Optimal. Leaf size=312 \[ \frac{x \left (a+b x^2\right ) (b c-2 a d)}{3 b^2 \left (c+d x^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\sqrt{c} \left (a+b x^2\right ) (b c-2 a d) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 b^2 \sqrt{d} \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{c^{3/2} \left (a+b x^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 b \sqrt{d} \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{x \left (a+b x^2\right )}{3 b \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]
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Rubi [A] time = 0.338106, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {6719, 478, 531, 418, 492, 411} \[ \frac{x \left (a+b x^2\right ) (b c-2 a d)}{3 b^2 \left (c+d x^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\sqrt{c} \left (a+b x^2\right ) (b c-2 a d) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 b^2 \sqrt{d} \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{c^{3/2} \left (a+b x^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 b \sqrt{d} \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{x \left (a+b x^2\right )}{3 b \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 6719
Rule 478
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \, dx &=\frac{\sqrt{a+b x^2} \int \frac{x^2 \sqrt{c+d x^2}}{\sqrt{a+b x^2}} \, dx}{\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=\frac{x \left (a+b x^2\right )}{3 b \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\sqrt{a+b x^2} \int \frac{a c+(-b c+2 a d) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 b \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=\frac{x \left (a+b x^2\right )}{3 b \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\left (a c \sqrt{a+b x^2}\right ) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 b \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}-\frac{\left ((-b c+2 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 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=\frac{x \left (a+b x^2\right )}{3 b \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{(b c-2 a d) x \left (a+b x^2\right )}{3 b^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac{c^{3/2} \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 b \sqrt{d} \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 (-b c+2 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^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=\frac{x \left (a+b x^2\right )}{3 b \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{(b c-2 a d) x \left (a+b x^2\right )}{3 b^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac{\sqrt{c} (b c-2 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^2 \sqrt{d} \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} \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 b \sqrt{d} \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 = 0.272699, size = 212, normalized size = 0.68 \[ \frac{d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (2 a d-b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{3 b d \sqrt{\frac{b}{a}} \left (c+d x^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 358, normalized size = 1.2 \begin{align*}{\frac{b{x}^{2}+a}{3\,bd} \left ( \sqrt{-{\frac{b}{a}}}{x}^{5}b{d}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{3}a{d}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{3}bcd+ac\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 ) d-\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}-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 ) acd+\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}+\sqrt{-{\frac{b}{a}}}xacd \right ){\frac{1}{\sqrt{{\frac{e \left ( b{x}^{2}+a \right ) }{d{x}^{2}+c}}}}}{\frac{1}{\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \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{x^{2}}{\sqrt{\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}\,{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 x^{4} + c x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{b e x^{2} + a e}, 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}}{\sqrt{\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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