3.10.54 \(\int \frac {x^{10}}{(1+x^4)^{3/2}} \, dx\) [954]

Optimal. Leaf size=140 \[ -\frac {x^7}{2 \sqrt {1+x^4}}+\frac {7}{10} x^3 \sqrt {1+x^4}-\frac {21 x \sqrt {1+x^4}}{10 \left (1+x^2\right )}+\frac {21 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{10 \sqrt {1+x^4}}-\frac {21 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{20 \sqrt {1+x^4}} \]

[Out]

-1/2*x^7/(x^4+1)^(1/2)+7/10*x^3*(x^4+1)^(1/2)-21/10*x*(x^4+1)^(1/2)/(x^2+1)+21/10*(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)-21/20*
(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 = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {294, 327, 311, 226, 1210} \begin {gather*} -\frac {21 \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 )}{20 \sqrt {x^4+1}}+\frac {21 \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 )}{10 \sqrt {x^4+1}}-\frac {x^7}{2 \sqrt {x^4+1}}+\frac {7}{10} \sqrt {x^4+1} x^3-\frac {21 \sqrt {x^4+1} x}{10 \left (x^2+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^10/(1 + x^4)^(3/2),x]

[Out]

-1/2*x^7/Sqrt[1 + x^4] + (7*x^3*Sqrt[1 + x^4])/10 - (21*x*Sqrt[1 + x^4])/(10*(1 + x^2)) + (21*(1 + x^2)*Sqrt[(
1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(10*Sqrt[1 + x^4]) - (21*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)
^2]*EllipticF[2*ArcTan[x], 1/2])/(20*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 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 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}}{\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 x \sqrt {1+x^4}}{10 \left (1+x^2\right )}+\frac {21 \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 )}{10 \sqrt {1+x^4}}-\frac {21 \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 )}{20 \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 = 3.84, size = 47, normalized size = 0.34 \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]

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

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Maple [C] Result contains complex when optimal does not.
time = 0.15, size = 107, normalized size = 0.76

method	result	size
meijerg	\(\frac {x^{11} \hypergeom \left (\left [\frac {3}{2}, \frac {11}{4}\right ], \left [\frac {15}{4}\right ], -x^{4}\right )}{11}\)	\(17\)
risch	\(\frac {x^{3} \left (2 x^{4}+7\right )}{10 \sqrt {x^{4}+1}}-\frac {21 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (\EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-\EllipticE \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{10 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(102\)
default	\(\frac {x^{3}}{2 \sqrt {x^{4}+1}}+\frac {x^{3} \sqrt {x^{4}+1}}{5}-\frac {21 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (\EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-\EllipticE \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{10 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(107\)
elliptic	\(\frac {x^{3}}{2 \sqrt {x^{4}+1}}+\frac {x^{3} \sqrt {x^{4}+1}}{5}-\frac {21 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (\EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-\EllipticE \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{10 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^10/(x^4 + 1)^(3/2), x)

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Fricas [C] Result contains complex when optimal does not.
time = 0.08, size = 79, normalized size = 0.56 \begin {gather*} -\frac {21 \, \sqrt {i} {\left (i \, x^{5} + i \, x\right )} E(\arcsin \left (\frac {\sqrt {i}}{x}\right )\,|\,-1) + 21 \, \sqrt {i} {\left (-i \, x^{5} - i \, x\right )} F(\arcsin \left (\frac {\sqrt {i}}{x}\right )\,|\,-1) - {\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*(21*sqrt(I)*(I*x^5 + I*x)*elliptic_e(arcsin(sqrt(I)/x), -1) + 21*sqrt(I)*(-I*x^5 - I*x)*elliptic_f(arcsi
n(sqrt(I)/x), -1) - (2*x^8 - 14*x^4 - 21)*sqrt(x^4 + 1))/(x^5 + x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.50, size = 29, normalized size = 0.21 \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^{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(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)^(3/2),x, algorithm="giac")

[Out]

integrate(x^10/(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^{10}}{{\left (x^4+1\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10/(x^4 + 1)^(3/2),x)

[Out]

int(x^10/(x^4 + 1)^(3/2), x)

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