Optimal. Leaf size=307 \[ \frac{e \left (c+d x^2\right ) (b c-2 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^2 d x}-\frac{e x (b c-2 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^2}+\frac{e (b c-2 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{c^{3/2} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{e (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x}+\frac{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Rubi [A] time = 0.449728, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {6719, 468, 583, 531, 418, 492, 411} \[ \frac{e \left (c+d x^2\right ) (b c-2 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^2 d x}-\frac{e x (b c-2 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^2}+\frac{e (b c-2 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{c^{3/2} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{e (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x}+\frac{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 6719
Rule 468
Rule 583
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^2} \, dx &=\frac{\left (e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{\left (a+b x^2\right )^{3/2}}{x^2 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x}-\frac{\left (e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{a (b c-2 a d)-a b d x^2}{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{c d \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x}+\frac{(b c-2 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{c^2 d x}+\frac{\left (e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{a^2 b c d-a b d (b c-2 a d) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{a c^2 d \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x}+\frac{(b c-2 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{c^2 d x}+\frac{\left (a b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{c \sqrt{a+b x^2}}-\frac{\left (b (b c-2 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{c^2 \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x}-\frac{(b c-2 a d) e x \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^2}+\frac{(b c-2 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{c^2 d x}+\frac{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{\left ((b c-2 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{c \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x}-\frac{(b c-2 a d) e x \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c^2}+\frac{(b c-2 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{c^2 d x}+\frac{(b c-2 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{c^{3/2} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\\ \end{align*}
Mathematica [C] time = 0.35807, size = 228, normalized size = 0.74 \[ -\frac{e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (d \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (a c+2 a d x^2-b c x^2\right )-i b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (2 a d-b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )\right )}{c^2 d x \sqrt{\frac{b}{a}} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 670, normalized size = 2.2 \begin{align*} -{\frac{d{x}^{2}+c}{ \left ( b{x}^{2}+a \right ) ^{2}{c}^{2}dx} \left ({\frac{ \left ( b{x}^{2}+a \right ) e}{d{x}^{2}+c}} \right ) ^{{\frac{3}{2}}} \left ( \sqrt{-{\frac{b}{a}}}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{4}ab{d}^{2}-\sqrt{-{\frac{b}{a}}}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{4}{b}^{2}cd+\sqrt{-{\frac{b}{a}}}\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }{x}^{4}ab{d}^{2}+\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 ) \sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }xabcd-\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 ) \sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }x{b}^{2}{c}^{2}-2\,\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 ) \sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }xabcd+\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 ) \sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }x{b}^{2}{c}^{2}+\sqrt{-{\frac{b}{a}}}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{2}{a}^{2}{d}^{2}-\sqrt{-{\frac{b}{a}}}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{2}abcd+\sqrt{-{\frac{b}{a}}}\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }{x}^{2}{a}^{2}{d}^{2}+\sqrt{-{\frac{b}{a}}}\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }{x}^{2}abcd+\sqrt{-{\frac{b}{a}}}\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }{a}^{2}cd \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}}}} \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 (\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac{3}{2}}}{x^{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{{\left (b e x^{2} + a e\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{d x^{4} + c x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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