Optimal. Leaf size=237 \[ \frac{e \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}}+\frac{x \sqrt{a+b x^2} (c f+d e)}{c \sqrt{c-d x^2} (a d+b c)}-\frac{\sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2} (a d+b c)} \]
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Rubi [A] time = 0.218324, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {527, 524, 427, 426, 424, 421, 419} \[ \frac{x \sqrt{a+b x^2} (c f+d e)}{c \sqrt{c-d x^2} (a d+b c)}-\frac{\sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2} (a d+b c)}+\frac{e \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}} \]
Antiderivative was successfully verified.
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Rule 527
Rule 524
Rule 427
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{e+f x^2}{\sqrt{a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx &=\frac{(d e+c f) x \sqrt{a+b x^2}}{c (b c+a d) \sqrt{c-d x^2}}+\frac{\int \frac{c (b e-a f)-b (d e+c f) x^2}{\sqrt{a+b x^2} \sqrt{c-d x^2}} \, dx}{c (b c+a d)}\\ &=\frac{(d e+c f) x \sqrt{a+b x^2}}{c (b c+a d) \sqrt{c-d x^2}}+\frac{e \int \frac{1}{\sqrt{a+b x^2} \sqrt{c-d x^2}} \, dx}{c}-\frac{(d e+c f) \int \frac{\sqrt{a+b x^2}}{\sqrt{c-d x^2}} \, dx}{c (b c+a d)}\\ &=\frac{(d e+c f) x \sqrt{a+b x^2}}{c (b c+a d) \sqrt{c-d x^2}}+\frac{\left (e \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{1}{\sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}}} \, dx}{c \sqrt{c-d x^2}}-\frac{\left ((d e+c f) \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{\sqrt{a+b x^2}}{\sqrt{1-\frac{d x^2}{c}}} \, dx}{c (b c+a d) \sqrt{c-d x^2}}\\ &=\frac{(d e+c f) x \sqrt{a+b x^2}}{c (b c+a d) \sqrt{c-d x^2}}-\frac{\left ((d e+c f) \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{\sqrt{1+\frac{b x^2}{a}}}{\sqrt{1-\frac{d x^2}{c}}} \, dx}{c (b c+a d) \sqrt{1+\frac{b x^2}{a}} \sqrt{c-d x^2}}+\frac{\left (e \sqrt{1+\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{1}{\sqrt{1+\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}}} \, dx}{c \sqrt{a+b x^2} \sqrt{c-d x^2}}\\ &=\frac{(d e+c f) x \sqrt{a+b x^2}}{c (b c+a d) \sqrt{c-d x^2}}-\frac{(d e+c f) \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} (b c+a d) \sqrt{1+\frac{b x^2}{a}} \sqrt{c-d x^2}}+\frac{e \sqrt{1+\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}}\\ \end{align*}
Mathematica [C] time = 0.432011, size = 213, normalized size = 0.9 \[ \frac{i c f \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{b}{a}}\right ),-\frac{a d}{b c}\right )+d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) (c f+d e)-i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (c f+d e) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{c d \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c-d x^2} (a d+b c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 345, normalized size = 1.5 \begin{align*}{\frac{1}{c \left ( ad+bc \right ) \left ( bd{x}^{4}+ad{x}^{2}-bc{x}^{2}-ac \right ) } \left ( -{x}^{3}bcf\sqrt{{\frac{d}{c}}}-{x}^{3}bde\sqrt{{\frac{d}{c}}}-{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) ade\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}}-{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) bce\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}}+{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) acf\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}}+{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) ade\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}}-xacf\sqrt{{\frac{d}{c}}}-xade\sqrt{{\frac{d}{c}}} \right ) \sqrt{b{x}^{2}+a}\sqrt{-d{x}^{2}+c}{\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{f x^{2} + e}{\sqrt{b x^{2} + a}{\left (-d x^{2} + c\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}{\left (f x^{2} + e\right )}}{b d^{2} x^{6} -{\left (2 \, b c d - a d^{2}\right )} x^{4} + a c^{2} +{\left (b c^{2} - 2 \, a c d\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{e + f x^{2}}{\sqrt{a + b x^{2}} \left (c - d 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{f x^{2} + e}{\sqrt{b x^{2} + a}{\left (-d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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