Optimal. Leaf size=58 \[ \frac {a^3}{3 b^4 (a+b x)^3}-\frac {3 a^2}{2 b^4 (a+b x)^2}+\frac {3 a}{b^4 (a+b x)}+\frac {\log (a+b x)}{b^4} \]
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Rubi [A] time = 0.03, antiderivative size = 58, 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^3}{3 b^4 (a+b x)^3}-\frac {3 a^2}{2 b^4 (a+b x)^2}+\frac {3 a}{b^4 (a+b x)}+\frac {\log (a+b x)}{b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^3}{(a+b x)^4} \, dx &=\int \left (-\frac {a^3}{b^3 (a+b x)^4}+\frac {3 a^2}{b^3 (a+b x)^3}-\frac {3 a}{b^3 (a+b x)^2}+\frac {1}{b^3 (a+b x)}\right ) \, dx\\ &=\frac {a^3}{3 b^4 (a+b x)^3}-\frac {3 a^2}{2 b^4 (a+b x)^2}+\frac {3 a}{b^4 (a+b x)}+\frac {\log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.76 \begin {gather*} \frac {\frac {a \left (11 a^2+27 a b x+18 b^2 x^2\right )}{(a+b x)^3}+6 \log (a+b x)}{6 b^4} \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^3}{(a+b x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.10, size = 94, normalized size = 1.62 \begin {gather*} \frac {18 \, a b^{2} x^{2} + 27 \, a^{2} b x + 11 \, a^{3} + 6 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 46, normalized size = 0.79 \begin {gather*} \frac {\log \left ({\left | b x + a \right |}\right )}{b^{4}} + \frac {18 \, a b x^{2} + 27 \, a^{2} x + \frac {11 \, a^{3}}{b}}{6 \, {\left (b x + a\right )}^{3} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 55, normalized size = 0.95 \begin {gather*} \frac {a^{3}}{3 \left (b x +a \right )^{3} b^{4}}-\frac {3 a^{2}}{2 \left (b x +a \right )^{2} b^{4}}+\frac {3 a}{\left (b x +a \right ) b^{4}}+\frac {\ln \left (b x +a \right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 70, normalized size = 1.21 \begin {gather*} \frac {18 \, a b^{2} x^{2} + 27 \, a^{2} b x + 11 \, a^{3}}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac {\log \left (b x + a\right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 45, normalized size = 0.78 \begin {gather*} \frac {\ln \left (a+b\,x\right )+\frac {3\,a}{a+b\,x}-\frac {3\,a^2}{2\,{\left (a+b\,x\right )}^2}+\frac {a^3}{3\,{\left (a+b\,x\right )}^3}}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 70, normalized size = 1.21 \begin {gather*} \frac {11 a^{3} + 27 a^{2} b x + 18 a b^{2} x^{2}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {\log {\left (a + b x \right )}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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