Optimal. Leaf size=78 \[ \frac{x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{2 a \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
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Rubi [A] time = 0.0719037, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1585, 1114, 722, 618, 206} \[ \frac{x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{2 a \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 1114
Rule 722
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{x^7}{\left (a x+b x^3+c x^5\right )^2} \, dx &=\int \frac{x^5}{\left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{b^2-4 a c}\\ &=\frac{x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{b^2-4 a c}\\ &=\frac{x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{2 a \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0911856, size = 93, normalized size = 1.19 \[ \frac{a \left (b-2 c x^2\right )+b^2 x^2}{2 c \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )}+\frac{2 a \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 104, normalized size = 1.3 \begin{align*}{\frac{1}{2\,c{x}^{4}+2\,b{x}^{2}+2\,a} \left ( -{\frac{ \left ( 2\,ac-{b}^{2} \right ){x}^{2}}{c \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{ab}{c \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+2\,{\frac{a}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.36568, size = 864, normalized size = 11.08 \begin{align*} \left [-\frac{a b^{3} - 4 \, a^{2} b c +{\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (a c^{2} x^{4} + a b c x^{2} + a^{2} c\right )} \sqrt{b^{2} - 4 \, a c} \log \left (\frac{2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c -{\left (2 \, c x^{2} + b\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right )}{2 \,{\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} +{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2}\right )}}, -\frac{a b^{3} - 4 \, a^{2} b c +{\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x^{2} - 4 \,{\left (a c^{2} x^{4} + a b c x^{2} + a^{2} c\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{2 \,{\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} +{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.74624, size = 282, normalized size = 3.62 \begin{align*} - a \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x^{2} + \frac{- 16 a^{3} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a^{2} b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - a b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + a b}{2 a c} \right )} + a \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x^{2} + \frac{16 a^{3} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a^{2} b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + a b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + a b}{2 a c} \right )} - \frac{- a b + x^{2} \left (2 a c - b^{2}\right )}{8 a^{2} c^{2} - 2 a b^{2} c + x^{4} \left (8 a c^{3} - 2 b^{2} c^{2}\right ) + x^{2} \left (8 a b c^{2} - 2 b^{3} c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 29.8736, size = 130, normalized size = 1.67 \begin{align*} -\frac{2 \, a \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{b^{2} x^{2} - 2 \, a c x^{2} + a b}{2 \,{\left (c x^{4} + b x^{2} + a\right )}{\left (b^{2} c - 4 \, a c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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