Optimal. Leaf size=207 \[ \frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (6 b c-5 a d)+3 b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{6 c^{9/4} d^{5/4} e^{5/2} \sqrt{c+d x^2}}-\frac{\sqrt{e x} \left (5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{3 c^2 d e^3 \sqrt{c+d x^2}}-\frac{2 a^2}{3 c e (e x)^{3/2} \sqrt{c+d x^2}} \]
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Rubi [A] time = 0.181223, antiderivative size = 207, 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, 457, 329, 220} \[ -\frac{\sqrt{e x} \left (5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{3 c^2 d e^3 \sqrt{c+d x^2}}-\frac{2 a^2}{3 c e (e x)^{3/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (6 b c-5 a d)+3 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{6 c^{9/4} d^{5/4} e^{5/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 462
Rule 457
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{(e x)^{5/2} \left (c+d x^2\right )^{3/2}} \, dx &=-\frac{2 a^2}{3 c e (e x)^{3/2} \sqrt{c+d x^2}}+\frac{2 \int \frac{\frac{1}{2} a (6 b c-5 a d)+\frac{3}{2} b^2 c x^2}{\sqrt{e x} \left (c+d x^2\right )^{3/2}} \, dx}{3 c e^2}\\ &=-\frac{2 a^2}{3 c e (e x)^{3/2} \sqrt{c+d x^2}}-\frac{\left (3 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt{e x}}{3 c^2 d e^3 \sqrt{c+d x^2}}+\frac{\left (3 b^2 c^2+a d (6 b c-5 a d)\right ) \int \frac{1}{\sqrt{e x} \sqrt{c+d x^2}} \, dx}{6 c^2 d e^2}\\ &=-\frac{2 a^2}{3 c e (e x)^{3/2} \sqrt{c+d x^2}}-\frac{\left (3 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt{e x}}{3 c^2 d e^3 \sqrt{c+d x^2}}+\frac{\left (3 b^2 c^2+a d (6 b c-5 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3 c^2 d e^3}\\ &=-\frac{2 a^2}{3 c e (e x)^{3/2} \sqrt{c+d x^2}}-\frac{\left (3 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt{e x}}{3 c^2 d e^3 \sqrt{c+d x^2}}+\frac{\left (3 b^2 c^2+a d (6 b c-5 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 )}{6 c^{9/4} d^{5/4} e^{5/2} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.180499, size = 181, normalized size = 0.87 \[ \frac{x \left (-i x^{5/2} \sqrt{\frac{c}{d x^2}+1} \left (5 a^2 d^2-6 a b c d-3 b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right ),-1\right )-\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (a^2 d \left (2 c+5 d x^2\right )-6 a b c d x^2+3 b^2 c^2 x^2\right )\right )}{3 c^2 d \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} (e x)^{5/2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 353, normalized size = 1.7 \begin{align*} -{\frac{1}{6\,x{c}^{2}{e}^{2}{d}^{2}} \left ( 5\,\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 ) 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-3\,\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 ) x{b}^{2}{c}^{2}+10\,{x}^{2}{a}^{2}{d}^{3}-12\,{x}^{2}abc{d}^{2}+6\,{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{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^{2} e^{3} x^{7} + 2 \, c d e^{3} x^{5} + c^{2} e^{3} x^{3}}, 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{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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