\(\int \frac {x^{15}}{\sqrt {1+x^8}} \, dx\) [1515]

Optimal result

Integrand size = 13, antiderivative size = 27 \[ \int \frac {x^{15}}{\sqrt {1+x^8}} \, dx=-\frac {1}{4} \sqrt {1+x^8}+\frac {1}{12} \left (1+x^8\right )^{3/2} \]

[Out]

1/12*(x^8+1)^(3/2)-1/4*(x^8+1)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, 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} \[ \int \frac {x^{15}}{\sqrt {1+x^8}} \, dx=\frac {1}{12} \left (x^8+1\right )^{3/2}-\frac {\sqrt {x^8+1}}{4} \]

[In]

Int[x^15/Sqrt[1 + x^8],x]

[Out]

-1/4*Sqrt[1 + x^8] + (1 + x^8)^(3/2)/12

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}{8} \text {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,x^8\right ) \\ & = \frac {1}{8} \text {Subst}\left (\int \left (-\frac {1}{\sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,x^8\right ) \\ & = -\frac {1}{4} \sqrt {1+x^8}+\frac {1}{12} \left (1+x^8\right )^{3/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {x^{15}}{\sqrt {1+x^8}} \, dx=\frac {1}{12} \left (-2+x^8\right ) \sqrt {1+x^8} \]

[In]

Integrate[x^15/Sqrt[1 + x^8],x]

[Out]

((-2 + x^8)*Sqrt[1 + x^8])/12

Maple [A] (verified)

Time = 3.33 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56

method	result	size
gosper	\(\frac {\sqrt {x^{8}+1}\, \left (x^{8}-2\right )}{12}\)	\(15\)
risch	\(\frac {\sqrt {x^{8}+1}\, \left (x^{8}-2\right )}{12}\)	\(15\)
pseudoelliptic	\(\frac {\sqrt {x^{8}+1}\, \left (x^{8}-2\right )}{12}\)	\(15\)
trager	\(\sqrt {x^{8}+1}\, \left (\frac {x^{8}}{12}-\frac {1}{6}\right )\)	\(16\)
meijerg	\(\frac {\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-4 x^{8}+8\right ) \sqrt {x^{8}+1}}{6}}{8 \sqrt {\pi }}\)	\(31\)

[In]

int(x^15/(x^8+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/12*(x^8+1)^(1/2)*(x^8-2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {x^{15}}{\sqrt {1+x^8}} \, dx=\frac {1}{12} \, \sqrt {x^{8} + 1} {\left (x^{8} - 2\right )} \]

[In]

integrate(x^15/(x^8+1)^(1/2),x, algorithm="fricas")

[Out]

1/12*sqrt(x^8 + 1)*(x^8 - 2)

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {x^{15}}{\sqrt {1+x^8}} \, dx=\frac {x^{8} \sqrt {x^{8} + 1}}{12} - \frac {\sqrt {x^{8} + 1}}{6} \]

[In]

integrate(x**15/(x**8+1)**(1/2),x)

[Out]

x**8*sqrt(x**8 + 1)/12 - sqrt(x**8 + 1)/6

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {x^{15}}{\sqrt {1+x^8}} \, dx=\frac {1}{12} \, {\left (x^{8} + 1\right )}^{\frac {3}{2}} - \frac {1}{4} \, \sqrt {x^{8} + 1} \]

[In]

integrate(x^15/(x^8+1)^(1/2),x, algorithm="maxima")

[Out]

1/12*(x^8 + 1)^(3/2) - 1/4*sqrt(x^8 + 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {x^{15}}{\sqrt {1+x^8}} \, dx=\frac {1}{12} \, {\left (x^{8} + 1\right )}^{\frac {3}{2}} - \frac {1}{4} \, \sqrt {x^{8} + 1} \]

[In]

integrate(x^15/(x^8+1)^(1/2),x, algorithm="giac")

[Out]

1/12*(x^8 + 1)^(3/2) - 1/4*sqrt(x^8 + 1)

Mupad [B] (verification not implemented)

Time = 6.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {x^{15}}{\sqrt {1+x^8}} \, dx=\frac {\sqrt {x^8+1}\,\left (x^8-2\right )}{12} \]

[In]

int(x^15/(x^8 + 1)^(1/2),x)

[Out]

((x^8 + 1)^(1/2)*(x^8 - 2))/12