Optimal. Leaf size=41 \[ -\frac {a x^3}{3 b^2}-\frac {x^6}{6 b}-\frac {a^2 \log \left (a-b x^3\right )}{3 b^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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 = 41, 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 = 0.14, size = 36, normalized size = 0.88
method | result | size |
default | \(-\frac {\frac {1}{2} b \,x^{6}+a \,x^{3}}{3 b^{2}}-\frac {a^{2} \ln \left (-b \,x^{3}+a \right )}{3 b^{3}}\) | \(36\) |
norman | \(-\frac {a \,x^{3}}{3 b^{2}}-\frac {x^{6}}{6 b}-\frac {a^{2} \ln \left (-b \,x^{3}+a \right )}{3 b^{3}}\) | \(36\) |
risch | \(-\frac {x^{6}}{6 b}-\frac {a \,x^{3}}{3 b^{2}}-\frac {a^{2}}{6 b^{3}}-\frac {a^{2} \ln \left (-b \,x^{3}+a \right )}{3 b^{3}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 36, normalized size = 0.88 \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 = 0.34, size = 35, normalized size = 0.85 \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 = 34, normalized size = 0.83 \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 = 1.79, size = 37, normalized size = 0.90 \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 = 0.06, size = 35, normalized size = 0.85 \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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