Optimal. Leaf size=163 \[ \frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Rubi [A] time = 0.133839, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {553, 537} \[ \frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 553
Rule 537
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} \sqrt{e+f x^2}} \, dx &=\frac{\left (c \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-d x^2\right ) \sqrt{1-\frac{(-b c+a d) x^2}{a}} \sqrt{1-\frac{(d e-c f) x^2}{e}}} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{a \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{e+f x^2}}\\ &=\frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{c+d x^2}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{e+f x^2}}\\ \end{align*}
Mathematica [A] time = 0.100159, size = 162, normalized size = 0.99 \[ \frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|\frac{(a d-b c) e}{a (d e-c f)}\right )}{a \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{f{x}^{2}+e}}}}\, dx \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{b x^{2} + a} \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{\sqrt{c + d x^{2}}}{\sqrt{a + b x^{2}} \sqrt{e + f x^{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{b x^{2} + a} \sqrt{f x^{2} + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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