Optimal. Leaf size=282 \[ -\frac{\sqrt{e} \sqrt{c+d x^2} (b e-3 a f) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f-b c f+2 b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 d f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} (-3 a d f-b c f+2 b d e)}{3 d f \sqrt{e+f x^2}}+\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 f} \]
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Rubi [A] time = 0.18439, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {528, 531, 418, 492, 411} \[ -\frac{\sqrt{e} \sqrt{c+d x^2} (b e-3 a f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f-b c f+2 b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 d f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} (-3 a d f-b c f+2 b d e)}{3 d f \sqrt{e+f x^2}}+\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 f} \]
Antiderivative was successfully verified.
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Rule 528
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx &=\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 f}+\frac{\int \frac{-c (b e-3 a f)+(-2 b d e+b c f+3 a d f) x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 f}\\ &=\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 f}-\frac{(c (b e-3 a f)) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 f}+\frac{(-2 b d e+b c f+3 a d f) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 f}\\ &=-\frac{(2 b d e-b c f-3 a d f) x \sqrt{c+d x^2}}{3 d f \sqrt{e+f x^2}}+\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 f}-\frac{\sqrt{e} (b e-3 a 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 )}{3 f^{3/2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}-\frac{(e (-2 b d e+b c f+3 a d f)) \int \frac{\sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 d f}\\ &=-\frac{(2 b d e-b c f-3 a d f) x \sqrt{c+d x^2}}{3 d f \sqrt{e+f x^2}}+\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 f}+\frac{\sqrt{e} (2 b d e-b c f-3 a d 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 )}{3 d f^{3/2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}-\frac{\sqrt{e} (b e-3 a 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 )}{3 f^{3/2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}\\ \end{align*}
Mathematica [C] time = 0.424289, size = 215, normalized size = 0.76 \[ \frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 b e-3 a f) (c f-d e) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (3 a d f+b c f-2 b d e) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+b f x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right )}{3 f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 501, normalized size = 1.8 \begin{align*}{\frac{1}{ \left ( 3\,df{x}^{4}+3\,cf{x}^{2}+3\,de{x}^{2}+3\,ce \right ){f}^{2}}\sqrt{d{x}^{2}+c}\sqrt{f{x}^{2}+e} \left ( \sqrt{-{\frac{d}{c}}}{x}^{5}bd{f}^{2}+\sqrt{-{\frac{d}{c}}}{x}^{3}bc{f}^{2}+\sqrt{-{\frac{d}{c}}}{x}^{3}bdef+3\,\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) ac{f}^{2}-3\,\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) adef-2\,\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bcef+2\,\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bd{e}^{2}+3\,\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) adef+\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bcef-2\,\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bd{e}^{2}+\sqrt{-{\frac{d}{c}}}xbcef \right ){\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \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{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + c}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + c}}{\sqrt{f x^{2} + e}}, 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{\left (a + b x^{2}\right ) \sqrt{c + d 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{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + c}}{\sqrt{f x^{2} + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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