Optimal. Leaf size=296 \[ -\frac{5 c^{3/4} e^{7/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{462 d^{17/4} \sqrt{c+d x^2}}+\frac{5 e^3 \sqrt{e x} \sqrt{c+d x^2} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right )}{231 d^4}-\frac{e (e x)^{5/2} \sqrt{c+d x^2} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right )}{77 c d^3}+\frac{(e x)^{9/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{9/2} \sqrt{c+d x^2}}{11 d^2 e} \]
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Rubi [A] time = 0.248713, antiderivative size = 296, 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 = {463, 459, 321, 329, 220} \[ \frac{5 e^3 \sqrt{e x} \sqrt{c+d x^2} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right )}{231 d^4}-\frac{5 c^{3/4} e^{7/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{462 d^{17/4} \sqrt{c+d x^2}}-\frac{e (e x)^{5/2} \sqrt{c+d x^2} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right )}{77 c d^3}+\frac{(e x)^{9/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{9/2} \sqrt{c+d x^2}}{11 d^2 e} \]
Antiderivative was successfully verified.
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Rule 463
Rule 459
Rule 321
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx &=\frac{(b c-a d)^2 (e x)^{9/2}}{c d^2 e \sqrt{c+d x^2}}-\frac{\int \frac{(e x)^{7/2} \left (\frac{1}{2} \left (-2 a^2 d^2+9 (b c-a d)^2\right )-b^2 c d x^2\right )}{\sqrt{c+d x^2}} \, dx}{c d^2}\\ &=\frac{(b c-a d)^2 (e x)^{9/2}}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{9/2} \sqrt{c+d x^2}}{11 d^2 e}-\frac{\left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) \int \frac{(e x)^{7/2}}{\sqrt{c+d x^2}} \, dx}{22 c d^2}\\ &=\frac{(b c-a d)^2 (e x)^{9/2}}{c d^2 e \sqrt{c+d x^2}}-\frac{\left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e (e x)^{5/2} \sqrt{c+d x^2}}{77 c d^3}+\frac{2 b^2 (e x)^{9/2} \sqrt{c+d x^2}}{11 d^2 e}+\frac{\left (5 \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^2\right ) \int \frac{(e x)^{3/2}}{\sqrt{c+d x^2}} \, dx}{154 d^3}\\ &=\frac{(b c-a d)^2 (e x)^{9/2}}{c d^2 e \sqrt{c+d x^2}}+\frac{5 \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^3 \sqrt{e x} \sqrt{c+d x^2}}{231 d^4}-\frac{\left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e (e x)^{5/2} \sqrt{c+d x^2}}{77 c d^3}+\frac{2 b^2 (e x)^{9/2} \sqrt{c+d x^2}}{11 d^2 e}-\frac{\left (5 c \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^4\right ) \int \frac{1}{\sqrt{e x} \sqrt{c+d x^2}} \, dx}{462 d^4}\\ &=\frac{(b c-a d)^2 (e x)^{9/2}}{c d^2 e \sqrt{c+d x^2}}+\frac{5 \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^3 \sqrt{e x} \sqrt{c+d x^2}}{231 d^4}-\frac{\left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e (e x)^{5/2} \sqrt{c+d x^2}}{77 c d^3}+\frac{2 b^2 (e x)^{9/2} \sqrt{c+d x^2}}{11 d^2 e}-\frac{\left (5 c \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{231 d^4}\\ &=\frac{(b c-a d)^2 (e x)^{9/2}}{c d^2 e \sqrt{c+d x^2}}+\frac{5 \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^3 \sqrt{e x} \sqrt{c+d x^2}}{231 d^4}-\frac{\left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e (e x)^{5/2} \sqrt{c+d x^2}}{77 c d^3}+\frac{2 b^2 (e x)^{9/2} \sqrt{c+d x^2}}{11 d^2 e}-\frac{5 c^{3/4} \left (117 b^2 c^2-198 a b c d+77 a^2 d^2\right ) e^{7/2} \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 )}{462 d^{17/4} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.243443, size = 226, normalized size = 0.76 \[ \frac{e^3 \sqrt{e x} \left (\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (77 a^2 d^2 \left (5 c+2 d x^2\right )+66 a b d \left (-15 c^2-6 c d x^2+2 d^2 x^4\right )+3 b^2 \left (78 c^2 d x^2+195 c^3-26 c d^2 x^4+14 d^3 x^6\right )\right )-5 i c \sqrt{x} \sqrt{\frac{c}{d x^2}+1} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right ),-1\right )\right )}{231 d^4 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 407, normalized size = 1.4 \begin{align*} -{\frac{{e}^{3}}{462\,x{d}^{5}}\sqrt{ex} \left ( -84\,{x}^{7}{b}^{2}{d}^{4}+385\,\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}-990\,\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+585\,\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}-264\,{x}^{5}ab{d}^{4}+156\,{x}^{5}{b}^{2}c{d}^{3}-308\,{x}^{3}{a}^{2}{d}^{4}+792\,{x}^{3}abc{d}^{3}-468\,{x}^{3}{b}^{2}{c}^{2}{d}^{2}-770\,x{a}^{2}c{d}^{3}+1980\,xab{c}^{2}{d}^{2}-1170\,x{b}^{2}{c}^{3}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \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 (e x\right )^{\frac{7}{2}}}{{\left (d x^{2} + c\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} e^{3} x^{7} + 2 \, a b e^{3} x^{5} + a^{2} e^{3} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{e x}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (e x\right )^{\frac{7}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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