1}{2} \left (-10 b a^{4} x-9 a^{5}\right )}{b^{6} \left (x b+a\right )^{2}}-\frac {10 a^{3} \ln \left |x b+a\right |}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 67, normalized size = 0.87 \begin {gather*} -\frac {\frac {5\,a\,{\left (a+b\,x\right )}^2}{2}-\frac {{\left (a+b\,x\right )}^3}{3}+\frac {5\,a^4}{a+b\,x}-\frac {a^5}{2\,{\left (a+b\,x\right )}^2}+10\,a^3\,\ln \left (a+b\,x\right )-10\,a^2\,b\,x}{b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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