Optimal. Leaf size=378 \[ \frac{c^{3/2} \left (a+b x^2\right ) (3 b c-4 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a b^2 \sqrt{d} e \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{4 d x \left (a+b x^2\right )}{3 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{d x \left (a+b x^2\right ) (7 b c-8 a d)}{3 b^3 e \left (c+d x^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\sqrt{c} \sqrt{d} \left (a+b x^2\right ) (7 b c-8 a d) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 b^3 e \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{x \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]
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Rubi [A] time = 0.441961, antiderivative size = 378, 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, 467, 528, 531, 418, 492, 411} \[ \frac{c^{3/2} \left (a+b x^2\right ) (3 b c-4 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a b^2 \sqrt{d} e \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{4 d x \left (a+b x^2\right )}{3 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{d x \left (a+b x^2\right ) (7 b c-8 a d)}{3 b^3 e \left (c+d x^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\sqrt{c} \sqrt{d} \left (a+b x^2\right ) (7 b c-8 a d) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 b^3 e \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{x \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 6719
Rule 467
Rule 528
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{x^2}{\left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx &=\frac{\sqrt{a+b x^2} \int \frac{x^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2}} \, dx}{e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=-\frac{x \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{\sqrt{a+b x^2} \int \frac{\sqrt{c+d x^2} \left (c+4 d x^2\right )}{\sqrt{a+b x^2}} \, dx}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=\frac{4 d x \left (a+b x^2\right )}{3 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{x \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{\sqrt{a+b x^2} \int \frac{c (3 b c-4 a d)+d (7 b c-8 a d) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=\frac{4 d x \left (a+b x^2\right )}{3 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{x \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{\left (d (7 b c-8 a d) \sqrt{a+b x^2}\right ) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}+\frac{\left (c (3 b c-4 a d) \sqrt{a+b x^2}\right ) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=\frac{4 d x \left (a+b x^2\right )}{3 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{d (7 b c-8 a d) x \left (a+b x^2\right )}{3 b^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac{x \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{c^{3/2} (3 b c-4 a d) \left (a+b x^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a b^2 \sqrt{d} e \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac{\left (c d (7 b c-8 a d) \sqrt{a+b x^2}\right ) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 b^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=\frac{4 d x \left (a+b x^2\right )}{3 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{d (7 b c-8 a d) x \left (a+b x^2\right )}{3 b^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac{x \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\sqrt{c} \sqrt{d} (7 b c-8 a d) \left (a+b x^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 b^3 e \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}+\frac{c^{3/2} (3 b c-4 a d) \left (a+b x^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a b^2 \sqrt{d} e \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}\\ \end{align*}
Mathematica [C] time = 0.389404, size = 219, normalized size = 0.58 \[ \frac{\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (x \sqrt{\frac{b}{a}} \left (c+d x^2\right ) \left (4 a d-3 b c+b d x^2\right )-4 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (8 a d-7 b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )\right )}{3 a^2 e^2 \left (\frac{b}{a}\right )^{5/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 643, normalized size = 1.7 \begin{align*}{\frac{b{x}^{2}+a}{3\,{b}^{2} \left ( d{x}^{2}+c \right ) ^{2}} \left ( \sqrt{-{\frac{b}{a}}}\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }{x}^{5}b{d}^{2}+3\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{-{\frac{b}{a}}}{x}^{3}a{d}^{2}-3\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{-{\frac{b}{a}}}{x}^{3}bcd+\sqrt{-{\frac{b}{a}}}\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }{x}^{3}a{d}^{2}+\sqrt{-{\frac{b}{a}}}\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }{x}^{3}bcd+4\,\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 ) }acd-4\,\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 ) }b{c}^{2}-8\,\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 ) }acd+7\,\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 ) }b{c}^{2}+3\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{-{\frac{b}{a}}}xacd-3\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{-{\frac{b}{a}}}xb{c}^{2}+\sqrt{-{\frac{b}{a}}}\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }xacd \right ) \left ({\frac{e \left ( b{x}^{2}+a \right ) }{d{x}^{2}+c}} \right ) ^{-{\frac{3}{2}}}{\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{x^{2}}{\left (\frac{{\left (b x^{2} + a\right )} e}{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{{\left (d^{2} x^{6} + 2 \, c d x^{4} + c^{2} x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{b^{2} e^{2} x^{4} + 2 \, a b e^{2} x^{2} + a^{2} e^{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{x^{2}}{\left (\frac{{\left (b x^{2} + a\right )} e}{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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