Optimal. Leaf size=76 \[ -\frac{64 \left (1-x^2\right )^{11/4}}{231 e (e x)^{11/2}}+\frac{16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0211333, antiderivative size = 76, normalized size of antiderivative = 1., 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} \[ -\frac{64 \left (1-x^2\right )^{11/4}}{231 e (e x)^{11/2}}+\frac{16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 125
Rule 273
Rule 264
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*}
Mathematica [A] time = 0.0162375, size = 40, normalized size = 0.53 \[ -\frac{2 \left (1-x^2\right )^{3/4} \left (32 x^4+24 x^2+21\right ) \sqrt{e x}}{231 e^7 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 33, normalized size = 0.4 \begin{align*} -{\frac{2\,x \left ( 32\,{x}^{4}+24\,{x}^{2}+21 \right ) }{231} \left ( 1+x \right ) ^{{\frac{3}{4}}} \left ( 1-x \right ) ^{{\frac{3}{4}}} \left ( ex \right ) ^{-{\frac{13}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.522, size = 108, normalized size = 1.42 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]