Optimal. Leaf size=122 \[ \frac{b \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} (b c-a d) \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d) \sqrt{d e-c f}} \]
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Rubi [A] time = 0.110828, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {532, 377, 205} \[ \frac{b \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} (b c-a d) \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d) \sqrt{d e-c f}} \]
Antiderivative was successfully verified.
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Rule 532
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \sqrt{e+f x^2}} \, dx &=\frac{b \int \frac{1}{\left (a+b x^2\right ) \sqrt{e+f x^2}} \, dx}{b c-a d}-\frac{d \int \frac{1}{\left (c+d x^2\right ) \sqrt{e+f x^2}} \, dx}{b c-a d}\\ &=\frac{b \operatorname{Subst}\left (\int \frac{1}{a-(-b e+a f) x^2} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{b c-a d}-\frac{d \operatorname{Subst}\left (\int \frac{1}{c-(-d e+c f) x^2} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{b c-a d}\\ &=\frac{b \tan ^{-1}\left (\frac{\sqrt{b e-a f} x}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} (b c-a d) \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d) \sqrt{d e-c f}}\\ \end{align*}
Mathematica [A] time = 0.176689, size = 113, normalized size = 0.93 \[ \frac{\frac{b \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}}}{b c-a d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.037, size = 782, normalized size = 6.4 \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*} \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )} \sqrt{f x^{2} + e}}\,{d x} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right ) \sqrt{e + f x^{2}}}\, dx \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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