Optimal. Leaf size=44 \[ -\frac {a^3 \log (a+b x)}{b^4}+\frac {a^2 x}{b^3}-\frac {a x^2}{2 b^2}+\frac {x^3}{3 b} \]
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Rubi [A] time = 0.02, antiderivative size = 44, 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} \[ \frac {a^2 x}{b^3}-\frac {a^3 \log (a+b x)}{b^4}-\frac {a x^2}{2 b^2}+\frac {x^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^3}{a+b x} \, dx &=\int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx\\ &=\frac {a^2 x}{b^3}-\frac {a x^2}{2 b^2}+\frac {x^3}{3 b}-\frac {a^3 \log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 44, normalized size = 1.00 \[ -\frac {a^3 \log (a+b x)}{b^4}+\frac {a^2 x}{b^3}-\frac {a x^2}{2 b^2}+\frac {x^3}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 41, normalized size = 0.93 \[ \frac {2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )}{6 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 43, normalized size = 0.98 \[ -\frac {a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4}} + \frac {2 \, b^{2} x^{3} - 3 \, a b x^{2} + 6 \, a^{2} x}{6 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 41, normalized size = 0.93 \[ \frac {x^{3}}{3 b}-\frac {a \,x^{2}}{2 b^{2}}-\frac {a^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {a^{2} x}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 42, normalized size = 0.95 \[ -\frac {a^{3} \log \left (b x + a\right )}{b^{4}} + \frac {2 \, b^{2} x^{3} - 3 \, a b x^{2} + 6 \, a^{2} x}{6 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 40, normalized size = 0.91 \[ \frac {x^3}{3\,b}-\frac {a^3\,\ln \left (a+b\,x\right )}{b^4}-\frac {a\,x^2}{2\,b^2}+\frac {a^2\,x}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 37, normalized size = 0.84 \[ - \frac {a^{3} \log {\left (a + b x \right )}}{b^{4}} + \frac {a^{2} x}{b^{3}} - \frac {a x^{2}}{2 b^{2}} + \frac {x^{3}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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