3.9.90 \(\int \frac {x^6}{\sqrt {1-x^4}} \, dx\) [890]

Optimal. Leaf size=35 \[ -\frac {1}{5} x^3 \sqrt {1-x^4}+\frac {3}{5} E\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {327, 313, 227, 1195, 435} \begin {gather*} -\frac {3}{5} F(\text {ArcSin}(x)|-1)+\frac {3}{5} E(\text {ArcSin}(x)|-1)-\frac {1}{5} \sqrt {1-x^4} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/Sqrt[1 - x^4],x]

[Out]

-1/5*(x^3*Sqrt[1 - x^4]) + (3*EllipticE[ArcSin[x], -1])/5 - (3*EllipticF[ArcSin[x], -1])/5

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 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^6}{\sqrt {1-x^4}} \, dx &=-\frac {1}{5} x^3 \sqrt {1-x^4}+\frac {3}{5} \int \frac {x^2}{\sqrt {1-x^4}} \, dx\\ &=-\frac {1}{5} x^3 \sqrt {1-x^4}-\frac {3}{5} \int \frac {1}{\sqrt {1-x^4}} \, dx+\frac {3}{5} \int \frac {1+x^2}{\sqrt {1-x^4}} \, dx\\ &=-\frac {1}{5} x^3 \sqrt {1-x^4}-\frac {3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {3}{5} \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\\ &=-\frac {1}{5} x^3 \sqrt {1-x^4}+\frac {3}{5} E\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {3}{5} 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 = 10.03, size = 34, normalized size = 0.97 \begin {gather*} \frac {1}{5} x^3 \left (-\sqrt {1-x^4}+\, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/Sqrt[1 - x^4],x]

[Out]

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

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Maple [A]
time = 0.16, size = 54, normalized size = 1.54

method	result	size
meijerg	\(\frac {x^{7} \hypergeom \left (\left [\frac {1}{2}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], x^{4}\right )}{7}\)	\(15\)
default	\(-\frac {x^{3} \sqrt {-x^{4}+1}}{5}-\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{5 \sqrt {-x^{4}+1}}\)	\(54\)
elliptic	\(-\frac {x^{3} \sqrt {-x^{4}+1}}{5}-\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{5 \sqrt {-x^{4}+1}}\)	\(54\)
risch	\(\frac {x^{3} \left (x^{4}-1\right )}{5 \sqrt {-x^{4}+1}}-\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{5 \sqrt {-x^{4}+1}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(-x^4+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/5*x^3*(-x^4+1)^(1/2)-3/5*(-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^6/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^6/sqrt(-x^4 + 1), x)

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Fricas [A]
time = 0.07, size = 19, normalized size = 0.54 \begin {gather*} -\frac {{\left (x^{4} + 3\right )} \sqrt {-x^{4} + 1}}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

-1/5*(x^4 + 3)*sqrt(-x^4 + 1)/x

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Sympy [A]
time = 0.37, size = 31, normalized size = 0.89 \begin {gather*} \frac {x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(-x**4+1)**(1/2),x)

[Out]

x**7*gamma(7/4)*hyper((1/2, 7/4), (11/4,), x**4*exp_polar(2*I*pi))/(4*gamma(11/4))

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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^6/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

integrate(x^6/sqrt(-x^4 + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^6}{\sqrt {1-x^4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(1 - x^4)^(1/2),x)

[Out]

int(x^6/(1 - x^4)^(1/2), x)

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