Optimal. Leaf size=76 \[ -\frac {64 \left (1-x^2\right )^{11/4}}{231 e (e x)^{11/2}}+\frac {16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}-\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {125, 273, 264} \begin {gather*} -\frac {64 \left (1-x^2\right )^{11/4}}{231 e (e x)^{11/2}}+\frac {16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}-\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 125
Rule 264
Rule 273
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx &=\int \frac {1}{(e x)^{13/2} \sqrt [4]{1-x^2}} \, dx\\ &=-\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}}-\frac {8}{3} \int \frac {\left (1-x^2\right )^{3/4}}{(e x)^{13/2}} \, dx\\ &=-\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}}+\frac {16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}+\frac {32}{21} \int \frac {\left (1-x^2\right )^{7/4}}{(e x)^{13/2}} \, dx\\ &=-\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}}+\frac {16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}-\frac {64 \left (1-x^2\right )^{11/4}}{231 e (e x)^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.53 \begin {gather*} -\frac {2 \left (1-x^2\right )^{3/4} \left (32 x^4+24 x^2+21\right ) \sqrt {e x}}{231 e^7 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 14.68, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.10, size = 37, normalized size = 0.49 \begin {gather*} -\frac {2 \, {\left (32 \, x^{4} + 24 \, x^{2} + 21\right )} \sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{231 \, e^{7} x^{6}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (e x\right )^{\frac {13}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.43 \begin {gather*} -\frac {2 \left (x +1\right )^{\frac {3}{4}} \left (32 x^{4}+24 x^{2}+21\right ) \left (-x +1\right )^{\frac {3}{4}} x}{231 \left (e x \right )^{\frac {13}{2}}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (e x\right )^{\frac {13}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 52, normalized size = 0.68 \begin {gather*} -\frac {\sqrt {e\,x}\,\left (\frac {2}{11\,e^7}+\frac {2\,x^2}{77\,e^7}+\frac {16\,x^4}{231\,e^7}-\frac {64\,x^6}{231\,e^7}\right )}{x^6\,{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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