3.5.46 \(\int x \sqrt {x+x^6} \, dx\)

Optimal. Leaf size=35 \[ \frac {1}{5} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6+x}}\right )+\frac {1}{5} \sqrt {x^6+x} x^2 \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2021, 2029, 206} \begin {gather*} \frac {1}{5} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6+x}}\right )+\frac {1}{5} \sqrt {x^6+x} x^2 \end {gather*}

Antiderivative was successfully verified.

[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 206

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

Rule 2021

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 2029

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*} \int x \sqrt {x+x^6} \, dx &=\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} \operatorname {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} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x+x^6}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 40, normalized size = 1.14 \begin {gather*} \frac {\sqrt {x^6+x} \left (x^{5/2}+\frac {\sinh ^{-1}\left (x^{5/2}\right )}{\sqrt {x^5+1}}\right )}{5 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[x + x^6],x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.25, size = 35, normalized size = 1.00 \begin {gather*} \frac {1}{5} x^2 \sqrt {x+x^6}+\frac {1}{5} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x*Sqrt[x + x^6],x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 35, normalized size = 1.00 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{6} + x} x\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^6 + x)*x, x)

________________________________________________________________________________________

maple [A]  time = 0.28, size = 27, normalized size = 0.77

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{6} + x} x\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^6 + x)*x, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x\,\sqrt {x^6+x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x + x^6)^(1/2),x)

[Out]

int(x*(x + x^6)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________