Optimal. Leaf size=52 \[ -\frac {a^2}{6 b^3 \left (a+b x^3\right )^2}+\frac {2 a}{3 b^3 \left (a+b x^3\right )}+\frac {\log \left (a+b x^3\right )}{3 b^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} -\frac {a^2}{6 b^3 \left (a+b x^3\right )^2}+\frac {2 a}{3 b^3 \left (a+b x^3\right )}+\frac {\log \left (a+b x^3\right )}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a+b x^3\right )^3} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^3}-\frac {2 a}{b^2 (a+b x)^2}+\frac {1}{b^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^2}{6 b^3 \left (a+b x^3\right )^2}+\frac {2 a}{3 b^3 \left (a+b x^3\right )}+\frac {\log \left (a+b x^3\right )}{3 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 39, normalized size = 0.75 \begin {gather*} \frac {\frac {a \left (3 a+4 b x^3\right )}{\left (a+b x^3\right )^2}+2 \log \left (a+b x^3\right )}{6 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 47, normalized size = 0.90
method | result | size |
norman | \(\frac {\frac {a^{2}}{2 b^{3}}+\frac {2 a \,x^{3}}{3 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\ln \left (b \,x^{3}+a \right )}{3 b^{3}}\) | \(43\) |
risch | \(\frac {\frac {a^{2}}{2 b^{3}}+\frac {2 a \,x^{3}}{3 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\ln \left (b \,x^{3}+a \right )}{3 b^{3}}\) | \(43\) |
default | \(-\frac {a^{2}}{6 b^{3} \left (b \,x^{3}+a \right )^{2}}+\frac {2 a}{3 b^{3} \left (b \,x^{3}+a \right )}+\frac {\ln \left (b \,x^{3}+a \right )}{3 b^{3}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 55, normalized size = 1.06 \begin {gather*} \frac {4 \, a b x^{3} + 3 \, a^{2}}{6 \, {\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac {\log \left (b x^{3} + a\right )}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 69, normalized size = 1.33 \begin {gather*} \frac {4 \, a b x^{3} + 3 \, a^{2} + 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (b x^{3} + a\right )}{6 \, {\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 53, normalized size = 1.02 \begin {gather*} \frac {3 a^{2} + 4 a b x^{3}}{6 a^{2} b^{3} + 12 a b^{4} x^{3} + 6 b^{5} x^{6}} + \frac {\log {\left (a + b x^{3} \right )}}{3 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.89, size = 42, normalized size = 0.81 \begin {gather*} \frac {\log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac {3 \, b x^{6} + 2 \, a x^{3}}{6 \, {\left (b x^{3} + a\right )}^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 53, normalized size = 1.02 \begin {gather*} \frac {\frac {a^2}{2\,b^3}+\frac {2\,a\,x^3}{3\,b^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\frac {\ln \left (b\,x^3+a\right )}{3\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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