Optimal. Leaf size=64 \[ -\frac {a^4}{2 b^5 (a+b x)^2}+\frac {4 a^3}{b^5 (a+b x)}+\frac {6 a^2 \log (a+b x)}{b^5}-\frac {3 a x}{b^4}+\frac {x^2}{2 b^3} \]
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Rubi [A] time = 0.04, antiderivative size = 64, 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^4}{2 b^5 (a+b x)^2}+\frac {4 a^3}{b^5 (a+b x)}+\frac {6 a^2 \log (a+b x)}{b^5}-\frac {3 a x}{b^4}+\frac {x^2}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^4}{(a+b x)^3} \, dx &=\int \left (-\frac {3 a}{b^4}+\frac {x}{b^3}+\frac {a^4}{b^4 (a+b x)^3}-\frac {4 a^3}{b^4 (a+b x)^2}+\frac {6 a^2}{b^4 (a+b x)}\right ) \, dx\\ &=-\frac {3 a x}{b^4}+\frac {x^2}{2 b^3}-\frac {a^4}{2 b^5 (a+b x)^2}+\frac {4 a^3}{b^5 (a+b x)}+\frac {6 a^2 \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 55, normalized size = 0.86 \begin {gather*} \frac {-\frac {a^4}{(a+b x)^2}+\frac {8 a^3}{a+b x}+12 a^2 \log (a+b x)-6 a b x+b^2 x^2}{2 b^5} \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^4}{(a+b x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.88, size = 95, normalized size = 1.48 \begin {gather*} \frac {b^{4} x^{4} - 4 \, a b^{3} x^{3} - 11 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b x + 7 \, a^{4} + 12 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 61, normalized size = 0.95 \begin {gather*} \frac {6 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {b^{3} x^{2} - 6 \, a b^{2} x}{2 \, b^{6}} + \frac {8 \, a^{3} b x + 7 \, a^{4}}{2 \, {\left (b x + a\right )}^{2} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 0.95 \begin {gather*} -\frac {a^{4}}{2 \left (b x +a \right )^{2} b^{5}}+\frac {x^{2}}{2 b^{3}}+\frac {4 a^{3}}{\left (b x +a \right ) b^{5}}+\frac {6 a^{2} \ln \left (b x +a \right )}{b^{5}}-\frac {3 a x}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 69, normalized size = 1.08 \begin {gather*} \frac {8 \, a^{3} b x + 7 \, a^{4}}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {6 \, a^{2} \log \left (b x + a\right )}{b^{5}} + \frac {b x^{2} - 6 \, a x}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 54, normalized size = 0.84 \begin {gather*} \frac {\frac {{\left (a+b\,x\right )}^2}{2}+\frac {4\,a^3}{a+b\,x}-\frac {a^4}{2\,{\left (a+b\,x\right )}^2}+6\,a^2\,\ln \left (a+b\,x\right )-4\,a\,b\,x}{b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 70, normalized size = 1.09 \begin {gather*} \frac {6 a^{2} \log {\left (a + b x \right )}}{b^{5}} - \frac {3 a x}{b^{4}} + \frac {7 a^{4} + 8 a^{3} b x}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac {x^{2}}{2 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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