3.10.32 \(\int \frac {x^{10}}{\sqrt {1+x^4}} \, dx\) [932]

Optimal. Leaf size=140 \[ -\frac {7}{45} x^3 \sqrt {1+x^4}+\frac {1}{9} x^7 \sqrt {1+x^4}+\frac {7 x \sqrt {1+x^4}}{15 \left (1+x^2\right )}-\frac {7 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{15 \sqrt {1+x^4}}+\frac {7 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{30 \sqrt {1+x^4}} \]

[Out]

-7/45*x^3*(x^4+1)^(1/2)+1/9*x^7*(x^4+1)^(1/2)+7/15*x*(x^4+1)^(1/2)/(x^2+1)-7/15*(x^2+1)*(cos(2*arctan(x))^2)^(
1/2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),1/2*2^(1/2))*((x^4+1)/(x^2+1)^2)^(1/2)/(x^4+1)^(1/2)+7/30*(x^
2+1)*(cos(2*arctan(x))^2)^(1/2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),1/2*2^(1/2))*((x^4+1)/(x^2+1)^2)^(
1/2)/(x^4+1)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {327, 311, 226, 1210} \begin {gather*} \frac {7 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \text {ArcTan}(x)\left |\frac {1}{2}\right .\right )}{30 \sqrt {x^4+1}}-\frac {7 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \text {ArcTan}(x)\left |\frac {1}{2}\right .\right )}{15 \sqrt {x^4+1}}+\frac {1}{9} \sqrt {x^4+1} x^7-\frac {7}{45} \sqrt {x^4+1} x^3+\frac {7 \sqrt {x^4+1} x}{15 \left (x^2+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^10/Sqrt[1 + x^4],x]

[Out]

(-7*x^3*Sqrt[1 + x^4])/45 + (x^7*Sqrt[1 + x^4])/9 + (7*x*Sqrt[1 + x^4])/(15*(1 + x^2)) - (7*(1 + x^2)*Sqrt[(1
+ x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(15*Sqrt[1 + x^4]) + (7*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]
*EllipticF[2*ArcTan[x], 1/2])/(30*Sqrt[1 + x^4])

Rule 226

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

Rule 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, 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] && PosQ[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 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin {align*} \int \frac {x^{10}}{\sqrt {1+x^4}} \, dx &=\frac {1}{9} x^7 \sqrt {1+x^4}-\frac {7}{9} \int \frac {x^6}{\sqrt {1+x^4}} \, dx\\ &=-\frac {7}{45} x^3 \sqrt {1+x^4}+\frac {1}{9} x^7 \sqrt {1+x^4}+\frac {7}{15} \int \frac {x^2}{\sqrt {1+x^4}} \, dx\\ &=-\frac {7}{45} x^3 \sqrt {1+x^4}+\frac {1}{9} x^7 \sqrt {1+x^4}+\frac {7}{15} \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {7}{15} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\\ &=-\frac {7}{45} x^3 \sqrt {1+x^4}+\frac {1}{9} x^7 \sqrt {1+x^4}+\frac {7 x \sqrt {1+x^4}}{15 \left (1+x^2\right )}-\frac {7 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{15 \sqrt {1+x^4}}+\frac {7 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{30 \sqrt {1+x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.02, size = 42, normalized size = 0.30 \begin {gather*} \frac {1}{45} x^3 \left (\sqrt {1+x^4} \left (-7+5 x^4\right )+7 \, _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^10/Sqrt[1 + x^4],x]

[Out]

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

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Maple [C] Result contains complex when optimal does not.
time = 0.17, size = 107, normalized size = 0.76

method	result	size
meijerg	\(\frac {x^{11} \hypergeom \left (\left [\frac {1}{2}, \frac {11}{4}\right ], \left [\frac {15}{4}\right ], -x^{4}\right )}{11}\)	\(17\)
risch	\(\frac {x^{3} \left (5 x^{4}-7\right ) \sqrt {x^{4}+1}}{45}+\frac {7 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (\EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-\EllipticE \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{15 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(102\)
default	\(\frac {x^{7} \sqrt {x^{4}+1}}{9}-\frac {7 x^{3} \sqrt {x^{4}+1}}{45}+\frac {7 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (\EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-\EllipticE \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{15 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(107\)
elliptic	\(\frac {x^{7} \sqrt {x^{4}+1}}{9}-\frac {7 x^{3} \sqrt {x^{4}+1}}{45}+\frac {7 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (\EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-\EllipticE \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{15 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^7*(x^4+1)^(1/2)-7/45*x^3*(x^4+1)^(1/2)+7/15*I/(1/2*2^(1/2)+1/2*I*2^(1/2))*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2
)/(x^4+1)^(1/2)*(EllipticF(x*(1/2*2^(1/2)+1/2*I*2^(1/2)),I)-EllipticE(x*(1/2*2^(1/2)+1/2*I*2^(1/2)),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^10/(x^4+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^10/sqrt(x^4 + 1), x)

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Fricas [C] Result contains complex when optimal does not.
time = 0.08, size = 58, normalized size = 0.41 \begin {gather*} \frac {21 i \, \sqrt {i} x E(\arcsin \left (\frac {\sqrt {i}}{x}\right )\,|\,-1) - 21 i \, \sqrt {i} x F(\arcsin \left (\frac {\sqrt {i}}{x}\right )\,|\,-1) + {\left (5 \, x^{8} - 7 \, x^{4} + 21\right )} \sqrt {x^{4} + 1}}{45 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45*(21*I*sqrt(I)*x*elliptic_e(arcsin(sqrt(I)/x), -1) - 21*I*sqrt(I)*x*elliptic_f(arcsin(sqrt(I)/x), -1) + (5
*x^8 - 7*x^4 + 21)*sqrt(x^4 + 1))/x

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Sympy [C] Result contains complex when optimal does not.
time = 0.46, size = 29, normalized size = 0.21 \begin {gather*} \frac {x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {x^{4} e^{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)**(1/2),x)

[Out]

x**11*gamma(11/4)*hyper((1/2, 11/4), (15/4,), x**4*exp_polar(I*pi))/(4*gamma(15/4))

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

[Out]

integrate(x^10/sqrt(x^4 + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{10}}{\sqrt {x^4+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10/(x^4 + 1)^(1/2),x)

[Out]

int(x^10/(x^4 + 1)^(1/2), x)

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