Optimal. Leaf size=70 \[ \frac {a^4 x}{b^5}-\frac {a^3 x^2}{2 b^4}+\frac {a^2 x^3}{3 b^3}-\frac {a x^4}{4 b^2}+\frac {x^5}{5 b}-\frac {a^5 \log (a+b x)}{b^6} \]
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Rubi [A]
time = 0.03, antiderivative size = 70, 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 \log (a+b x)}{b^6}+\frac {a^4 x}{b^5}-\frac {a^3 x^2}{2 b^4}+\frac {a^2 x^3}{3 b^3}-\frac {a x^4}{4 b^2}+\frac {x^5}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^5}{a+b x} \, dx &=\int \left (\frac {a^4}{b^5}-\frac {a^3 x}{b^4}+\frac {a^2 x^2}{b^3}-\frac {a x^3}{b^2}+\frac {x^4}{b}-\frac {a^5}{b^5 (a+b x)}\right ) \, dx\\ &=\frac {a^4 x}{b^5}-\frac {a^3 x^2}{2 b^4}+\frac {a^2 x^3}{3 b^3}-\frac {a x^4}{4 b^2}+\frac {x^5}{5 b}-\frac {a^5 \log (a+b x)}{b^6}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 70, normalized size = 1.00 \begin {gather*} \frac {a^4 x}{b^5}-\frac {a^3 x^2}{2 b^4}+\frac {a^2 x^3}{3 b^3}-\frac {a x^4}{4 b^2}+\frac {x^5}{5 b}-\frac {a^5 \log (a+b x)}{b^6} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.00, size = 61, normalized size = 0.87 \begin {gather*} \frac {-a^5 \text {Log}\left [a+b x\right ]+a^4 b x-\frac {a^3 b^2 x^2}{2}+\frac {a^2 b^3 x^3}{3}-\frac {a b^4 x^4}{4}+\frac {b^5 x^5}{5}}{b^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 63, normalized size = 0.90
method | result | size |
default | \(\frac {\frac {1}{5} b^{4} x^{5}-\frac {1}{4} a \,b^{3} x^{4}+\frac {1}{3} a^{2} b^{2} x^{3}-\frac {1}{2} a^{3} b \,x^{2}+a^{4} x}{b^{5}}-\frac {a^{5} \ln \left (b x +a \right )}{b^{6}}\) | \(63\) |
norman | \(\frac {a^{4} x}{b^{5}}-\frac {a^{3} x^{2}}{2 b^{4}}+\frac {a^{2} x^{3}}{3 b^{3}}-\frac {a \,x^{4}}{4 b^{2}}+\frac {x^{5}}{5 b}-\frac {a^{5} \ln \left (b x +a \right )}{b^{6}}\) | \(63\) |
risch | \(\frac {a^{4} x}{b^{5}}-\frac {a^{3} x^{2}}{2 b^{4}}+\frac {a^{2} x^{3}}{3 b^{3}}-\frac {a \,x^{4}}{4 b^{2}}+\frac {x^{5}}{5 b}-\frac {a^{5} \ln \left (b x +a \right )}{b^{6}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 64, normalized size = 0.91 \begin {gather*} -\frac {a^{5} \log \left (b x + a\right )}{b^{6}} + \frac {12 \, b^{4} x^{5} - 15 \, a b^{3} x^{4} + 20 \, a^{2} b^{2} x^{3} - 30 \, a^{3} b x^{2} + 60 \, a^{4} x}{60 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 63, normalized size = 0.90 \begin {gather*} \frac {12 \, b^{5} x^{5} - 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} - 30 \, a^{3} b^{2} x^{2} + 60 \, a^{4} b x - 60 \, a^{5} \log \left (b x + a\right )}{60 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 61, normalized size = 0.87 \begin {gather*} - \frac {a^{5} \log {\left (a + b x \right )}}{b^{6}} + \frac {a^{4} x}{b^{5}} - \frac {a^{3} x^{2}}{2 b^{4}} + \frac {a^{2} x^{3}}{3 b^{3}} - \frac {a x^{4}}{4 b^{2}} + \frac {x^{5}}{5 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 75, normalized size = 1.07 \begin {gather*} \frac {\frac {1}{5} x^{5} b^{4}-\frac {1}{4} x^{4} b^{3} a+\frac {1}{3} x^{3} b^{2} a^{2}-\frac {1}{2} x^{2} b a^{3}+x a^{4}}{b^{5}}-\frac {a^{5} \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 = 0.08, size = 62, normalized size = 0.89 \begin {gather*} \frac {x^5}{5\,b}-\frac {a^5\,\ln \left (a+b\,x\right )}{b^6}-\frac {a\,x^4}{4\,b^2}+\frac {a^4\,x}{b^5}+\frac {a^2\,x^3}{3\,b^3}-\frac {a^3\,x^2}{2\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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