Optimal. Leaf size=107 \[ \frac{2 \sqrt{b} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-3 a d) \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{3 a^{3/2} e^4 \left (a+b x^2\right )^{3/4}}-\frac{2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}} \]
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Rubi [A] time = 0.0972325, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {453, 329, 237, 335, 275, 231} \[ \frac{2 \sqrt{b} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-3 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} e^4 \left (a+b x^2\right )^{3/4}}-\frac{2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 329
Rule 237
Rule 335
Rule 275
Rule 231
Rubi steps
\begin{align*} \int \frac{c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac{2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}-\frac{(2 b c-3 a d) \int \frac{1}{\sqrt{e x} \left (a+b x^2\right )^{3/4}} \, dx}{3 a e^2}\\ &=-\frac{2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}-\frac{(2 (2 b c-3 a d)) \operatorname{Subst}\left (\int \frac{1}{\left (a+\frac{b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt{e x}\right )}{3 a e^3}\\ &=-\frac{2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}-\frac{\left (2 (2 b c-3 a d) \left (1+\frac{a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a e^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt{e x}\right )}{3 a e^3 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}+\frac{\left (2 (2 b c-3 a d) \left (1+\frac{a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a e^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{\sqrt{e x}}\right )}{3 a e^3 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}+\frac{\left ((2 b c-3 a d) \left (1+\frac{a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a e^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{e x}\right )}{3 a e^3 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}+\frac{2 \sqrt{b} (2 b c-3 a d) \left (1+\frac{a}{b x^2}\right )^{3/4} (e x)^{3/2} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} e^4 \left (a+b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0441215, size = 84, normalized size = 0.79 \[ \frac{x \left (2 x^2 \left (\frac{b x^2}{a}+1\right )^{3/4} (3 a d-2 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )-2 c \left (a+b x^2\right )\right )}{3 a (e x)^{5/2} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{5}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \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{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \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 x^{2} + a\right )}^{\frac{1}{4}}{\left (d x^{2} + c\right )} \sqrt{e x}}{b e^{3} x^{5} + a e^{3} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 111.809, size = 82, normalized size = 0.77 \begin{align*} - \frac{d{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{b^{\frac{3}{4}} e^{\frac{5}{2}} x} + \frac{c \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{4} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} e^{\frac{5}{2}} x^{\frac{3}{2}} \Gamma \left (\frac{1}{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{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \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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