Optimal. Leaf size=393 \[ \frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{2 c^{7/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{\sqrt{e x} \sqrt{c+d x^2} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{c^2 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{(e x)^{3/2} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{c^2 d e^3 \sqrt{c+d x^2}}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{c^{7/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{2 a^2}{c e \sqrt{e x} \sqrt{c+d x^2}} \]
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Rubi [A] time = 0.361189, antiderivative size = 393, 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, 457, 329, 305, 220, 1196} \[ \frac{\sqrt{e x} \sqrt{c+d x^2} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{c^2 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{(e x)^{3/2} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{c^2 d e^3 \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 c^{7/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{c^{7/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{2 a^2}{c e \sqrt{e x} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 462
Rule 457
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} \left (c+d x^2\right )^{3/2}} \, dx &=-\frac{2 a^2}{c e \sqrt{e x} \sqrt{c+d x^2}}+\frac{2 \int \frac{\sqrt{e x} \left (\frac{1}{2} a (2 b c-3 a d)+\frac{1}{2} b^2 c x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{c e^2}\\ &=-\frac{2 a^2}{c e \sqrt{e x} \sqrt{c+d x^2}}-\frac{\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt{c+d x^2}}-\frac{\left (2 a b-\frac{3 b^2 c}{d}-\frac{3 a^2 d}{c}\right ) \int \frac{\sqrt{e x}}{\sqrt{c+d x^2}} \, dx}{2 c e^2}\\ &=-\frac{2 a^2}{c e \sqrt{e x} \sqrt{c+d x^2}}-\frac{\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt{c+d x^2}}-\frac{\left (2 a b-\frac{3 b^2 c}{d}-\frac{3 a^2 d}{c}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{c e^3}\\ &=-\frac{2 a^2}{c e \sqrt{e x} \sqrt{c+d x^2}}-\frac{\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt{c+d x^2}}+\frac{\left (3 b^2 c^2-2 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 )}{c^{3/2} d^{3/2} e^2}-\frac{\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\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 )}{c^{3/2} d^{3/2} e^2}\\ &=-\frac{2 a^2}{c e \sqrt{e x} \sqrt{c+d x^2}}-\frac{\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt{c+d x^2}}+\frac{\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt{e x} \sqrt{c+d x^2}}{c^2 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{\left (3 b^2 c^2-2 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}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{c^{7/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{\left (3 b^2 c^2-2 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 )}{2 c^{7/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.107433, size = 126, normalized size = 0.32 \[ \frac{x \left (x^2 \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{d x^2}{c}\right )-3 a^2 d \left (2 c+3 d x^2\right )+6 a b c d x^2-3 b^2 c^2 x^2\right )}{3 c^2 d (e x)^{3/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 594, normalized size = 1.5 \begin{align*}{\frac{1}{2\,{d}^{2}e{c}^{2}} \left ( 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 ){a}^{2}c{d}^{2}-4\,\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}-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 ){a}^{2}c{d}^{2}+2\,\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}-6\,{x}^{2}{a}^{2}{d}^{3}+4\,{x}^{2}abc{d}^{2}-2\,{x}^{2}{b}^{2}{c}^{2}d-4\,{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}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}} \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^{2} e^{2} x^{6} + 2 \, c d e^{2} x^{4} + c^{2} e^{2} x^{2}}, 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}}{\left (e x\right )^{\frac{3}{2}} \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}} \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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