Optimal. Leaf size=50 \[ \frac {a^3}{2 b^4 (a+b x)^2}-\frac {3 a^2}{b^4 (a+b x)}-\frac {3 a \log (a+b x)}{b^4}+\frac {x}{b^3} \]
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Rubi [A] time = 0.03, antiderivative size = 50, 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}{2 b^4 (a+b x)^2}-\frac {3 a^2}{b^4 (a+b x)}-\frac {3 a \log (a+b x)}{b^4}+\frac {x}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^3}{(a+b x)^3} \, dx &=\int \left (\frac {1}{b^3}-\frac {a^3}{b^3 (a+b x)^3}+\frac {3 a^2}{b^3 (a+b x)^2}-\frac {3 a}{b^3 (a+b x)}\right ) \, dx\\ &=\frac {x}{b^3}+\frac {a^3}{2 b^4 (a+b x)^2}-\frac {3 a^2}{b^4 (a+b x)}-\frac {3 a \log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 40, normalized size = 0.80 \begin {gather*} -\frac {\frac {a^2 (5 a+6 b x)}{(a+b x)^2}+6 a \log (a+b x)-2 b x}{2 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)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.84, size = 83, normalized size = 1.66 \begin {gather*} \frac {2 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} - 4 \, a^{2} b x - 5 \, a^{3} - 6 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 44, normalized size = 0.88 \begin {gather*} \frac {x}{b^{3}} - \frac {3 \, a \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac {6 \, a^{2} b x + 5 \, a^{3}}{2 \, {\left (b x + a\right )}^{2} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 49, normalized size = 0.98 \begin {gather*} \frac {a^{3}}{2 \left (b x +a \right )^{2} b^{4}}-\frac {3 a^{2}}{\left (b x +a \right ) b^{4}}-\frac {3 a \ln \left (b x +a \right )}{b^{4}}+\frac {x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 57, normalized size = 1.14 \begin {gather*} -\frac {6 \, a^{2} b x + 5 \, a^{3}}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {x}{b^{3}} - \frac {3 \, a \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 = 0.15, size = 43, normalized size = 0.86 \begin {gather*} -\frac {3\,a\,\ln \left (a+b\,x\right )-b\,x+\frac {3\,a^2}{a+b\,x}-\frac {a^3}{2\,{\left (a+b\,x\right )}^2}}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 58, normalized size = 1.16 \begin {gather*} - \frac {3 a \log {\left (a + b x \right )}}{b^{4}} + \frac {- 5 a^{3} - 6 a^{2} b x}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac {x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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