Optimal. Leaf size=247 \[ \frac{\sqrt{a} f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}}+\frac{x \sqrt{a-b x^2} (d e-c f)}{c \sqrt{c+d x^2} (a d+b c)}+\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} (d e-c f) E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{c d \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1} (a d+b c)} \]
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Rubi [A] time = 0.239852, antiderivative size = 247, 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} \[ \frac{x \sqrt{a-b x^2} (d e-c f)}{c \sqrt{c+d x^2} (a d+b c)}+\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} (d e-c f) E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{c d \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1} (a d+b c)}+\frac{\sqrt{a} f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}} \]
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{f \int \frac{1}{\sqrt{a-b x^2} \sqrt{c+d x^2}} \, dx}{d}+\frac{(b (d e-c f)) \int \frac{\sqrt{c+d x^2}}{\sqrt{a-b x^2}} \, dx}{c d (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 (b (d e-c f) \sqrt{1-\frac{b x^2}{a}}\right ) \int \frac{\sqrt{c+d x^2}}{\sqrt{1-\frac{b x^2}{a}}} \, dx}{c d (b c+a d) \sqrt{a-b x^2}}+\frac{\left (f \sqrt{1+\frac{d x^2}{c}}\right ) \int \frac{1}{\sqrt{a-b x^2} \sqrt{1+\frac{d x^2}{c}}} \, dx}{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 (b (d e-c f) \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2}\right ) \int \frac{\sqrt{1+\frac{d x^2}{c}}}{\sqrt{1-\frac{b x^2}{a}}} \, dx}{c d (b c+a d) \sqrt{a-b x^2} \sqrt{1+\frac{d x^2}{c}}}+\frac{\left (f \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}{d \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{\sqrt{a} \sqrt{b} (d e-c f) \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{c d (b c+a d) \sqrt{a-b x^2} \sqrt{1+\frac{d x^2}{c}}}+\frac{\sqrt{a} f \sqrt{1-\frac{b x^2}{a}} \sqrt{1+\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.69438, size = 220, normalized size = 0.89 \[ \frac{-i c f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{-\frac{b}{a}}\right ),-\frac{a d}{b c}\right )+d x \sqrt{-\frac{b}{a}} \left (a-b x^2\right ) (d e-c f)+i b c \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (c f-d e) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{c d \sqrt{-\frac{b}{a}} \sqrt{a-b x^2} \sqrt{c+d x^2} (a d+b c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 359, normalized size = 1.5 \begin{align*}{\frac{1}{cd \left ( ad+bc \right ) \left ( bd{x}^{4}-ad{x}^{2}+bc{x}^{2}-ac \right ) } \left ( -{x}^{3}bcdf\sqrt{{\frac{b}{a}}}+{x}^{3}b{d}^{2}e\sqrt{{\frac{b}{a}}}-{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) acdf\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) b{c}^{2}f\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) b{c}^{2}f\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) bcde\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+xacdf\sqrt{{\frac{b}{a}}}-xa{d}^{2}e\sqrt{{\frac{b}{a}}} \right ) \sqrt{-b{x}^{2}+a}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{{\frac{b}{a}}}}}} \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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