3.10.48 \(\int \frac {x^8}{(1+x^4)^{3/2}} \, dx\) [948]

Optimal. Leaf size=74 \[ -\frac {x^5}{2 \sqrt {1+x^4}}+\frac {5}{6} x \sqrt {1+x^4}-\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}} \]

[Out]

-1/2*x^5/(x^4+1)^(1/2)+5/6*x*(x^4+1)^(1/2)-5/12*(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 = 0.01, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {294, 327, 226} \begin {gather*} -\frac {5 \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 )}{12 \sqrt {x^4+1}}+\frac {5}{6} \sqrt {x^4+1} x-\frac {x^5}{2 \sqrt {x^4+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8/(1 + x^4)^(3/2),x]

[Out]

-1/2*x^5/Sqrt[1 + x^4] + (5*x*Sqrt[1 + x^4])/6 - (5*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x
], 1/2])/(12*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 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]

Rubi steps

\begin {align*} \int \frac {x^8}{\left (1+x^4\right )^{3/2}} \, dx &=-\frac {x^5}{2 \sqrt {1+x^4}}+\frac {5}{2} \int \frac {x^4}{\sqrt {1+x^4}} \, dx\\ &=-\frac {x^5}{2 \sqrt {1+x^4}}+\frac {5}{6} x \sqrt {1+x^4}-\frac {5}{6} \int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=-\frac {x^5}{2 \sqrt {1+x^4}}+\frac {5}{6} x \sqrt {1+x^4}-\frac {5 \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 )}{12 \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 = 3.96, size = 47, normalized size = 0.64 \begin {gather*} \frac {x \left (5+2 x^4-5 \sqrt {1+x^4} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-x^4\right )\right )}{6 \sqrt {1+x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8/(1 + x^4)^(3/2),x]

[Out]

(x*(5 + 2*x^4 - 5*Sqrt[1 + x^4]*Hypergeometric2F1[1/4, 1/2, 5/4, -x^4]))/(6*Sqrt[1 + x^4])

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Maple [C] Result contains complex when optimal does not.
time = 0.16, size = 82, normalized size = 1.11

method	result	size
meijerg	\(\frac {x^{9} \hypergeom \left (\left [\frac {3}{2}, \frac {9}{4}\right ], \left [\frac {13}{4}\right ], -x^{4}\right )}{9}\)	\(17\)
risch	\(\frac {x \left (2 x^{4}+5\right )}{6 \sqrt {x^{4}+1}}-\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{6 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(79\)
default	\(\frac {x}{2 \sqrt {x^{4}+1}}+\frac {x \sqrt {x^{4}+1}}{3}-\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{6 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(82\)
elliptic	\(\frac {x}{2 \sqrt {x^{4}+1}}+\frac {x \sqrt {x^{4}+1}}{3}-\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{6 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*x/(x^4+1)^(1/2)+1/3*x*(x^4+1)^(1/2)-5/6/(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)

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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^8/(x^4+1)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^8/(x^4 + 1)^(3/2), x)

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Fricas [C] Result contains complex when optimal does not.
time = 0.08, size = 50, normalized size = 0.68 \begin {gather*} -\frac {5 \, \sqrt {i} {\left (i \, x^{4} + i\right )} F(\arcsin \left (\frac {\sqrt {i}}{x}\right )\,|\,-1) - {\left (2 \, x^{5} + 5 \, x\right )} \sqrt {x^{4} + 1}}{6 \, {\left (x^{4} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(x^4+1)^(3/2),x, algorithm="fricas")

[Out]

-1/6*(5*sqrt(I)*(I*x^4 + I)*elliptic_f(arcsin(sqrt(I)/x), -1) - (2*x^5 + 5*x)*sqrt(x^4 + 1))/(x^4 + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 0.44, size = 29, normalized size = 0.39 \begin {gather*} \frac {x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(x**4+1)**(3/2),x)

[Out]

x**9*gamma(9/4)*hyper((3/2, 9/4), (13/4,), x**4*exp_polar(I*pi))/(4*gamma(13/4))

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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^8/(x^4+1)^(3/2),x, algorithm="giac")

[Out]

integrate(x^8/(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^8}{{\left (x^4+1\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(x^4 + 1)^(3/2),x)

[Out]

int(x^8/(x^4 + 1)^(3/2), x)

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