Optimal. Leaf size=50 \[ \frac {x}{b^3}+\frac {a^3}{2 b^4 (a+b x)^2}-\frac {3 a^2}{b^4 (a+b x)}-\frac {3 a \log (a+b x)}{b^4} \]
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Rubi [A]
time = 0.02, 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 = {45}
\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 45
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 = 0.03, size = 40, normalized size = 0.80 \begin {gather*} -\frac {-2 b x+\frac {a^2 (5 a+6 b x)}{(a+b x)^2}+6 a \log (a+b x)}{2 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 49, normalized size = 0.98
method | result | size |
risch | \(\frac {x}{b^{3}}+\frac {-3 a^{2} x -\frac {5 a^{3}}{2 b}}{b^{3} \left (b x +a \right )^{2}}-\frac {3 a \ln \left (b x +a \right )}{b^{4}}\) | \(45\) |
norman | \(\frac {\frac {x^{3}}{b}-\frac {9 a^{3}}{2 b^{4}}-\frac {6 a^{2} x}{b^{3}}}{\left (b x +a \right )^{2}}-\frac {3 a \ln \left (b x +a \right )}{b^{4}}\) | \(47\) |
default | \(\frac {x}{b^{3}}+\frac {a^{3}}{2 b^{4} \left (b x +a \right )^{2}}-\frac {3 a^{2}}{b^{4} \left (b x +a \right )}-\frac {3 a \ln \left (b x +a \right )}{b^{4}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, 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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Fricas [A]
time = 0.76, 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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Sympy [A]
time = 0.11, 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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Giac [A]
time = 0.70, 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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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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