Optimal. Leaf size=53 \[ \frac {b^2 x^2}{2 c^3}-\frac {b x^4}{4 c^2}+\frac {x^6}{6 c}-\frac {b^3 \log \left (b+c x^2\right )}{2 c^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1598, 272, 45}
\begin {gather*} -\frac {b^3 \log \left (b+c x^2\right )}{2 c^4}+\frac {b^2 x^2}{2 c^3}-\frac {b x^4}{4 c^2}+\frac {x^6}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^9}{b x^2+c x^4} \, dx &=\int \frac {x^7}{b+c x^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{b+c x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {b^2}{c^3}-\frac {b x}{c^2}+\frac {x^2}{c}-\frac {b^3}{c^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {b^2 x^2}{2 c^3}-\frac {b x^4}{4 c^2}+\frac {x^6}{6 c}-\frac {b^3 \log \left (b+c x^2\right )}{2 c^4}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 53, normalized size = 1.00 \begin {gather*} \frac {b^2 x^2}{2 c^3}-\frac {b x^4}{4 c^2}+\frac {x^6}{6 c}-\frac {b^3 \log \left (b+c x^2\right )}{2 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 46, normalized size = 0.87
method | result | size |
default | \(\frac {\frac {1}{3} c^{2} x^{6}-\frac {1}{2} b c \,x^{4}+b^{2} x^{2}}{2 c^{3}}-\frac {b^{3} \ln \left (c \,x^{2}+b \right )}{2 c^{4}}\) | \(46\) |
risch | \(\frac {b^{2} x^{2}}{2 c^{3}}-\frac {b \,x^{4}}{4 c^{2}}+\frac {x^{6}}{6 c}-\frac {b^{3} \ln \left (c \,x^{2}+b \right )}{2 c^{4}}\) | \(46\) |
norman | \(\frac {\frac {x^{7}}{6 c}-\frac {b \,x^{5}}{4 c^{2}}+\frac {b^{2} x^{3}}{2 c^{3}}}{x}-\frac {b^{3} \ln \left (c \,x^{2}+b \right )}{2 c^{4}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 46, normalized size = 0.87 \begin {gather*} -\frac {b^{3} \log \left (c x^{2} + b\right )}{2 \, c^{4}} + \frac {2 \, c^{2} x^{6} - 3 \, b c x^{4} + 6 \, b^{2} x^{2}}{12 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 45, normalized size = 0.85 \begin {gather*} \frac {2 \, c^{3} x^{6} - 3 \, b c^{2} x^{4} + 6 \, b^{2} c x^{2} - 6 \, b^{3} \log \left (c x^{2} + b\right )}{12 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 44, normalized size = 0.83 \begin {gather*} - \frac {b^{3} \log {\left (b + c x^{2} \right )}}{2 c^{4}} + \frac {b^{2} x^{2}}{2 c^{3}} - \frac {b x^{4}}{4 c^{2}} + \frac {x^{6}}{6 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.54, size = 47, normalized size = 0.89 \begin {gather*} -\frac {b^{3} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{4}} + \frac {2 \, c^{2} x^{6} - 3 \, b c x^{4} + 6 \, b^{2} x^{2}}{12 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 45, normalized size = 0.85 \begin {gather*} \frac {x^6}{6\,c}-\frac {b\,x^4}{4\,c^2}-\frac {b^3\,\ln \left (c\,x^2+b\right )}{2\,c^4}+\frac {b^2\,x^2}{2\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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