Optimal. Leaf size=33 \[ \frac {b}{2 c^2 \left (b+c x^2\right )}+\frac {\log \left (b+c x^2\right )}{2 c^2} \]
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Rubi [A] time = 0.03, antiderivative size = 33, 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} \[ \frac {b}{2 c^2 \left (b+c x^2\right )}+\frac {\log \left (b+c x^2\right )}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^7}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^3}{\left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(b+c x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {b}{c (b+c x)^2}+\frac {1}{c (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {b}{2 c^2 \left (b+c x^2\right )}+\frac {\log \left (b+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.82 \[ \frac {\frac {b}{b+c x^2}+\log \left (b+c x^2\right )}{2 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 35, normalized size = 1.06 \[ \frac {{\left (c x^{2} + b\right )} \log \left (c x^{2} + b\right ) + b}{2 \, {\left (c^{3} x^{2} + b c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 32, normalized size = 0.97 \[ -\frac {x^{2}}{2 \, {\left (c x^{2} + b\right )} c} + \frac {\log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 0.91 \[ \frac {b}{2 \left (c \,x^{2}+b \right ) c^{2}}+\frac {\ln \left (c \,x^{2}+b \right )}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 32, normalized size = 0.97 \[ \frac {b}{2 \, {\left (c^{3} x^{2} + b c^{2}\right )}} + \frac {\log \left (c x^{2} + b\right )}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.18, size = 29, normalized size = 0.88 \[ \frac {\ln \left (c\,x^2+b\right )}{2\,c^2}+\frac {b}{2\,c^2\,\left (c\,x^2+b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 29, normalized size = 0.88 \[ \frac {b}{2 b c^{2} + 2 c^{3} x^{2}} + \frac {\log {\left (b + c x^{2} \right )}}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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