Optimal. Leaf size=200 \[ -\frac{2 \left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt [4]{-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{\sqrt [4]{a+c x^2} \sqrt{d+e x} \left (\sqrt{-a} e+\sqrt{c} d\right )} \]
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Rubi [A] time = 0.0944947, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {727} \[ -\frac{2 \left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt [4]{-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{\sqrt [4]{a+c x^2} \sqrt{d+e x} \left (\sqrt{-a} e+\sqrt{c} d\right )} \]
Antiderivative was successfully verified.
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Rule 727
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{3/2} \sqrt [4]{a+c x^2}} \, dx &=-\frac{2 \left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt [4]{-\frac{\left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\sqrt{-a}+\sqrt{c} x\right )}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{\left (\sqrt{c} d+\sqrt{-a} e\right ) \sqrt{d+e x} \sqrt [4]{a+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.517099, size = 108, normalized size = 0.54 \[ \frac{\left (a+c x^2\right )^{3/4} (c d x-a e) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{(a e-c d x)^2}{a c (d+e x)^2}\right )}{a c (d+e x)^{5/2} \left (\frac{\left (a+c x^2\right ) \left (a e^2+c d^2\right )}{a c (d+e x)^2}\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.668, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{c{x}^{2}+a}}}}\, 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{1}{{\left (c x^{2} + a\right )}^{\frac{1}{4}}{\left (e x + d\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 (c x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x + d}}{c e^{2} x^{4} + 2 \, c d e x^{3} + 2 \, a d e x + a d^{2} +{\left (c d^{2} + a e^{2}\right )} 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{1}{\sqrt [4]{a + c x^{2}} \left (d + e x\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{1}{{\left (c x^{2} + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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