3.10.10 \(\int \frac {x^{14}}{(1-x^4)^{3/2}} \, dx\) [910]

Optimal. Leaf size=71 \[ \frac {x^{11}}{2 \sqrt {1-x^4}}+\frac {77}{90} x^3 \sqrt {1-x^4}+\frac {11}{18} x^7 \sqrt {1-x^4}-\frac {77}{30} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {77}{30} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

-77/30*EllipticE(x,I)+77/30*EllipticF(x,I)+1/2*x^11/(-x^4+1)^(1/2)+77/90*x^3*(-x^4+1)^(1/2)+11/18*x^7*(-x^4+1)
^(1/2)

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Rubi [A]
time = 0.02, 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 = {294, 327, 313, 227, 1195, 435} \begin {gather*} \frac {77}{30} F(\text {ArcSin}(x)|-1)-\frac {77}{30} E(\text {ArcSin}(x)|-1)+\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 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^14/(1 - x^4)^(3/2),x]

[Out]

x^11/(2*Sqrt[1 - x^4]) + (77*x^3*Sqrt[1 - x^4])/90 + (11*x^7*Sqrt[1 - x^4])/18 - (77*EllipticE[ArcSin[x], -1])
/30 + (77*EllipticF[ArcSin[x], -1])/30

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^{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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 6.45, size = 56, normalized size = 0.79 \begin {gather*} -\frac {x^3 \left (77+11 x^4+5 x^8-77 \sqrt {1-x^4} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};x^4\right )\right )}{45 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^14/(1 - x^4)^(3/2),x]

[Out]

-1/45*(x^3*(77 + 11*x^4 + 5*x^8 - 77*Sqrt[1 - x^4]*Hypergeometric2F1[3/4, 3/2, 7/4, x^4]))/Sqrt[1 - x^4]

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Maple [A]
time = 0.17, size = 82, normalized size = 1.15

method	result	size
meijerg	\(\frac {x^{15} \hypergeom \left (\left [\frac {3}{2}, \frac {15}{4}\right ], \left [\frac {19}{4}\right ], x^{4}\right )}{15}\)	\(15\)
risch	\(-\frac {x^{3} \left (10 x^{8}+22 x^{4}-77\right )}{90 \sqrt {-x^{4}+1}}+\frac {77 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{30 \sqrt {-x^{4}+1}}\)	\(66\)
default	\(\frac {x^{3}}{2 \sqrt {-x^{4}+1}}+\frac {x^{7} \sqrt {-x^{4}+1}}{9}+\frac {16 x^{3} \sqrt {-x^{4}+1}}{45}+\frac {77 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{30 \sqrt {-x^{4}+1}}\)	\(82\)
elliptic	\(\frac {x^{3}}{2 \sqrt {-x^{4}+1}}+\frac {x^{7} \sqrt {-x^{4}+1}}{9}+\frac {16 x^{3} \sqrt {-x^{4}+1}}{45}+\frac {77 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{30 \sqrt {-x^{4}+1}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14/(-x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*x^3/(-x^4+1)^(1/2)+1/9*x^7*(-x^4+1)^(1/2)+16/45*x^3*(-x^4+1)^(1/2)+77/30*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^
4+1)^(1/2)*(EllipticF(x,I)-EllipticE(x,I))

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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^14/(-x^4+1)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^14/(-x^4 + 1)^(3/2), x)

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Fricas [A]
time = 0.08, size = 37, normalized size = 0.52 \begin {gather*} \frac {{\left (10 \, x^{12} + 22 \, x^{8} + 154 \, x^{4} - 231\right )} \sqrt {-x^{4} + 1}}{90 \, {\left (x^{5} - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(-x^4+1)^(3/2),x, algorithm="fricas")

[Out]

1/90*(10*x^12 + 22*x^8 + 154*x^4 - 231)*sqrt(-x^4 + 1)/(x^5 - x)

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Sympy [A]
time = 0.72, size = 31, normalized size = 0.44 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**14/(-x**4+1)**(3/2),x)

[Out]

x**15*gamma(15/4)*hyper((3/2, 15/4), (19/4,), x**4*exp_polar(2*I*pi))/(4*gamma(19/4))

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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^14/(-x^4+1)^(3/2),x, algorithm="giac")

[Out]

integrate(x^14/(-x^4 + 1)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{14}}{{\left (1-x^4\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14/(1 - x^4)^(3/2),x)

[Out]

int(x^14/(1 - x^4)^(3/2), x)

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