Optimal. Leaf size=244 \[ \frac{2 c^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{231 d^{9/4} \sqrt{e} \sqrt{c+d x^2}}+\frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right )}{231 d^2 e}-\frac{2 b \sqrt{e x} \left (c+d x^2\right )^{3/2} (5 b c-22 a d)}{77 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3} \]
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Rubi [A] time = 0.203183, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {464, 459, 279, 329, 220} \[ \frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right )}{231 d^2 e}+\frac{2 c^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 d^{9/4} \sqrt{e} \sqrt{c+d x^2}}-\frac{2 b \sqrt{e x} \left (c+d x^2\right )^{3/2} (5 b c-22 a d)}{77 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3} \]
Antiderivative was successfully verified.
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Rule 464
Rule 459
Rule 279
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{\sqrt{e x}} \, dx &=\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}+\frac{2 \int \frac{\sqrt{c+d x^2} \left (\frac{11 a^2 d}{2}-\frac{1}{2} b (5 b c-22 a d) x^2\right )}{\sqrt{e x}} \, dx}{11 d}\\ &=-\frac{2 b (5 b c-22 a d) \sqrt{e x} \left (c+d x^2\right )^{3/2}}{77 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}-\frac{1}{77} \left (-77 a^2-\frac{b c (5 b c-22 a d)}{d^2}\right ) \int \frac{\sqrt{c+d x^2}}{\sqrt{e x}} \, dx\\ &=\frac{2 \left (77 a^2+\frac{b c (5 b c-22 a d)}{d^2}\right ) \sqrt{e x} \sqrt{c+d x^2}}{231 e}-\frac{2 b (5 b c-22 a d) \sqrt{e x} \left (c+d x^2\right )^{3/2}}{77 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}+\frac{1}{231} \left (2 c \left (77 a^2+\frac{b c (5 b c-22 a d)}{d^2}\right )\right ) \int \frac{1}{\sqrt{e x} \sqrt{c+d x^2}} \, dx\\ &=\frac{2 \left (77 a^2+\frac{b c (5 b c-22 a d)}{d^2}\right ) \sqrt{e x} \sqrt{c+d x^2}}{231 e}-\frac{2 b (5 b c-22 a d) \sqrt{e x} \left (c+d x^2\right )^{3/2}}{77 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}+\frac{\left (4 c \left (77 a^2+\frac{b c (5 b c-22 a d)}{d^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{231 e}\\ &=\frac{2 \left (77 a^2+\frac{b c (5 b c-22 a d)}{d^2}\right ) \sqrt{e x} \sqrt{c+d x^2}}{231 e}-\frac{2 b (5 b c-22 a d) \sqrt{e x} \left (c+d x^2\right )^{3/2}}{77 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}+\frac{2 c^{3/4} \left (77 a^2+\frac{b c (5 b c-22 a d)}{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 )}{231 \sqrt [4]{d} \sqrt{e} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.196546, size = 189, normalized size = 0.77 \[ \frac{\sqrt{x} \left (\frac{2 \sqrt{x} \left (c+d x^2\right ) \left (77 a^2 d^2+22 a b d \left (2 c+3 d x^2\right )+b^2 \left (-10 c^2+6 c d x^2+21 d^2 x^4\right )\right )}{d^2}+\frac{4 i c x \sqrt{\frac{c}{d x^2}+1} \left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right ),-1\right )}{d^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{231 \sqrt{e x} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 401, normalized size = 1.6 \begin{align*}{\frac{2}{231\,{d}^{3}} \left ( 21\,{x}^{7}{b}^{2}{d}^{4}+77\,\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}{a}^{2}c{d}^{2}-22\,\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}ab{c}^{2}d+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 ) \sqrt{-cd}{b}^{2}{c}^{3}+66\,{x}^{5}ab{d}^{4}+27\,{x}^{5}{b}^{2}c{d}^{3}+77\,{x}^{3}{a}^{2}{d}^{4}+110\,{x}^{3}abc{d}^{3}-4\,{x}^{3}{b}^{2}{c}^{2}{d}^{2}+77\,x{a}^{2}c{d}^{3}+44\,xab{c}^{2}{d}^{2}-10\,x{b}^{2}{c}^{3}d \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}}{\sqrt{e x}}\,{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}}{e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 7.34902, size = 150, normalized size = 0.61 \begin{align*} \frac{a^{2} \sqrt{c} \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 \sqrt{e} \Gamma \left (\frac{5}{4}\right )} + \frac{a b \sqrt{c} 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 )}}{\sqrt{e} \Gamma \left (\frac{9}{4}\right )} + \frac{b^{2} \sqrt{c} 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 \sqrt{e} \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} \sqrt{d x^{2} + c}}{\sqrt{e x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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