Optimal. Leaf size=372 \[ -\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-5 a d (a d+2 b c)\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{5 c^{3/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{c e \sqrt{e x}}-\frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (3 b^2 c^2-5 a d (a d+2 b c)\right )}{5 c d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-5 a d (a d+2 b c)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{3/2} \sqrt{c+d x^2}}{5 d e^3} \]
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Rubi [A] time = 0.340267, antiderivative size = 372, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {462, 459, 329, 305, 220, 1196} \[ -\frac{2 a^2 \sqrt{c+d x^2}}{c e \sqrt{e x}}-\frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (3 b^2 c^2-5 a d (a d+2 b c)\right )}{5 c d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-5 a d (a d+2 b c)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-5 a d (a d+2 b c)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{3/2} \sqrt{c+d x^2}}{5 d e^3} \]
Antiderivative was successfully verified.
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Rule 462
Rule 459
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{(e x)^{3/2} \sqrt{c+d x^2}} \, dx &=-\frac{2 a^2 \sqrt{c+d x^2}}{c e \sqrt{e x}}+\frac{2 \int \frac{\sqrt{e x} \left (\frac{1}{2} a (2 b c+a d)+\frac{1}{2} b^2 c x^2\right )}{\sqrt{c+d x^2}} \, dx}{c e^2}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{c e \sqrt{e x}}+\frac{2 b^2 (e x)^{3/2} \sqrt{c+d x^2}}{5 d e^3}-\frac{\left (4 \left (\frac{3 b^2 c^2}{4}-\frac{5}{4} a d (2 b c+a d)\right )\right ) \int \frac{\sqrt{e x}}{\sqrt{c+d x^2}} \, dx}{5 c d e^2}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{c e \sqrt{e x}}+\frac{2 b^2 (e x)^{3/2} \sqrt{c+d x^2}}{5 d e^3}-\frac{\left (8 \left (\frac{3 b^2 c^2}{4}-\frac{5}{4} a d (2 b c+a d)\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 c d e^3}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{c e \sqrt{e x}}+\frac{2 b^2 (e x)^{3/2} \sqrt{c+d x^2}}{5 d e^3}-\frac{\left (2 \left (3 b^2 c^2-5 a d (2 b c+a d)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 \sqrt{c} d^{3/2} e^2}+\frac{\left (2 \left (3 b^2 c^2-5 a d (2 b c+a d)\right )\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{d} x^2}{\sqrt{c} e}}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 \sqrt{c} d^{3/2} e^2}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{c e \sqrt{e x}}+\frac{2 b^2 (e x)^{3/2} \sqrt{c+d x^2}}{5 d e^3}-\frac{2 \left (3 b^2 c^2-5 a d (2 b c+a d)\right ) \sqrt{e x} \sqrt{c+d x^2}}{5 c d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 \left (3 b^2 c^2-5 a d (2 b c+a d)\right ) \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{\left (3 b^2 c^2-5 a d (2 b c+a d)\right ) \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.111314, size = 115, normalized size = 0.31 \[ \frac{x \left (2 x^2 \sqrt{\frac{c}{d x^2}+1} \left (5 a^2 d^2+10 a b c d-3 b^2 c^2\right ) \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right )+2 \left (c+d x^2\right ) \left (b^2 c x^2-5 a^2 d\right )\right )}{5 c d (e x)^{3/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 595, normalized size = 1.6 \begin{align*}{\frac{1}{5\,e{d}^{2}c} \left ( 10\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}+20\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d-6\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}-5\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}-10\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d+3\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}+2\,{x}^{4}{b}^{2}c{d}^{2}-10\,{x}^{2}{a}^{2}{d}^{3}+2\,{x}^{2}{b}^{2}{c}^{2}d-10\,{a}^{2}c{d}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \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 (b x^{2} + a\right )}^{2}}{\sqrt{d x^{2} + c} \left (e x\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 (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c} \sqrt{e x}}{d e^{2} x^{4} + c e^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 6.53332, size = 148, normalized size = 0.4 \begin{align*} \frac{a^{2} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{c} e^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} + \frac{a b x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt{c} e^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} + \frac{b^{2} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{c} e^{\frac{3}{2}} \Gamma \left (\frac{11}{4}\right )} \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 (b x^{2} + a\right )}^{2}}{\sqrt{d x^{2} + c} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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