\(\int \frac {x^{11}}{(1+x^4)^{3/2}} \, dx\) [937]

Optimal result

Integrand size = 13, antiderivative size = 38 \[ \int \frac {x^{11}}{\left (1+x^4\right )^{3/2}} \, dx=-\frac {1}{2 \sqrt {1+x^4}}-\sqrt {1+x^4}+\frac {1}{6} \left (1+x^4\right )^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {x^{11}}{\left (1+x^4\right )^{3/2}} \, dx=\frac {1}{6} \left (x^4+1\right )^{3/2}-\sqrt {x^4+1}-\frac {1}{2 \sqrt {x^4+1}} \]

[In]

Int[x^11/(1 + x^4)^(3/2),x]

[Out]

-1/2*1/Sqrt[1 + x^4] - Sqrt[1 + x^4] + (1 + x^4)^(3/2)/6

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {x^2}{(1+x)^{3/2}} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (\frac {1}{(1+x)^{3/2}}-\frac {2}{\sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,x^4\right ) \\ & = -\frac {1}{2 \sqrt {1+x^4}}-\sqrt {1+x^4}+\frac {1}{6} \left (1+x^4\right )^{3/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61 \[ \int \frac {x^{11}}{\left (1+x^4\right )^{3/2}} \, dx=\frac {-8-4 x^4+x^8}{6 \sqrt {1+x^4}} \]

[In]

Integrate[x^11/(1 + x^4)^(3/2),x]

[Out]

(-8 - 4*x^4 + x^8)/(6*Sqrt[1 + x^4])

Maple [A] (verified)

Time = 4.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.53

method	result	size
gosper	\(\frac {x^{8}-4 x^{4}-8}{6 \sqrt {x^{4}+1}}\)	\(20\)
default	\(\frac {x^{8}-4 x^{4}-8}{6 \sqrt {x^{4}+1}}\)	\(20\)
trager	\(\frac {x^{8}-4 x^{4}-8}{6 \sqrt {x^{4}+1}}\)	\(20\)
risch	\(\frac {x^{8}-4 x^{4}-8}{6 \sqrt {x^{4}+1}}\)	\(20\)
elliptic	\(\frac {x^{8}-4 x^{4}-8}{6 \sqrt {x^{4}+1}}\)	\(20\)
pseudoelliptic	\(\frac {x^{8}-4 x^{4}-8}{6 \sqrt {x^{4}+1}}\)	\(20\)
meijerg	\(\frac {\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 x^{8}+8 x^{4}+16\right )}{6 \sqrt {x^{4}+1}}}{2 \sqrt {\pi }}\)	\(36\)

[In]

int(x^11/(x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/6*(x^8-4*x^4-8)/(x^4+1)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.50 \[ \int \frac {x^{11}}{\left (1+x^4\right )^{3/2}} \, dx=\frac {x^{8} - 4 \, x^{4} - 8}{6 \, \sqrt {x^{4} + 1}} \]

[In]

integrate(x^11/(x^4+1)^(3/2),x, algorithm="fricas")

[Out]

1/6*(x^8 - 4*x^4 - 8)/sqrt(x^4 + 1)

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.03 \[ \int \frac {x^{11}}{\left (1+x^4\right )^{3/2}} \, dx=\frac {x^{8}}{6 \sqrt {x^{4} + 1}} - \frac {2 x^{4}}{3 \sqrt {x^{4} + 1}} - \frac {4}{3 \sqrt {x^{4} + 1}} \]

[In]

integrate(x**11/(x**4+1)**(3/2),x)

[Out]

x**8/(6*sqrt(x**4 + 1)) - 2*x**4/(3*sqrt(x**4 + 1)) - 4/(3*sqrt(x**4 + 1))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {x^{11}}{\left (1+x^4\right )^{3/2}} \, dx=\frac {1}{6} \, {\left (x^{4} + 1\right )}^{\frac {3}{2}} - \sqrt {x^{4} + 1} - \frac {1}{2 \, \sqrt {x^{4} + 1}} \]

[In]

integrate(x^11/(x^4+1)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {x^{11}}{\left (1+x^4\right )^{3/2}} \, dx=\frac {1}{6} \, {\left (x^{4} + 1\right )}^{\frac {3}{2}} - \sqrt {x^{4} + 1} - \frac {1}{2 \, \sqrt {x^{4} + 1}} \]

[In]

integrate(x^11/(x^4+1)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.90 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66 \[ \int \frac {x^{11}}{\left (1+x^4\right )^{3/2}} \, dx=-\frac {6\,x^4-{\left (x^4+1\right )}^2+9}{6\,\sqrt {x^4+1}} \]

[In]

int(x^11/(x^4 + 1)^(3/2),x)

[Out]

-(6*x^4 - (x^4 + 1)^2 + 9)/(6*(x^4 + 1)^(1/2))