Optimal. Leaf size=71 \[ \frac{77}{30} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )+\frac{x^{11}}{2 \sqrt{1-x^4}}+\frac{11}{18} \sqrt{1-x^4} x^7+\frac{77}{90} \sqrt{1-x^4} x^3-\frac{77}{30} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0268503, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {288, 321, 307, 221, 1181, 424} \[ \frac{x^{11}}{2 \sqrt{1-x^4}}+\frac{11}{18} \sqrt{1-x^4} x^7+\frac{77}{90} \sqrt{1-x^4} x^3+\frac{77}{30} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{77}{30} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 288
Rule 321
Rule 307
Rule 221
Rule 1181
Rule 424
Rubi steps
\begin{align*} \int \frac{x^{14}}{\left (1-x^4\right )^{3/2}} \, dx &=\frac{x^{11}}{2 \sqrt{1-x^4}}-\frac{11}{2} \int \frac{x^{10}}{\sqrt{1-x^4}} \, dx\\ &=\frac{x^{11}}{2 \sqrt{1-x^4}}+\frac{11}{18} x^7 \sqrt{1-x^4}-\frac{77}{18} \int \frac{x^6}{\sqrt{1-x^4}} \, dx\\ &=\frac{x^{11}}{2 \sqrt{1-x^4}}+\frac{77}{90} x^3 \sqrt{1-x^4}+\frac{11}{18} x^7 \sqrt{1-x^4}-\frac{77}{30} \int \frac{x^2}{\sqrt{1-x^4}} \, dx\\ &=\frac{x^{11}}{2 \sqrt{1-x^4}}+\frac{77}{90} x^3 \sqrt{1-x^4}+\frac{11}{18} x^7 \sqrt{1-x^4}+\frac{77}{30} \int \frac{1}{\sqrt{1-x^4}} \, dx-\frac{77}{30} \int \frac{1+x^2}{\sqrt{1-x^4}} \, dx\\ &=\frac{x^{11}}{2 \sqrt{1-x^4}}+\frac{77}{90} x^3 \sqrt{1-x^4}+\frac{11}{18} x^7 \sqrt{1-x^4}+\frac{77}{30} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{77}{30} \int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx\\ &=\frac{x^{11}}{2 \sqrt{1-x^4}}+\frac{77}{90} x^3 \sqrt{1-x^4}+\frac{11}{18} x^7 \sqrt{1-x^4}-\frac{77}{30} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac{77}{30} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.0123059, size = 56, normalized size = 0.79 \[ -\frac{x^3 \left (-77 \sqrt{1-x^4} \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};x^4\right )+5 x^8+11 x^4+77\right )}{45 \sqrt{1-x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 82, normalized size = 1.2 \begin{align*}{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}+{\frac{{x}^{7}}{9}\sqrt{-{x}^{4}+1}}+{\frac{16\,{x}^{3}}{45}\sqrt{-{x}^{4}+1}}+{\frac{77\,{\it EllipticF} \left ( x,i \right ) -77\,{\it EllipticE} \left ( x,i \right ) }{30}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \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{x^{14}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-x^{4} + 1} x^{14}}{x^{8} - 2 \, x^{4} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.87774, size = 31, normalized size = 0.44 \begin{align*} \frac{x^{15} \Gamma \left (\frac{15}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{15}{4} \\ \frac{19}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{19}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{14}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]