Optimal. Leaf size=121 \[ -\frac{6 a^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{x^6 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{3 a x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \]
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Rubi [A] time = 0.111931, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1114, 722, 618, 206} \[ -\frac{6 a^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{x^6 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{3 a x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 722
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{x^9}{\left (a+b x^2+c x^4\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac{x^6 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac{x^6 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{3 a x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{\left (b^2-4 a c\right )^2}\\ &=\frac{x^6 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{3 a x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{\left (6 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{\left (b^2-4 a c\right )^2}\\ &=\frac{x^6 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{3 a x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{6 a^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.175756, size = 194, normalized size = 1.6 \[ \frac{1}{4} \left (\frac{a^2 c \left (2 c x^2-3 b\right )+a b^2 \left (b-4 c x^2\right )+b^4 x^2}{c^3 \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2}+\frac{22 a^2 b c^2-20 a^2 c^3 x^2+16 a b^2 c^2 x^2-8 a b^3 c-2 b^4 c x^2+b^5}{c^3 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{24 a^2 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.179, size = 267, normalized size = 2.2 \begin{align*}{\frac{1}{2\, \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}} \left ( -{\frac{ \left ( 10\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ){x}^{6}}{c \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) }}+{\frac{b \left ( 2\,{a}^{2}{c}^{2}+8\,ac{b}^{2}-{b}^{4} \right ){x}^{4}}{2\,{c}^{2} \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) }}-{\frac{a \left ( 6\,{a}^{2}{c}^{2}-10\,ac{b}^{2}+{b}^{4} \right ){x}^{2}}{{c}^{2} \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) }}+{\frac{b{a}^{2} \left ( 10\,ac-{b}^{2} \right ) }{2\,{c}^{2} \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) }} \right ) }+6\,{\frac{{a}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{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 = 1.59675, size = 2033, normalized size = 16.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 12.3407, size = 554, normalized size = 4.58 \begin{align*} - 3 a^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x^{2} + \frac{- 192 a^{5} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 144 a^{4} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 36 a^{3} b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a^{2} b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a^{2} b}{6 a^{2} c} \right )} + 3 a^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x^{2} + \frac{192 a^{5} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 144 a^{4} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 36 a^{3} b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 3 a^{2} b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a^{2} b}{6 a^{2} c} \right )} - \frac{- 10 a^{3} b c + a^{2} b^{3} + x^{6} \left (20 a^{2} c^{3} - 16 a b^{2} c^{2} + 2 b^{4} c\right ) + x^{4} \left (- 2 a^{2} b c^{2} - 8 a b^{3} c + b^{5}\right ) + x^{2} \left (12 a^{3} c^{2} - 20 a^{2} b^{2} c + 2 a b^{4}\right )}{64 a^{4} c^{4} - 32 a^{3} b^{2} c^{3} + 4 a^{2} b^{4} c^{2} + x^{8} \left (64 a^{2} c^{6} - 32 a b^{2} c^{5} + 4 b^{4} c^{4}\right ) + x^{6} \left (128 a^{2} b c^{5} - 64 a b^{3} c^{4} + 8 b^{5} c^{3}\right ) + x^{4} \left (128 a^{3} c^{5} - 24 a b^{4} c^{3} + 4 b^{6} c^{2}\right ) + x^{2} \left (128 a^{3} b c^{4} - 64 a^{2} b^{3} c^{3} + 8 a b^{5} c^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 27.3263, size = 286, normalized size = 2.36 \begin{align*} \frac{6 \, a^{2} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, b^{4} c x^{6} - 16 \, a b^{2} c^{2} x^{6} + 20 \, a^{2} c^{3} x^{6} + b^{5} x^{4} - 8 \, a b^{3} c x^{4} - 2 \, a^{2} b c^{2} x^{4} + 2 \, a b^{4} x^{2} - 20 \, a^{2} b^{2} c x^{2} + 12 \, a^{3} c^{2} x^{2} + a^{2} b^{3} - 10 \, a^{3} b c}{4 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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