3.10.11 \(\int \frac {x^{10}}{(1-x^4)^{3/2}} \, dx\) [911]

Optimal. Leaf size=53 \[ \frac {x^7}{2 \sqrt {1-x^4}}+\frac {7}{10} x^3 \sqrt {1-x^4}-\frac {21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

-21/10*EllipticE(x,I)+21/10*EllipticF(x,I)+1/2*x^7/(-x^4+1)^(1/2)+7/10*x^3*(-x^4+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {294, 327, 313, 227, 1195, 435} \begin {gather*} \frac {21}{10} F(\text {ArcSin}(x)|-1)-\frac {21}{10} E(\text {ArcSin}(x)|-1)+\frac {x^7}{2 \sqrt {1-x^4}}+\frac {7}{10} \sqrt {1-x^4} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^10/(1 - x^4)^(3/2),x]

[Out]

x^7/(2*Sqrt[1 - x^4]) + (7*x^3*Sqrt[1 - x^4])/10 - (21*EllipticE[ArcSin[x], -1])/10 + (21*EllipticF[ArcSin[x],
-1])/10

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 1195

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c], Int
[(d + e*x^2)/(Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c, 0]

Rubi steps

\begin {align*} \int \frac {x^{10}}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {x^7}{2 \sqrt {1-x^4}}-\frac {7}{2} \int \frac {x^6}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^7}{2 \sqrt {1-x^4}}+\frac {7}{10} x^3 \sqrt {1-x^4}-\frac {21}{10} \int \frac {x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^7}{2 \sqrt {1-x^4}}+\frac {7}{10} x^3 \sqrt {1-x^4}+\frac {21}{10} \int \frac {1}{\sqrt {1-x^4}} \, dx-\frac {21}{10} \int \frac {1+x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {x^7}{2 \sqrt {1-x^4}}+\frac {7}{10} x^3 \sqrt {1-x^4}+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {21}{10} \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\\ &=\frac {x^7}{2 \sqrt {1-x^4}}+\frac {7}{10} x^3 \sqrt {1-x^4}-\frac {21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 4.36, size = 49, normalized size = 0.92 \begin {gather*} -\frac {x^3 \left (7+x^4-7 \sqrt {1-x^4} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};x^4\right )\right )}{5 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^10/(1 - x^4)^(3/2),x]

[Out]

-1/5*(x^3*(7 + x^4 - 7*Sqrt[1 - x^4]*Hypergeometric2F1[3/4, 3/2, 7/4, x^4]))/Sqrt[1 - x^4]

________________________________________________________________________________________

Maple [A]
time = 0.16, size = 68, normalized size = 1.28

method	result	size
meijerg	\(\frac {x^{11} \hypergeom \left (\left [\frac {3}{2}, \frac {11}{4}\right ], \left [\frac {15}{4}\right ], x^{4}\right )}{11}\)	\(15\)
risch	\(-\frac {x^{3} \left (2 x^{4}-7\right )}{10 \sqrt {-x^{4}+1}}+\frac {21 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{10 \sqrt {-x^{4}+1}}\)	\(61\)
default	\(\frac {x^{3}}{2 \sqrt {-x^{4}+1}}+\frac {x^{3} \sqrt {-x^{4}+1}}{5}+\frac {21 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{10 \sqrt {-x^{4}+1}}\)	\(68\)
elliptic	\(\frac {x^{3}}{2 \sqrt {-x^{4}+1}}+\frac {x^{3} \sqrt {-x^{4}+1}}{5}+\frac {21 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{10 \sqrt {-x^{4}+1}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10/(-x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*x^3/(-x^4+1)^(1/2)+1/5*x^3*(-x^4+1)^(1/2)+21/10*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)*(EllipticF(x,I
)-EllipticE(x,I))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(-x^4+1)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^10/(-x^4 + 1)^(3/2), x)

________________________________________________________________________________________

Fricas [A]
time = 0.08, size = 32, normalized size = 0.60 \begin {gather*} \frac {{\left (2 \, x^{8} + 14 \, x^{4} - 21\right )} \sqrt {-x^{4} + 1}}{10 \, {\left (x^{5} - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(-x^4+1)^(3/2),x, algorithm="fricas")

[Out]

1/10*(2*x^8 + 14*x^4 - 21)*sqrt(-x^4 + 1)/(x^5 - x)

________________________________________________________________________________________

Sympy [A]
time = 0.51, size = 31, normalized size = 0.58 \begin {gather*} \frac {x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10/(-x**4+1)**(3/2),x)

[Out]

x**11*gamma(11/4)*hyper((3/2, 11/4), (15/4,), x**4*exp_polar(2*I*pi))/(4*gamma(15/4))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(-x^4+1)^(3/2),x, algorithm="giac")

[Out]

integrate(x^10/(-x^4 + 1)^(3/2), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{10}}{{\left (1-x^4\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10/(1 - x^4)^(3/2),x)

[Out]

int(x^10/(1 - x^4)^(3/2), x)

________________________________________________________________________________________