Optimal. Leaf size=49 \[ -\frac {b^2}{4 c^3 \left (b+c x^2\right )^2}+\frac {b}{c^3 \left (b+c x^2\right )}+\frac {\log \left (b+c x^2\right )}{2 c^3} \]
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Rubi [A] time = 0.04, antiderivative size = 49, 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} \[ -\frac {b^2}{4 c^3 \left (b+c x^2\right )^2}+\frac {b}{c^3 \left (b+c x^2\right )}+\frac {\log \left (b+c x^2\right )}{2 c^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^5}{\left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{(b+c x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {b^2}{c^2 (b+c x)^3}-\frac {2 b}{c^2 (b+c x)^2}+\frac {1}{c^2 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b^2}{4 c^3 \left (b+c x^2\right )^2}+\frac {b}{c^3 \left (b+c x^2\right )}+\frac {\log \left (b+c x^2\right )}{2 c^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.80 \[ \frac {\frac {b \left (3 b+4 c x^2\right )}{\left (b+c x^2\right )^2}+2 \log \left (b+c x^2\right )}{4 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 69, normalized size = 1.41 \[ \frac {4 \, b c x^{2} + 3 \, b^{2} + 2 \, {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \log \left (c x^{2} + b\right )}{4 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 42, normalized size = 0.86 \[ \frac {\log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{3}} - \frac {3 \, c x^{4} + 2 \, b x^{2}}{4 \, {\left (c x^{2} + b\right )}^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 0.94 \[ -\frac {b^{2}}{4 \left (c \,x^{2}+b \right )^{2} c^{3}}+\frac {b}{\left (c \,x^{2}+b \right ) c^{3}}+\frac {\ln \left (c \,x^{2}+b \right )}{2 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 55, normalized size = 1.12 \[ \frac {4 \, b c x^{2} + 3 \, b^{2}}{4 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} + \frac {\log \left (c x^{2} + b\right )}{2 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.18, size = 52, normalized size = 1.06 \[ \frac {\frac {3\,b^2}{4\,c^3}+\frac {b\,x^2}{c^2}}{b^2+2\,b\,c\,x^2+c^2\,x^4}+\frac {\ln \left (c\,x^2+b\right )}{2\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 53, normalized size = 1.08 \[ \frac {3 b^{2} + 4 b c x^{2}}{4 b^{2} c^{3} + 8 b c^{4} x^{2} + 4 c^{5} x^{4}} + \frac {\log {\left (b + c x^{2} \right )}}{2 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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