Optimal. Leaf size=298 \[ -\frac{\sqrt{c} \left (\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} \left (-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} a \sqrt{\sqrt{b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}-\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac{1}{a d x} \]
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Rubi [A] time = 0.960295, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1287, 205, 1166} \[ -\frac{\sqrt{c} \left (\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} \left (-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} a \sqrt{\sqrt{b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}-\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac{1}{a d x} \]
Antiderivative was successfully verified.
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Rule 1287
Rule 205
Rule 1166
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx &=\int \left (\frac{1}{a d x^2}-\frac{e^3}{d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )}+\frac{-b c d+b^2 e-a c e-c (c d-b e) x^2}{a \left (c d^2-b d e+a e^2\right ) \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=-\frac{1}{a d x}+\frac{\int \frac{-b c d+b^2 e-a c e-c (c d-b e) x^2}{a+b x^2+c x^4} \, dx}{a \left (c d^2-b d e+a e^2\right )}-\frac{e^3 \int \frac{1}{d+e x^2} \, dx}{d \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{1}{a d x}-\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{3/2} \left (c d^2-b d e+a e^2\right )}-\frac{\left (c \left (c d-b e-\frac{b c d-b^2 e+2 a c e}{\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}{2 a \left (c d^2-b d e+a e^2\right )}-\frac{\left (c \left (c d-b e+\frac{b c d-b^2 e+2 a c e}{\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}{2 a \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{1}{a d x}-\frac{\sqrt{c} \left (c d-b e+\frac{b c d-b^2 e+2 a c e}{\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 )}{\sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac{\sqrt{c} \left (c d-b e-\frac{b c d-b^2 e+2 a c e}{\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 )}{\sqrt{2} a \sqrt{b+\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{3/2} \left (c d^2-b d e+a e^2\right )}\\ \end{align*}
Mathematica [A] time = 0.433557, size = 340, normalized size = 1.14 \[ -\frac{\sqrt{c} \left (c d \sqrt{b^2-4 a c}-b e \sqrt{b^2-4 a c}+2 a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}} \left (e (a e-b d)+c d^2\right )}+\frac{\sqrt{c} \left (-c d \sqrt{b^2-4 a c}+b e \sqrt{b^2-4 a c}+2 a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} a \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b} \left (e (a e-b d)+c d^2\right )}-\frac{e^{5/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac{1}{a d x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.028, size = 817, normalized size = 2.7 \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(-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 [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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