\(\int x \sqrt {x+x^6} \, dx\) [446]

Optimal result

Integrand size = 11, antiderivative size = 35 \[ \int x \sqrt {x+x^6} \, dx=\frac {1}{5} x^2 \sqrt {x+x^6}+\frac {1}{5} \text {arctanh}\left (\frac {x^3}{\sqrt {x+x^6}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2046, 2054, 212} \[ \int x \sqrt {x+x^6} \, dx=\frac {1}{5} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6+x}}\right )+\frac {1}{5} \sqrt {x^6+x} x^2 \]

[In]

Int[x*Sqrt[x + x^6],x]

[Out]

(x^2*Sqrt[x + x^6])/5 + ArcTanh[x^3/Sqrt[x + x^6]]/5

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2046

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a*x^j + b
*x^n)^p/(c*(m + n*p + 1))), x] + Dist[a*(n - j)*(p/(c^j*(m + n*p + 1))), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2054

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^2 \sqrt {x+x^6}+\frac {1}{2} \int \frac {x^2}{\sqrt {x+x^6}} \, dx \\ & = \frac {1}{5} x^2 \sqrt {x+x^6}+\frac {1}{5} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {x+x^6}}\right ) \\ & = \frac {1}{5} x^2 \sqrt {x+x^6}+\frac {1}{5} \text {arctanh}\left (\frac {x^3}{\sqrt {x+x^6}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.71 \[ \int x \sqrt {x+x^6} \, dx=\frac {1}{5} x^2 \sqrt {x+x^6}+\frac {\sqrt {x+x^6} \log \left (x^{5/2}+\sqrt {1+x^5}\right )}{5 \sqrt {x} \sqrt {1+x^5}} \]

[In]

Integrate[x*Sqrt[x + x^6],x]

[Out]

(x^2*Sqrt[x + x^6])/5 + (Sqrt[x + x^6]*Log[x^(5/2) + Sqrt[1 + x^5]])/(5*Sqrt[x]*Sqrt[1 + x^5])

Maple [A] (verified)

Time = 1.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77

method	result	size
risch	\(\frac {x^{3} \left (x^{5}+1\right )}{5 \sqrt {x \left (x^{5}+1\right )}}+\frac {\operatorname {arcsinh}\left (x^{\frac {5}{2}}\right )}{5}\)	\(27\)
meijerg	\(-\frac {-2 \sqrt {\pi }\, x^{\frac {5}{2}} \sqrt {x^{5}+1}-2 \sqrt {\pi }\, \operatorname {arcsinh}\left (x^{\frac {5}{2}}\right )}{10 \sqrt {\pi }}\)	\(31\)
trager	\(\frac {x^{2} \sqrt {x^{6}+x}}{5}+\frac {\ln \left (2 x^{5}+2 x^{2} \sqrt {x^{6}+x}+1\right )}{10}\)	\(36\)
pseudoelliptic	\(\frac {x^{2} \sqrt {x^{6}+x}}{5}-\frac {\ln \left (\frac {-x^{3}+\sqrt {x^{6}+x}}{x^{3}}\right )}{10}+\frac {\ln \left (\frac {x^{3}+\sqrt {x^{6}+x}}{x^{3}}\right )}{10}\)	\(52\)

[In]

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

[Out]

1/5*x^3*(x^5+1)/(x*(x^5+1))^(1/2)+1/5*arcsinh(x^(5/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int x \sqrt {x+x^6} \, dx=\frac {1}{5} \, \sqrt {x^{6} + x} x^{2} + \frac {1}{10} \, \log \left (2 \, x^{5} + 2 \, \sqrt {x^{6} + x} x^{2} + 1\right ) \]

[In]

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

[Out]

1/5*sqrt(x^6 + x)*x^2 + 1/10*log(2*x^5 + 2*sqrt(x^6 + x)*x^2 + 1)

Sympy [F]

\[ \int x \sqrt {x+x^6} \, dx=\int x \sqrt {x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}\, dx \]

[In]

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

[Out]

Integral(x*sqrt(x*(x + 1)*(x**4 - x**3 + x**2 - x + 1)), x)

Maxima [F]

\[ \int x \sqrt {x+x^6} \, dx=\int { \sqrt {x^{6} + x} x \,d x } \]

[In]

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

[Out]

integrate(sqrt(x^6 + x)*x, x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int x \sqrt {x+x^6} \, dx=\frac {1}{5} \, \sqrt {x^{6} + x} x^{2} + \frac {\log \left (\sqrt {\frac {1}{x^{5}} + 1} + 1\right ) - \log \left ({\left | \sqrt {\frac {1}{x^{5}} + 1} - 1 \right |}\right )}{10 \, \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

1/5*sqrt(x^6 + x)*x^2 + 1/10*(log(sqrt(1/x^5 + 1) + 1) - log(abs(sqrt(1/x^5 + 1) - 1)))/sgn(x)

Mupad [F(-1)]

Timed out. \[ \int x \sqrt {x+x^6} \, dx=\int x\,\sqrt {x^6+x} \,d x \]

[In]

int(x*(x + x^6)^(1/2),x)

[Out]

int(x*(x + x^6)^(1/2), x)