Optimal. Leaf size=72 \[ -\frac {4 a^3 x}{b^5}+\frac {3 a^2 x^2}{2 b^4}-\frac {2 a x^3}{3 b^3}+\frac {x^4}{4 b^2}+\frac {a^5}{b^6 (a+b x)}+\frac {5 a^4 \log (a+b x)}{b^6} \]
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Rubi [A]
time = 0.03, antiderivative size = 72, 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}{b^6 (a+b x)}+\frac {5 a^4 \log (a+b x)}{b^6}-\frac {4 a^3 x}{b^5}+\frac {3 a^2 x^2}{2 b^4}-\frac {2 a x^3}{3 b^3}+\frac {x^4}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^5}{(a+b x)^2} \, dx &=\int \left (-\frac {4 a^3}{b^5}+\frac {3 a^2 x}{b^4}-\frac {2 a x^2}{b^3}+\frac {x^3}{b^2}-\frac {a^5}{b^5 (a+b x)^2}+\frac {5 a^4}{b^5 (a+b x)}\right ) \, dx\\ &=-\frac {4 a^3 x}{b^5}+\frac {3 a^2 x^2}{2 b^4}-\frac {2 a x^3}{3 b^3}+\frac {x^4}{4 b^2}+\frac {a^5}{b^6 (a+b x)}+\frac {5 a^4 \log (a+b x)}{b^6}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 66, normalized size = 0.92 \begin {gather*} \frac {-48 a^3 b x+18 a^2 b^2 x^2-8 a b^3 x^3+3 b^4 x^4+\frac {12 a^5}{a+b x}+60 a^4 \log (a+b x)}{12 b^6} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.22, size = 89, normalized size = 1.24 \begin {gather*} \frac {60 a^4 \text {Log}\left [a+b x\right ] \left (a+b x\right )+12 a^5-48 a^3 b x \left (a+b x\right )+18 a^2 b^2 x^2 \left (a+b x\right )-8 a b^3 x^3 \left (a+b x\right )+3 b^4 x^4 \left (a+b x\right )}{12 b^6 \left (a+b x\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 68, normalized size = 0.94
method | result | size |
risch | \(-\frac {4 a^{3} x}{b^{5}}+\frac {3 a^{2} x^{2}}{2 b^{4}}-\frac {2 a \,x^{3}}{3 b^{3}}+\frac {x^{4}}{4 b^{2}}+\frac {a^{5}}{b^{6} \left (b x +a \right )}+\frac {5 a^{4} \ln \left (b x +a \right )}{b^{6}}\) | \(67\) |
default | \(-\frac {-\frac {1}{4} b^{3} x^{4}+\frac {2}{3} a \,b^{2} x^{3}-\frac {3}{2} a^{2} b \,x^{2}+4 a^{3} x}{b^{5}}+\frac {a^{5}}{b^{6} \left (b x +a \right )}+\frac {5 a^{4} \ln \left (b x +a \right )}{b^{6}}\) | \(68\) |
norman | \(\frac {\frac {5 a^{5}}{b^{6}}+\frac {x^{5}}{4 b}-\frac {5 a \,x^{4}}{12 b^{2}}+\frac {5 a^{2} x^{3}}{6 b^{3}}-\frac {5 a^{3} x^{2}}{2 b^{4}}}{b x +a}+\frac {5 a^{4} \ln \left (b x +a \right )}{b^{6}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 70, normalized size = 0.97 \begin {gather*} \frac {a^{5}}{b^{7} x + a b^{6}} + \frac {5 \, a^{4} \log \left (b x + a\right )}{b^{6}} + \frac {3 \, b^{3} x^{4} - 8 \, a b^{2} x^{3} + 18 \, a^{2} b x^{2} - 48 \, a^{3} x}{12 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 85, normalized size = 1.18 \begin {gather*} \frac {3 \, b^{5} x^{5} - 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} - 30 \, a^{3} b^{2} x^{2} - 48 \, a^{4} b x + 12 \, a^{5} + 60 \, {\left (a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{12 \, {\left (b^{7} x + a b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 71, normalized size = 0.99 \begin {gather*} \frac {a^{5}}{a b^{6} + b^{7} x} + \frac {5 a^{4} \log {\left (a + b x \right )}}{b^{6}} - \frac {4 a^{3} x}{b^{5}} + \frac {3 a^{2} x^{2}}{2 b^{4}} - \frac {2 a x^{3}}{3 b^{3}} + \frac {x^{4}}{4 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 82, normalized size = 1.14 \begin {gather*} \frac {\frac {1}{4} x^{4} b^{6}-\frac {2}{3} x^{3} b^{5} a+\frac {3}{2} x^{2} b^{4} a^{2}-4 x b^{3} a^{3}}{b^{8}}+\frac {a^{5}}{b^{6} \left (x b+a\right )}+\frac {5 a^{4} \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 = 0.07, size = 72, normalized size = 1.00 \begin {gather*} \frac {x^4}{4\,b^2}+\frac {5\,a^4\,\ln \left (a+b\,x\right )}{b^6}-\frac {2\,a\,x^3}{3\,b^3}-\frac {4\,a^3\,x}{b^5}+\frac {3\,a^2\,x^2}{2\,b^4}+\frac {a^5}{b\,\left (x\,b^6+a\,b^5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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