Optimal. Leaf size=71 \[ \frac {x^{11}}{2 \sqrt {1-x^4}}+\frac {11}{18} \sqrt {1-x^4} x^7+\frac {77}{90} \sqrt {1-x^4} x^3+\frac {77}{30} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {77}{30} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 221
Rule 288
Rule 307
Rule 321
Rule 424
Rule 1181
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 = 0.02, 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]
________________________________________________________________________________________
fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{4} + 1} x^{14}}{x^{8} - 2 \, x^{4} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{14}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 82, normalized size = 1.15 \[ \frac {\sqrt {-x^{4}+1}\, x^{7}}{9}+\frac {x^{3}}{2 \sqrt {-x^{4}+1}}+\frac {16 \sqrt {-x^{4}+1}\, x^{3}}{45}+\frac {77 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (x , i\right )+\EllipticF \left (x , i\right )\right )}{30 \sqrt {-x^{4}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{14}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^{14}}{{\left (1-x^4\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.66, size = 31, normalized size = 0.44 \[ \frac {x^{15} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {19}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________