Optimal. Leaf size=184 \[ -\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a^2 d^2-6 a b c d+b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{3 c^{5/4} d^{5/4} e^{5/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{3 c e (e x)^{3/2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d e^3} \]
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Rubi [A] time = 0.138926, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {462, 459, 329, 220} \[ -\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a^2 d^2-6 a b c d+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3 c^{5/4} d^{5/4} e^{5/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{3 c e (e x)^{3/2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d e^3} \]
Antiderivative was successfully verified.
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Rule 462
Rule 459
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{(e x)^{5/2} \sqrt{c+d x^2}} \, dx &=-\frac{2 a^2 \sqrt{c+d x^2}}{3 c e (e x)^{3/2}}+\frac{2 \int \frac{\frac{1}{2} a (6 b c-a d)+\frac{3}{2} b^2 c x^2}{\sqrt{e x} \sqrt{c+d x^2}} \, dx}{3 c e^2}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{3 c e (e x)^{3/2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d e^3}-\frac{\left (b^2 c^2-6 a b c d+a^2 d^2\right ) \int \frac{1}{\sqrt{e x} \sqrt{c+d x^2}} \, dx}{3 c d e^2}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{3 c e (e x)^{3/2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d e^3}-\frac{\left (2 \left (b^2 c^2-6 a b c d+a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3 c d e^3}\\ &=-\frac{2 a^2 \sqrt{c+d x^2}}{3 c e (e x)^{3/2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d e^3}-\frac{\left (b^2 c^2-6 a b c d+a^2 d^2\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 )}{3 c^{5/4} d^{5/4} e^{5/2} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.167646, size = 165, normalized size = 0.9 \[ \frac{x \left (2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (c+d x^2\right ) \left (b^2 c x^2-a^2 d\right )-2 i x^{5/2} \sqrt{\frac{c}{d x^2}+1} \left (a^2 d^2-6 a b c d+b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right ),-1\right )\right )}{3 c d \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} (e x)^{5/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 352, normalized size = 1.9 \begin{align*} -{\frac{1}{3\,cx{e}^{2}{d}^{2}} \left ( \sqrt{-cd}\sqrt{{ \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{ \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{ \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ) x{a}^{2}{d}^{2}-6\,\sqrt{-cd}\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 ) xabcd+\sqrt{-cd}\sqrt{{ \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{ \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{ \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ) x{b}^{2}{c}^{2}-2\,{x}^{4}{b}^{2}c{d}^{2}+2\,{x}^{2}{a}^{2}{d}^{3}-2\,{x}^{2}{b}^{2}{c}^{2}d+2\,{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{5}{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^{3} x^{5} + c e^{3} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 34.4227, size = 148, normalized size = 0.8 \begin{align*} \frac{a^{2} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{c} e^{\frac{5}{2}} x^{\frac{3}{2}} \Gamma \left (\frac{1}{4}\right )} + \frac{a b \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt{c} e^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right )} + \frac{b^{2} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{c} e^{\frac{5}{2}} \Gamma \left (\frac{9}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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