Optimal. Leaf size=381 \[ \frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{2 b^2 c \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (b^2 c e-a^2 d f\right ) \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{2 a^2 b^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}-\frac{f x \sqrt{c+d x^2}}{2 a b \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 a b \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]
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Rubi [A] time = 0.291334, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {548, 531, 418, 492, 411, 538, 537} \[ \frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (b^2 c e-a^2 d f\right ) \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{2 a^2 b^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}-\frac{f x \sqrt{c+d x^2}}{2 a b \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 a b \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 b^2 c \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 548
Rule 531
Rule 418
Rule 492
Rule 411
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{\left (a+b x^2\right )^2} \, dx &=\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}+\frac{(d f) \int \frac{a-b x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{2 a b^2}+\frac{1}{2} \left (\frac{c e}{a}-\frac{a d f}{b^2}\right ) \int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx\\ &=\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}+\frac{(d f) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{2 b^2}-\frac{(d f) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{2 a b}+\frac{\left (\left (\frac{c e}{a}-\frac{a d f}{b^2}\right ) \sqrt{1+\frac{d x^2}{c}}\right ) \int \frac{1}{\left (a+b x^2\right ) \sqrt{1+\frac{d x^2}{c}} \sqrt{e+f x^2}} \, dx}{2 \sqrt{c+d x^2}}\\ &=-\frac{f x \sqrt{c+d x^2}}{2 a b \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}+\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 b^2 c \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{(e f) \int \frac{\sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{2 a b}+\frac{\left (\left (\frac{c e}{a}-\frac{a d f}{b^2}\right ) \sqrt{1+\frac{d x^2}{c}} \sqrt{1+\frac{f x^2}{e}}\right ) \int \frac{1}{\left (a+b x^2\right ) \sqrt{1+\frac{d x^2}{c}} \sqrt{1+\frac{f x^2}{e}}} \, dx}{2 \sqrt{c+d x^2} \sqrt{e+f x^2}}\\ &=-\frac{f x \sqrt{c+d x^2}}{2 a b \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 a b \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 b^2 c \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{\sqrt{-c} \left (\frac{c e}{a}-\frac{a d f}{b^2}\right ) \sqrt{1+\frac{d x^2}{c}} \sqrt{1+\frac{f x^2}{e}} \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{2 a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}\\ \end{align*}
Mathematica [C] time = 1.99279, size = 401, normalized size = 1.05 \[ \frac{-\frac{i c \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (a f+b e) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )}{b^2}+\frac{i a c f \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{b^2}-\frac{i c e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{a \sqrt{\frac{d}{c}}}+\frac{c e x}{a+b x^2}+\frac{c f x^3}{a+b x^2}+\frac{d e x^3}{a+b x^2}+\frac{d f x^5}{a+b x^2}+\frac{i c e \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{b}}{2 a \sqrt{c+d x^2} \sqrt{e+f x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 765, normalized size = 2. \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{\sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}{{\left (b x^{2} + a\right )}^{2}}\,{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{\sqrt{c + d x^{2}} \sqrt{e + f x^{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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