Optimal. Leaf size=70 \[ -\frac {4 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{5 e^4 \sqrt [4]{1-x^2}}-\frac {2 \left (1-x^2\right )^{3/4}}{5 e (e x)^{5/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ -\frac {4 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{5 e^4 \sqrt [4]{1-x^2}}-\frac {2 \left (1-x^2\right )^{3/4}}{5 e (e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 125
Rule 228
Rule 317
Rule 325
Rule 335
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{1-x} (e x)^{7/2} \sqrt [4]{1+x}} \, dx &=\int \frac {1}{(e x)^{7/2} \sqrt [4]{1-x^2}} \, dx\\ &=-\frac {2 \left (1-x^2\right )^{3/4}}{5 e (e x)^{5/2}}+\frac {2 \int \frac {1}{(e x)^{3/2} \sqrt [4]{1-x^2}} \, dx}{5 e^2}\\ &=-\frac {2 \left (1-x^2\right )^{3/4}}{5 e (e x)^{5/2}}+\frac {\left (2 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x}\right ) \int \frac {1}{\sqrt [4]{1-\frac {1}{x^2}} x^2} \, dx}{5 e^4 \sqrt [4]{1-x^2}}\\ &=-\frac {2 \left (1-x^2\right )^{3/4}}{5 e (e x)^{5/2}}-\frac {\left (2 \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 )}{5 e^4 \sqrt [4]{1-x^2}}\\ &=-\frac {2 \left (1-x^2\right )^{3/4}}{5 e (e x)^{5/2}}-\frac {4 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{5 e^4 \sqrt [4]{1-x^2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 25, normalized size = 0.36 \[ -\frac {2 x \, _2F_1\left (-\frac {5}{4},\frac {1}{4};-\frac {1}{4};x^2\right )}{5 (e x)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{e^{4} x^{6} - e^{4} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e x\right )^{\frac {7}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-x +1\right )^{\frac {1}{4}} \left (e x \right )^{\frac {7}{2}} \left (x +1\right )^{\frac {1}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e x\right )^{\frac {7}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (e\,x\right )}^{7/2}\,{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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