Optimal. Leaf size=65 \[ \frac {b^3}{4 c^4 \left (b+c x^2\right )^2}-\frac {3 b^2}{2 c^4 \left (b+c x^2\right )}-\frac {3 b \log \left (b+c x^2\right )}{2 c^4}+\frac {x^2}{2 c^3} \]
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Rubi [A] time = 0.05, antiderivative size = 65, 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 = {1584, 266, 43} \begin {gather*} \frac {b^3}{4 c^4 \left (b+c x^2\right )^2}-\frac {3 b^2}{2 c^4 \left (b+c x^2\right )}-\frac {3 b \log \left (b+c x^2\right )}{2 c^4}+\frac {x^2}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{13}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^7}{\left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{(b+c x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{c^3}-\frac {b^3}{c^3 (b+c x)^3}+\frac {3 b^2}{c^3 (b+c x)^2}-\frac {3 b}{c^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 c^3}+\frac {b^3}{4 c^4 \left (b+c x^2\right )^2}-\frac {3 b^2}{2 c^4 \left (b+c x^2\right )}-\frac {3 b \log \left (b+c x^2\right )}{2 c^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 48, normalized size = 0.74 \begin {gather*} -\frac {\frac {b^2 \left (5 b+6 c x^2\right )}{\left (b+c x^2\right )^2}+6 b \log \left (b+c x^2\right )-2 c x^2}{4 c^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{13}}{\left (b x^2+c x^4\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.53, size = 91, normalized size = 1.40 \begin {gather*} \frac {2 \, c^{3} x^{6} + 4 \, b c^{2} x^{4} - 4 \, b^{2} c x^{2} - 5 \, b^{3} - 6 \, {\left (b c^{2} x^{4} + 2 \, b^{2} c x^{2} + b^{3}\right )} \log \left (c x^{2} + b\right )}{4 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 62, normalized size = 0.95 \begin {gather*} \frac {x^{2}}{2 \, c^{3}} - \frac {3 \, b \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{4}} + \frac {9 \, b c^{2} x^{4} + 12 \, b^{2} c x^{2} + 4 \, b^{3}}{4 \, {\left (c x^{2} + b\right )}^{2} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.89 \begin {gather*} \frac {b^{3}}{4 \left (c \,x^{2}+b \right )^{2} c^{4}}+\frac {x^{2}}{2 c^{3}}-\frac {3 b^{2}}{2 \left (c \,x^{2}+b \right ) c^{4}}-\frac {3 b \ln \left (c \,x^{2}+b \right )}{2 c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 66, normalized size = 1.02 \begin {gather*} -\frac {6 \, b^{2} c x^{2} + 5 \, b^{3}}{4 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}} + \frac {x^{2}}{2 \, c^{3}} - \frac {3 \, b \log \left (c x^{2} + b\right )}{2 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.26, size = 68, normalized size = 1.05 \begin {gather*} \frac {x^2}{2\,c^3}-\frac {\frac {5\,b^3}{4\,c}+\frac {3\,b^2\,x^2}{2}}{b^2\,c^3+2\,b\,c^4\,x^2+c^5\,x^4}-\frac {3\,b\,\ln \left (c\,x^2+b\right )}{2\,c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 68, normalized size = 1.05 \begin {gather*} - \frac {3 b \log {\left (b + c x^{2} \right )}}{2 c^{4}} + \frac {- 5 b^{3} - 6 b^{2} c x^{2}}{4 b^{2} c^{4} + 8 b c^{5} x^{2} + 4 c^{6} x^{4}} + \frac {x^{2}}{2 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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