Optimal. Leaf size=51 \[ \frac{8 \left (1-x^2\right )^{7/4}}{21 e (e x)^{7/2}}-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{7/2}} \]
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Rubi [A] time = 0.014609, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {125, 273, 264} \[ \frac{8 \left (1-x^2\right )^{7/4}}{21 e (e x)^{7/2}}-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 125
Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{1-x} (e x)^{9/2} \sqrt [4]{1+x}} \, dx &=\int \frac{1}{(e x)^{9/2} \sqrt [4]{1-x^2}} \, dx\\ &=-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{7/2}}-\frac{4}{3} \int \frac{\left (1-x^2\right )^{3/4}}{(e x)^{9/2}} \, dx\\ &=-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{7/2}}+\frac{8 \left (1-x^2\right )^{7/4}}{21 e (e x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0140084, size = 35, normalized size = 0.69 \[ -\frac{2 \left (1-x^2\right )^{3/4} \left (4 x^2+3\right ) \sqrt{e x}}{21 e^5 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 28, normalized size = 0.6 \begin{align*} -{\frac{2\,x \left ( 4\,{x}^{2}+3 \right ) }{21} \left ( 1+x \right ) ^{{\frac{3}{4}}} \left ( 1-x \right ) ^{{\frac{3}{4}}} \left ( ex \right ) ^{-{\frac{9}{2}}}} \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{9}{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 [A] time = 1.52466, size = 92, normalized size = 1.8 \begin{align*} -\frac{2 \, \sqrt{e x}{\left (4 \, x^{2} + 3\right )}{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{21 \, e^{5} x^{4}} \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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