Optimal. Leaf size=319 \[ \frac{c^{3/2} \sqrt{a+b x^2} \text{EllipticF}\left (\tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right ),-\frac{e (b c-a d)}{a (d e-c f)}\right )}{a e \sqrt{c+d x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} (d e-c f)}{e \sqrt{c+d x^2} \sqrt{e+f x^2} (b e-a f)}-\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{d e-c f} E\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{e \sqrt{c+d x^2} (b e-a f) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Rubi [A] time = 0.46667, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {554, 422, 418, 492, 411} \[ \frac{c^{3/2} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a e \sqrt{c+d x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} (d e-c f)}{e \sqrt{c+d x^2} \sqrt{e+f x^2} (b e-a f)}-\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{d e-c f} E\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{e \sqrt{c+d x^2} (b e-a f) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 554
Rule 422
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} \left (e+f x^2\right )^{3/2}} \, dx &=\frac{\left (\sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{(-d e+c f) x^2}{c}}}{\sqrt{1-\frac{(-b e+a f) x^2}{a}}} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{e \sqrt{a+b x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\\ &=\frac{\left (\sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{(-b e+a f) x^2}{a}} \sqrt{1-\frac{(-d e+c f) x^2}{c}}} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{e \sqrt{a+b x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\left ((-d e+c f) \sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{(-b e+a f) x^2}{a}} \sqrt{1-\frac{(-d e+c f) x^2}{c}}} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{c e \sqrt{a+b x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\\ &=\frac{(d e-c f) x \sqrt{a+b x^2}}{e (b e-a f) \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{c^{3/2} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a e \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{\left (a (-d e+c f) \sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{(-b e+a f) x^2}{a}}}{\left (1-\frac{(-d e+c f) x^2}{c}\right )^{3/2}} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{c e (b e-a f) \sqrt{a+b x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\\ &=\frac{(d e-c f) x \sqrt{a+b x^2}}{e (b e-a f) \sqrt{c+d x^2} \sqrt{e+f x^2}}-\frac{\sqrt{c} \sqrt{d e-c f} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{e (b e-a f) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{c^{3/2} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a e \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0987182, size = 148, normalized size = 0.46 \[ \frac{\sqrt{a} \sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{a f-b e} x}{\sqrt{a} \sqrt{f x^2+e}}\right )|\frac{a (c f-d e)}{c (a f-b e)}\right )}{e \sqrt{a+b x^2} \sqrt{a f-b e} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{b{x}^{2}+a}}} \left ( f{x}^{2}+e \right ) ^{-{\frac{3}{2}}}}\, 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}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}{b f^{2} x^{6} +{\left (2 \, b e f + a f^{2}\right )} x^{4} + a e^{2} +{\left (b e^{2} + 2 \, a e f\right )} x^{2}}, x\right ) \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}} \left (e + f x^{2}\right )^{\frac{3}{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}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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