Optimal. Leaf size=81 \[ \frac {a^5}{3 b^6 (a+b x)^3}-\frac {5 a^4}{2 b^6 (a+b x)^2}+\frac {10 a^3}{b^6 (a+b x)}+\frac {10 a^2 \log (a+b x)}{b^6}-\frac {4 a x}{b^5}+\frac {x^2}{2 b^4} \]
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Rubi [A] time = 0.05, antiderivative size = 81, 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 = {43} \begin {gather*} \frac {a^5}{3 b^6 (a+b x)^3}-\frac {5 a^4}{2 b^6 (a+b x)^2}+\frac {10 a^3}{b^6 (a+b x)}+\frac {10 a^2 \log (a+b x)}{b^6}-\frac {4 a x}{b^5}+\frac {x^2}{2 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^5}{(a+b x)^4} \, dx &=\int \left (-\frac {4 a}{b^5}+\frac {x}{b^4}-\frac {a^5}{b^5 (a+b x)^4}+\frac {5 a^4}{b^5 (a+b x)^3}-\frac {10 a^3}{b^5 (a+b x)^2}+\frac {10 a^2}{b^5 (a+b x)}\right ) \, dx\\ &=-\frac {4 a x}{b^5}+\frac {x^2}{2 b^4}+\frac {a^5}{3 b^6 (a+b x)^3}-\frac {5 a^4}{2 b^6 (a+b x)^2}+\frac {10 a^3}{b^6 (a+b x)}+\frac {10 a^2 \log (a+b x)}{b^6}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 68, normalized size = 0.84 \begin {gather*} \frac {\frac {2 a^5}{(a+b x)^3}-\frac {15 a^4}{(a+b x)^2}+\frac {60 a^3}{a+b x}+60 a^2 \log (a+b x)-24 a b x+3 b^2 x^2}{6 b^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5}{(a+b x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.79, size = 129, normalized size = 1.59 \begin {gather*} \frac {3 \, b^{5} x^{5} - 15 \, a b^{4} x^{4} - 63 \, a^{2} b^{3} x^{3} - 9 \, a^{3} b^{2} x^{2} + 81 \, a^{4} b x + 47 \, a^{5} + 60 \, {\left (a^{2} b^{3} x^{3} + 3 \, a^{3} b^{2} x^{2} + 3 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{6 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 72, normalized size = 0.89 \begin {gather*} \frac {10 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac {b^{4} x^{2} - 8 \, a b^{3} x}{2 \, b^{8}} + \frac {60 \, a^{3} b^{2} x^{2} + 105 \, a^{4} b x + 47 \, a^{5}}{6 \, {\left (b x + a\right )}^{3} b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 76, normalized size = 0.94 \begin {gather*} \frac {a^{5}}{3 \left (b x +a \right )^{3} b^{6}}-\frac {5 a^{4}}{2 \left (b x +a \right )^{2} b^{6}}+\frac {x^{2}}{2 b^{4}}+\frac {10 a^{3}}{\left (b x +a \right ) b^{6}}+\frac {10 a^{2} \ln \left (b x +a \right )}{b^{6}}-\frac {4 a x}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 91, normalized size = 1.12 \begin {gather*} \frac {60 \, a^{3} b^{2} x^{2} + 105 \, a^{4} b x + 47 \, a^{5}}{6 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} + \frac {10 \, a^{2} \log \left (b x + a\right )}{b^{6}} + \frac {b x^{2} - 8 \, a x}{2 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 66, normalized size = 0.81 \begin {gather*} \frac {\frac {{\left (a+b\,x\right )}^2}{2}+\frac {10\,a^3}{a+b\,x}-\frac {5\,a^4}{2\,{\left (a+b\,x\right )}^2}+\frac {a^5}{3\,{\left (a+b\,x\right )}^3}+10\,a^2\,\ln \left (a+b\,x\right )-5\,a\,b\,x}{b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 94, normalized size = 1.16 \begin {gather*} \frac {10 a^{2} \log {\left (a + b x \right )}}{b^{6}} - \frac {4 a x}{b^{5}} + \frac {47 a^{5} + 105 a^{4} b x + 60 a^{3} b^{2} x^{2}}{6 a^{3} b^{6} + 18 a^{2} b^{7} x + 18 a b^{8} x^{2} + 6 b^{9} x^{3}} + \frac {x^{2}}{2 b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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