Optimal. Leaf size=288 \[ -\frac{4 c^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{231 d^{5/4} e^{5/2} \sqrt{c+d x^2}}-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}-\frac{2 \sqrt{e x} \left (c+d x^2\right )^{3/2} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right )}{231 c d e^3}-\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right )}{231 d e^3}+\frac{2 b^2 \sqrt{e x} \left (c+d x^2\right )^{5/2}}{11 d e^3} \]
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Rubi [A] time = 0.241653, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {462, 459, 279, 329, 220} \[ -\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}-\frac{4 c^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 d^{5/4} e^{5/2} \sqrt{c+d x^2}}-\frac{2 \sqrt{e x} \left (c+d x^2\right )^{3/2} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right )}{231 c d e^3}-\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right )}{231 d e^3}+\frac{2 b^2 \sqrt{e x} \left (c+d x^2\right )^{5/2}}{11 d e^3} \]
Antiderivative was successfully verified.
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Rule 462
Rule 459
Rule 279
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx &=-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac{2 \int \frac{\left (\frac{1}{2} a (6 b c+7 a d)+\frac{3}{2} b^2 c x^2\right ) \left (c+d x^2\right )^{3/2}}{\sqrt{e x}} \, dx}{3 c e^2}\\ &=-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac{2 b^2 \sqrt{e x} \left (c+d x^2\right )^{5/2}}{11 d e^3}-\frac{\left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \int \frac{\left (c+d x^2\right )^{3/2}}{\sqrt{e x}} \, dx}{33 c d e^2}\\ &=-\frac{2 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt{e x} \left (c+d x^2\right )^{3/2}}{231 c d e^3}-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac{2 b^2 \sqrt{e x} \left (c+d x^2\right )^{5/2}}{11 d e^3}-\frac{\left (2 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right )\right ) \int \frac{\sqrt{c+d x^2}}{\sqrt{e x}} \, dx}{77 d e^2}\\ &=-\frac{4 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt{e x} \sqrt{c+d x^2}}{231 d e^3}-\frac{2 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt{e x} \left (c+d x^2\right )^{3/2}}{231 c d e^3}-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac{2 b^2 \sqrt{e x} \left (c+d x^2\right )^{5/2}}{11 d e^3}-\frac{\left (4 c \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right )\right ) \int \frac{1}{\sqrt{e x} \sqrt{c+d x^2}} \, dx}{231 d e^2}\\ &=-\frac{4 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt{e x} \sqrt{c+d x^2}}{231 d e^3}-\frac{2 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt{e x} \left (c+d x^2\right )^{3/2}}{231 c d e^3}-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac{2 b^2 \sqrt{e x} \left (c+d x^2\right )^{5/2}}{11 d e^3}-\frac{\left (8 c \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{231 d e^3}\\ &=-\frac{4 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt{e x} \sqrt{c+d x^2}}{231 d e^3}-\frac{2 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt{e x} \left (c+d x^2\right )^{3/2}}{231 c d e^3}-\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac{2 b^2 \sqrt{e x} \left (c+d x^2\right )^{5/2}}{11 d e^3}-\frac{4 c^{3/4} \left (3 b^2 c^2-11 a d (6 b c+7 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 )}{231 d^{5/4} e^{5/2} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.228935, size = 202, normalized size = 0.7 \[ \frac{x^{5/2} \left (\frac{2 \left (c+d x^2\right ) \left (77 a^2 d \left (d x^2-c\right )+66 a b d x^2 \left (3 c+d x^2\right )+3 b^2 x^2 \left (4 c^2+13 c d x^2+7 d^2 x^4\right )\right )}{d x^{3/2}}+\frac{8 i c x \sqrt{\frac{c}{d x^2}+1} \left (77 a^2 d^2+66 a b c d-3 b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right ),-1\right )}{d \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{231 (e x)^{5/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 415, normalized size = 1.4 \begin{align*}{\frac{2}{231\,x{d}^{2}{e}^{2}} \left ( 21\,{x}^{8}{b}^{2}{d}^{4}+154\,\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 ) \sqrt{-cd}x{a}^{2}c{d}^{2}+132\,\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 ) \sqrt{-cd}xab{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 EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}x{b}^{2}{c}^{3}+66\,{x}^{6}ab{d}^{4}+60\,{x}^{6}{b}^{2}c{d}^{3}+77\,{x}^{4}{a}^{2}{d}^{4}+264\,{x}^{4}abc{d}^{3}+51\,{x}^{4}{b}^{2}{c}^{2}{d}^{2}+198\,{x}^{2}ab{c}^{2}{d}^{2}+12\,{x}^{2}{b}^{2}{c}^{3}d-77\,{a}^{2}{c}^{2}{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}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}{\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} d x^{6} +{\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c +{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{e x}}{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 = 55.4485, size = 309, normalized size = 1.07 \begin{align*} \frac{a^{2} c^{\frac{3}{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 e^{\frac{5}{2}} x^{\frac{3}{2}} \Gamma \left (\frac{1}{4}\right )} + \frac{a^{2} \sqrt{c} d \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right )} + \frac{a b c^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right )} + \frac{a b \sqrt{c} d 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 )}}{e^{\frac{5}{2}} \Gamma \left (\frac{9}{4}\right )} + \frac{b^{2} c^{\frac{3}{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 e^{\frac{5}{2}} \Gamma \left (\frac{9}{4}\right )} + \frac{b^{2} \sqrt{c} d x^{\frac{9}{2}} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac{5}{2}} \Gamma \left (\frac{13}{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}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}{\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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