Optimal. Leaf size=563 \[ -\frac{\left (-b^2 \left (a e^3 \sqrt{b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (e \sqrt{b^2-4 a c}+2 c d\right )-b c \left (c d^2 \left (d \sqrt{b^2-4 a c}+12 a e\right )+a e^2 \left (3 d \sqrt{b^2-4 a c}+8 a e\right )\right )+a b^3 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (-b^2 \left (-a e^3 \sqrt{b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (2 c d-e \sqrt{b^2-4 a c}\right )+b c \left (c d^2 \left (d \sqrt{b^2-4 a c}-12 a e\right )+a e^2 \left (3 d \sqrt{b^2-4 a c}-8 a e\right )\right )+a b^3 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left (c \left (-\frac{a b e \left (a e^2+3 c d^2\right )}{c}-2 a d \left (c d^2-3 a e^2\right )+b^2 d^3\right )-x^2 \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
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Rubi [A] time = 3.51858, antiderivative size = 563, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1205, 1166, 205} \[ -\frac{\left (-b^2 \left (a e^3 \sqrt{b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (e \sqrt{b^2-4 a c}+2 c d\right )-b c \left (c d^2 \left (d \sqrt{b^2-4 a c}+12 a e\right )+a e^2 \left (3 d \sqrt{b^2-4 a c}+8 a e\right )\right )+a b^3 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (-b^2 \left (-a e^3 \sqrt{b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (2 c d-e \sqrt{b^2-4 a c}\right )+b c \left (c d^2 \left (d \sqrt{b^2-4 a c}-12 a e\right )+a e^2 \left (3 d \sqrt{b^2-4 a c}-8 a e\right )\right )+a b^3 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left (c \left (-\frac{a b e \left (a e^2+3 c d^2\right )}{c}-2 a d \left (c d^2-3 a e^2\right )+b^2 d^3\right )-x^2 \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1205
Rule 1166
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac{x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac{a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\int \frac{-b^2 d^3+6 a d \left (c d^2+a e^2\right )-\frac{a b e \left (3 c d^2+a e^2\right )}{c}+\left (-\frac{a b^2 e^3}{c}+6 a e \left (c d^2+a e^2\right )-b \left (c d^3+3 a d e^2\right )\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac{a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (a b^3 e^3+6 a c \left (2 c d-\sqrt{b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2-a \sqrt{b^2-4 a c} e^3\right )+b c \left (c d^2 \left (\sqrt{b^2-4 a c} d-12 a e\right )+a e^2 \left (3 \sqrt{b^2-4 a c} d-8 a e\right )\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a c \left (b^2-4 a c\right )^{3/2}}-\frac{\left (a b^3 e^3+6 a c \left (2 c d+\sqrt{b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2+a \sqrt{b^2-4 a c} e^3\right )-b c \left (a e^2 \left (3 \sqrt{b^2-4 a c} d+8 a e\right )+c d^2 \left (\sqrt{b^2-4 a c} d+12 a e\right )\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a c \left (b^2-4 a c\right )^{3/2}}\\ &=\frac{x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac{a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (a b^3 e^3+6 a c \left (2 c d+\sqrt{b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2+a \sqrt{b^2-4 a c} e^3\right )-b c \left (a e^2 \left (3 \sqrt{b^2-4 a c} d+8 a e\right )+c d^2 \left (\sqrt{b^2-4 a c} d+12 a e\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (a b^3 e^3+6 a c \left (2 c d-\sqrt{b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2-a \sqrt{b^2-4 a c} e^3\right )+b c \left (c d^2 \left (\sqrt{b^2-4 a c} d-12 a e\right )+a e^2 \left (3 \sqrt{b^2-4 a c} d-8 a e\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}
Mathematica [A] time = 1.71249, size = 540, normalized size = 0.96 \[ \frac{\frac{2 \sqrt{c} x \left (b \left (-a^2 e^3-3 a c d e \left (d-e x^2\right )+c^2 d^3 x^2\right )+b^2 \left (c d^3-a e^3 x^2\right )+2 a c \left (a e^2 \left (3 d+e x^2\right )-c d^2 \left (d+3 e x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \left (b^2 \left (a e^3 \sqrt{b^2-4 a c}-3 a c d e^2+c^2 d^3\right )-6 a c \left (a e^2+c d^2\right ) \left (e \sqrt{b^2-4 a c}+2 c d\right )+b c \left (c d^2 \left (d \sqrt{b^2-4 a c}+12 a e\right )+a e^2 \left (3 d \sqrt{b^2-4 a c}+8 a e\right )\right )-a b^3 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (b^2 \left (a e^3 \sqrt{b^2-4 a c}+3 a c d e^2-c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (2 c d-e \sqrt{b^2-4 a c}\right )+b c \left (c d^2 \left (d \sqrt{b^2-4 a c}-12 a e\right )+a e^2 \left (3 d \sqrt{b^2-4 a c}-8 a e\right )\right )+a b^3 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 a c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 1846, normalized size = 3.3 \begin{align*} \text{result too large to display} \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{{\left (b c^{2} d^{3} - 6 \, a c^{2} d^{2} e + 3 \, a b c d e^{2} -{\left (a b^{2} - 2 \, a^{2} c\right )} e^{3}\right )} x^{3} -{\left (3 \, a b c d^{2} e - 6 \, a^{2} c d e^{2} + a^{2} b e^{3} -{\left (b^{2} c - 2 \, a c^{2}\right )} d^{3}\right )} x}{2 \,{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2} +{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x^{4} +{\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} x^{2}\right )}} - \frac{-\int \frac{3 \, a b c d^{2} e - 6 \, a^{2} c d e^{2} + a^{2} b e^{3} +{\left (b^{2} c - 6 \, a c^{2}\right )} d^{3} +{\left (b c^{2} d^{3} - 6 \, a c^{2} d^{2} e + 3 \, a b c d e^{2} +{\left (a b^{2} - 6 \, a^{2} c\right )} e^{3}\right )} x^{2}}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )}} \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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