\(\int \frac {x^{23}}{\sqrt {2+x^6}} \, dx\) [1383]

Optimal result

Integrand size = 13, antiderivative size = 53 \[ \int \frac {x^{23}}{\sqrt {2+x^6}} \, dx=-\frac {8}{3} \sqrt {2+x^6}+\frac {4}{3} \left (2+x^6\right )^{3/2}-\frac {2}{5} \left (2+x^6\right )^{5/2}+\frac {1}{21} \left (2+x^6\right )^{7/2} \]

[Out]

4/3*(x^6+2)^(3/2)-2/5*(x^6+2)^(5/2)+1/21*(x^6+2)^(7/2)-8/3*(x^6+2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 53, 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} \[ \int \frac {x^{23}}{\sqrt {2+x^6}} \, dx=\frac {1}{21} \left (x^6+2\right )^{7/2}-\frac {2}{5} \left (x^6+2\right )^{5/2}+\frac {4}{3} \left (x^6+2\right )^{3/2}-\frac {8 \sqrt {x^6+2}}{3} \]

[In]

Int[x^23/Sqrt[2 + x^6],x]

[Out]

(-8*Sqrt[2 + x^6])/3 + (4*(2 + x^6)^(3/2))/3 - (2*(2 + x^6)^(5/2))/5 + (2 + x^6)^(7/2)/21

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}{6} \text {Subst}\left (\int \frac {x^3}{\sqrt {2+x}} \, dx,x,x^6\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \left (-\frac {8}{\sqrt {2+x}}+12 \sqrt {2+x}-6 (2+x)^{3/2}+(2+x)^{5/2}\right ) \, dx,x,x^6\right ) \\ & = -\frac {8}{3} \sqrt {2+x^6}+\frac {4}{3} \left (2+x^6\right )^{3/2}-\frac {2}{5} \left (2+x^6\right )^{5/2}+\frac {1}{21} \left (2+x^6\right )^{7/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.57 \[ \int \frac {x^{23}}{\sqrt {2+x^6}} \, dx=\frac {1}{105} \sqrt {2+x^6} \left (-128+32 x^6-12 x^{12}+5 x^{18}\right ) \]

[In]

Integrate[x^23/Sqrt[2 + x^6],x]

[Out]

(Sqrt[2 + x^6]*(-128 + 32*x^6 - 12*x^12 + 5*x^18))/105

Maple [A] (verified)

Time = 4.47 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.49

method	result	size
trager	\(\sqrt {x^{6}+2}\, \left (-\frac {128}{105}+\frac {32}{105} x^{6}-\frac {4}{35} x^{12}+\frac {1}{21} x^{18}\right )\)	\(26\)
gosper	\(\frac {\sqrt {x^{6}+2}\, \left (5 x^{18}-12 x^{12}+32 x^{6}-128\right )}{105}\)	\(27\)
risch	\(\frac {\sqrt {x^{6}+2}\, \left (5 x^{18}-12 x^{12}+32 x^{6}-128\right )}{105}\)	\(27\)
pseudoelliptic	\(\frac {\sqrt {x^{6}+2}\, \left (5 x^{18}-12 x^{12}+32 x^{6}-128\right )}{105}\)	\(27\)
meijerg	\(\frac {4 \sqrt {2}\, \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-5 x^{18}+12 x^{12}-32 x^{6}+128\right ) \sqrt {1+\frac {x^{6}}{2}}}{140}\right )}{3 \sqrt {\pi }}\)	\(46\)

[In]

int(x^23/(x^6+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(x^6+2)^(1/2)*(-128/105+32/105*x^6-4/35*x^12+1/21*x^18)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.49 \[ \int \frac {x^{23}}{\sqrt {2+x^6}} \, dx=\frac {1}{105} \, {\left (5 \, x^{18} - 12 \, x^{12} + 32 \, x^{6} - 128\right )} \sqrt {x^{6} + 2} \]

[In]

integrate(x^23/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

1/105*(5*x^18 - 12*x^12 + 32*x^6 - 128)*sqrt(x^6 + 2)

Sympy [A] (verification not implemented)

Time = 0.91 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02 \[ \int \frac {x^{23}}{\sqrt {2+x^6}} \, dx=\frac {x^{18} \sqrt {x^{6} + 2}}{21} - \frac {4 x^{12} \sqrt {x^{6} + 2}}{35} + \frac {32 x^{6} \sqrt {x^{6} + 2}}{105} - \frac {128 \sqrt {x^{6} + 2}}{105} \]

[In]

integrate(x**23/(x**6+2)**(1/2),x)

[Out]

x**18*sqrt(x**6 + 2)/21 - 4*x**12*sqrt(x**6 + 2)/35 + 32*x**6*sqrt(x**6 + 2)/105 - 128*sqrt(x**6 + 2)/105

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {x^{23}}{\sqrt {2+x^6}} \, dx=\frac {1}{21} \, {\left (x^{6} + 2\right )}^{\frac {7}{2}} - \frac {2}{5} \, {\left (x^{6} + 2\right )}^{\frac {5}{2}} + \frac {4}{3} \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - \frac {8}{3} \, \sqrt {x^{6} + 2} \]

[In]

integrate(x^23/(x^6+2)^(1/2),x, algorithm="maxima")

[Out]

1/21*(x^6 + 2)^(7/2) - 2/5*(x^6 + 2)^(5/2) + 4/3*(x^6 + 2)^(3/2) - 8/3*sqrt(x^6 + 2)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {x^{23}}{\sqrt {2+x^6}} \, dx=\frac {1}{21} \, {\left (x^{6} + 2\right )}^{\frac {7}{2}} - \frac {2}{5} \, {\left (x^{6} + 2\right )}^{\frac {5}{2}} + \frac {4}{3} \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - \frac {8}{3} \, \sqrt {x^{6} + 2} \]

[In]

integrate(x^23/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

1/21*(x^6 + 2)^(7/2) - 2/5*(x^6 + 2)^(5/2) + 4/3*(x^6 + 2)^(3/2) - 8/3*sqrt(x^6 + 2)

Mupad [B] (verification not implemented)

Time = 5.52 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.47 \[ \int \frac {x^{23}}{\sqrt {2+x^6}} \, dx=\sqrt {x^6+2}\,\left (\frac {x^{18}}{21}-\frac {4\,x^{12}}{35}+\frac {32\,x^6}{105}-\frac {128}{105}\right ) \]

[In]

int(x^23/(x^6 + 2)^(1/2),x)

[Out]

(x^6 + 2)^(1/2)*((32*x^6)/105 - (4*x^12)/35 + x^18/21 - 128/105)