Optimal. Leaf size=193 \[ -\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-3 a^2 d^2-6 a b c d+5 b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{6 c^{5/4} d^{9/4} \sqrt{e} \sqrt{c+d x^2}}+\frac{\sqrt{e x} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d^2 e} \]
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Rubi [A] time = 0.155666, antiderivative size = 193, 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 = {463, 459, 329, 220} \[ -\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (-3 a^2 d^2-6 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{6 c^{5/4} d^{9/4} \sqrt{e} \sqrt{c+d x^2}}+\frac{\sqrt{e x} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d^2 e} \]
Antiderivative was successfully verified.
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Rule 463
Rule 459
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{\sqrt{e x} \left (c+d x^2\right )^{3/2}} \, dx &=\frac{(b c-a d)^2 \sqrt{e x}}{c d^2 e \sqrt{c+d x^2}}-\frac{\int \frac{\frac{1}{2} \left (-2 a^2 d^2+(b c-a d)^2\right )-b^2 c d x^2}{\sqrt{e x} \sqrt{c+d x^2}} \, dx}{c d^2}\\ &=\frac{(b c-a d)^2 \sqrt{e x}}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d^2 e}-\frac{\left (5 b^2 c^2-6 a b c d-3 a^2 d^2\right ) \int \frac{1}{\sqrt{e x} \sqrt{c+d x^2}} \, dx}{6 c d^2}\\ &=\frac{(b c-a d)^2 \sqrt{e x}}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d^2 e}-\frac{\left (5 b^2 c^2-6 a b c d-3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3 c d^2 e}\\ &=\frac{(b c-a d)^2 \sqrt{e x}}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d^2 e}-\frac{\left (5 b^2 c^2-6 a b c d-3 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 )}{6 c^{5/4} d^{9/4} \sqrt{e} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.15125, size = 174, normalized size = 0.9 \[ \frac{i x^{3/2} \sqrt{\frac{c}{d x^2}+1} \left (3 a^2 d^2+6 a b c d-5 b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right ),-1\right )+x \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (3 a^2 d^2-6 a b c d+b^2 c \left (5 c+2 d x^2\right )\right )}{3 c d^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \sqrt{e x} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 341, normalized size = 1.8 \begin{align*}{\frac{1}{6\,c{d}^{3}} \left ( 3\,\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{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-cd}{a}^{2}{d}^{2}+6\,\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{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-cd}abcd-5\,\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{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-cd}{b}^{2}{c}^{2}+4\,{x}^{3}{b}^{2}c{d}^{2}+6\,x{a}^{2}{d}^{3}-12\,xabc{d}^{2}+10\,x{b}^{2}{c}^{2}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}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}} \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}}{d^{2} e x^{5} + 2 \, c d e x^{3} + c^{2} e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2}\right )^{2}}{\sqrt{e x} \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \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}} \sqrt{e x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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