Optimal. Leaf size=132 \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{x^4 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b x^2}{2 c \left (b^2-4 a c\right )}+\frac{\log \left (a+b x^2+c x^4\right )}{4 c^2} \]
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Rubi [A] time = 0.151706, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {1585, 1114, 738, 773, 634, 618, 206, 628} \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{x^4 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b x^2}{2 c \left (b^2-4 a c\right )}+\frac{\log \left (a+b x^2+c x^4\right )}{4 c^2} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 1114
Rule 738
Rule 773
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^9}{\left (a x+b x^3+c x^5\right )^2} \, dx &=\int \frac{x^7}{\left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{x^4 \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{\operatorname{Subst}\left (\int \frac{x (4 a+b x)}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=-\frac{b x^2}{2 c \left (b^2-4 a c\right )}+\frac{x^4 \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{\operatorname{Subst}\left (\int \frac{-a b+\left (-b^2+4 a c\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c \left (b^2-4 a c\right )}\\ &=-\frac{b x^2}{2 c \left (b^2-4 a c\right )}+\frac{x^4 \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{\operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2}-\frac{\left (b \left (b^2-6 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2 \left (b^2-4 a c\right )}\\ &=-\frac{b x^2}{2 c \left (b^2-4 a c\right )}+\frac{x^4 \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{\log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{\left (b \left (b^2-6 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^2 \left (b^2-4 a c\right )}\\ &=-\frac{b x^2}{2 c \left (b^2-4 a c\right )}+\frac{x^4 \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{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{\log \left (a+b x^2+c x^4\right )}{4 c^2}\\ \end{align*}
Mathematica [A] time = 0.185909, size = 121, normalized size = 0.92 \[ \frac{\frac{2 \left (-2 a^2 c+a b \left (b-3 c x^2\right )+b^3 x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{2 b \left (b^2-6 a c\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\log \left (a+b x^2+c x^4\right )}{4 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 222, normalized size = 1.7 \begin{align*}{\frac{1}{2\,c{x}^{4}+2\,b{x}^{2}+2\,a} \left ({\frac{b \left ( 3\,ac-{b}^{2} \right ){x}^{2}}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{a \left ( 2\,ac-{b}^{2} \right ) }{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) a}{c \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}}{4\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }}-3\,{\frac{ab}{c \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{3}}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ) \left ( 4\,ac-{b}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a b^{2} - 2 \, a^{2} c +{\left (b^{3} - 3 \, a b c\right )} x^{2}}{2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{4} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2}\right )}} - \frac{-\int \frac{{\left (b^{2} - 4 \, a c\right )} x^{3} + a b x}{c x^{4} + b x^{2} + a}\,{d x}}{b^{2} c - 4 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.38481, size = 1412, normalized size = 10.7 \begin{align*} \left [\frac{2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + 2 \,{\left (b^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} x^{2} +{\left ({\left (b^{3} c - 6 \, a b c^{2}\right )} x^{4} + a b^{3} - 6 \, a^{2} b c +{\left (b^{4} - 6 \, a b^{2} c\right )} x^{2}\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 ) +{\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} +{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} +{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{2}\right )}}, \frac{2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + 2 \,{\left (b^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} x^{2} + 2 \,{\left ({\left (b^{3} c - 6 \, a b c^{2}\right )} x^{4} + a b^{3} - 6 \, a^{2} b c +{\left (b^{4} - 6 \, a b^{2} c\right )} x^{2}\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 ) +{\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} +{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} +{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.50461, size = 745, normalized size = 5.64 \begin{align*} \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right ) \log{\left (x^{2} + \frac{- 32 a^{2} c^{3} \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right ) + 8 a^{2} c + 16 a b^{2} c^{2} \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right ) - a b^{2} - 2 b^{4} c \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right ) \log{\left (x^{2} + \frac{- 32 a^{2} c^{3} \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right ) + 8 a^{2} c + 16 a b^{2} c^{2} \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right ) - a b^{2} - 2 b^{4} c \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \frac{2 a^{2} c - a b^{2} + x^{2} \left (3 a b c - b^{3}\right )}{8 a^{2} c^{3} - 2 a b^{2} c^{2} + x^{4} \left (8 a c^{4} - 2 b^{2} c^{3}\right ) + x^{2} \left (8 a b c^{3} - 2 b^{3} c^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 25.6046, size = 205, normalized size = 1.55 \begin{align*} -\frac{{\left (b^{3} - 6 \, a b c\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{b^{2} c x^{4} - 4 \, a c^{2} x^{4} - b^{3} x^{2} + 2 \, a b c x^{2} - a b^{2}}{4 \,{\left (c x^{4} + b x^{2} + a\right )}{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}} + \frac{\log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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