Optimal. Leaf size=40 \[ -\frac {a x^3}{3 b^2}+\frac {x^6}{6 b}+\frac {a^2 \log \left (a+b x^3\right )}{3 b^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 40, 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 \log \left (a+b x^3\right )}{3 b^3}-\frac {a x^3}{3 b^2}+\frac {x^6}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^8}{a+b x^3} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{a+b x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {a x^3}{3 b^2}+\frac {x^6}{6 b}+\frac {a^2 \log \left (a+b x^3\right )}{3 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 40, normalized size = 1.00 \begin {gather*} -\frac {a x^3}{3 b^2}+\frac {x^6}{6 b}+\frac {a^2 \log \left (a+b x^3\right )}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 35, normalized size = 0.88
method | result | size |
default | \(-\frac {-\frac {1}{2} b \,x^{6}+a \,x^{3}}{3 b^{2}}+\frac {a^{2} \ln \left (b \,x^{3}+a \right )}{3 b^{3}}\) | \(35\) |
norman | \(-\frac {a \,x^{3}}{3 b^{2}}+\frac {x^{6}}{6 b}+\frac {a^{2} \ln \left (b \,x^{3}+a \right )}{3 b^{3}}\) | \(35\) |
risch | \(\frac {x^{6}}{6 b}-\frac {a \,x^{3}}{3 b^{2}}+\frac {a^{2}}{6 b^{3}}+\frac {a^{2} \ln \left (b \,x^{3}+a \right )}{3 b^{3}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 34, normalized size = 0.85 \begin {gather*} \frac {a^{2} \log \left (b x^{3} + a\right )}{3 \, b^{3}} + \frac {b x^{6} - 2 \, a x^{3}}{6 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 33, normalized size = 0.82 \begin {gather*} \frac {b^{2} x^{6} - 2 \, a b x^{3} + 2 \, a^{2} \log \left (b x^{3} + a\right )}{6 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 32, normalized size = 0.80 \begin {gather*} \frac {a^{2} \log {\left (a + b x^{3} \right )}}{3 b^{3}} - \frac {a x^{3}}{3 b^{2}} + \frac {x^{6}}{6 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.51, size = 35, normalized size = 0.88 \begin {gather*} \frac {a^{2} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac {b x^{6} - 2 \, a x^{3}}{6 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 33, normalized size = 0.82 \begin {gather*} \frac {2\,a^2\,\ln \left (b\,x^3+a\right )+b^2\,x^6-2\,a\,b\,x^3}{6\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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