Optimal. Leaf size=347 \[ \frac{x^2 \left (2 a c C+A b c+b^2 (-C)\right )+a (2 A c-b C)}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{(A b-2 a C) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{B x \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \left (b-\frac{4 a c+b^2}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{B \left (b \sqrt{b^2-4 a c}+4 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]
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Rubi [A] time = 0.617503, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321, Rules used = {1662, 1251, 777, 618, 206, 12, 1120, 1166, 205} \[ \frac{x^2 \left (2 a c C+A b c+b^2 (-C)\right )+a (2 A c-b C)}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{(A b-2 a C) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{B x \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \left (b-\frac{4 a c+b^2}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{B \left (b \sqrt{b^2-4 a c}+4 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]
Antiderivative was successfully verified.
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Rule 1662
Rule 1251
Rule 777
Rule 618
Rule 206
Rule 12
Rule 1120
Rule 1166
Rule 205
Rubi steps
\begin{align*} \int \frac{x^3 \left (A+B x+C x^2\right )}{\left (a+b x^2+c x^4\right )^2} \, dx &=\int \frac{B x^4}{\left (a+b x^2+c x^4\right )^2} \, dx+\int \frac{x^3 \left (A+C x^2\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (A+C x)}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )+B \int \frac{x^4}{\left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{B x \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 (2 A c-b C)+\left (A b c-b^2 C+2 a c C\right ) x^2}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{B \int \frac{2 a-b x^2}{a+b x^2+c x^4} \, dx}{2 \left (b^2-4 a c\right )}+\frac{(A b-2 a C) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac{B x \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 (2 A c-b C)+\left (A b c-b^2 C+2 a c C\right ) x^2}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (B \left (b^2+4 a c-b \sqrt{b^2-4 a c}\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 \left (b^2-4 a c\right )^{3/2}}+\frac{\left (B \left (b^2+4 a c+b \sqrt{b^2-4 a c}\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 \left (b^2-4 a c\right )^{3/2}}-\frac{(A b-2 a C) \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{B x \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 (2 A c-b C)+\left (A b c-b^2 C+2 a c C\right ) x^2}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{B \left (b^2+4 a c-b \sqrt{b^2-4 a c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{B \left (b^2+4 a c+b \sqrt{b^2-4 a c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{(A b-2 a C) \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 = 0.995371, size = 358, normalized size = 1.03 \[ \frac{1}{4} \left (-\frac{2 \left (a (2 A c-b C+2 c x (B+C x))+b x^2 (A c-b C+B c x)\right )}{c \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )}+\frac{2 (A b-2 a C) \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{2 (A b-2 a C) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\sqrt{2} B \left (b \sqrt{b^2-4 a c}-4 a c-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} B \left (b \sqrt{b^2-4 a c}+4 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \text{hanged} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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