Optimal. Leaf size=95 \[ -\frac{4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}-\frac{8 \sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{15 e^6 \sqrt [4]{1-x^2}}-\frac{2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}} \]
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Rubi [A] time = 0.0324452, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {125, 325, 317, 335, 228} \[ -\frac{4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}-\frac{8 \sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{15 e^6 \sqrt [4]{1-x^2}}-\frac{2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 125
Rule 325
Rule 317
Rule 335
Rule 228
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{1-x} (e x)^{11/2} \sqrt [4]{1+x}} \, dx &=\int \frac{1}{(e x)^{11/2} \sqrt [4]{1-x^2}} \, dx\\ &=-\frac{2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}}+\frac{2 \int \frac{1}{(e x)^{7/2} \sqrt [4]{1-x^2}} \, dx}{3 e^2}\\ &=-\frac{2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}}-\frac{4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}+\frac{4 \int \frac{1}{(e x)^{3/2} \sqrt [4]{1-x^2}} \, dx}{15 e^4}\\ &=-\frac{2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}}-\frac{4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}+\frac{\left (4 \sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x}\right ) \int \frac{1}{\sqrt [4]{1-\frac{1}{x^2}} x^2} \, dx}{15 e^6 \sqrt [4]{1-x^2}}\\ &=-\frac{2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}}-\frac{4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}-\frac{\left (4 \sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-x^2}} \, dx,x,\frac{1}{x}\right )}{15 e^6 \sqrt [4]{1-x^2}}\\ &=-\frac{2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}}-\frac{4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}-\frac{8 \sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{15 e^6 \sqrt [4]{1-x^2}}\\ \end{align*}
Mathematica [C] time = 0.0087668, size = 25, normalized size = 0.26 \[ -\frac{2 x \, _2F_1\left (-\frac{9}{4},\frac{1}{4};-\frac{5}{4};x^2\right )}{9 (e x)^{11/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [4]{1-x}}} \left ( ex \right ) ^{-{\frac{11}{2}}}{\frac{1}{\sqrt [4]{1+x}}}}\, 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 (e x\right )^{\frac{11}{2}}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}}\,{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{\sqrt{e x}{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{e^{6} x^{8} - e^{6} x^{6}}, 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(-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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