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

Optimal. Leaf size=39 \[ \frac {1}{5} x^2 \sqrt {x^6-x}-\frac {1}{5} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-x}}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2021, 2029, 206} \begin {gather*} \frac {1}{5} x^2 \sqrt {x^6-x}-\frac {1}{5} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-x}}\right ) \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*}

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Mathematica [A]  time = 0.03, size = 44, normalized size = 1.13 \begin {gather*} \frac {\sqrt {x \left (x^5-1\right )} \left (x^{5/2}+\frac {\sin ^{-1}\left (x^{5/2}\right )}{\sqrt {1-x^5}}\right )}{5 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[-x + x^6],x]

[Out]

(Sqrt[x*(-1 + x^5)]*(x^(5/2) + ArcSin[x^(5/2)]/Sqrt[1 - x^5]))/(5*Sqrt[x])

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IntegrateAlgebraic [A]  time = 0.26, size = 39, 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

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fricas [A]  time = 0.50, size = 39, 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)

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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)

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maple [A]  time = 0.32, size = 40, normalized size = 1.03

method	result	size
trager	\(\frac {x^{2} \sqrt {x^{6}-x}}{5}-\frac {\ln \left (2 x^{5}+2 x^{2} \sqrt {x^{6}-x}-1\right )}{10}\)	\(40\)
risch	\(\frac {x^{3} \left (x^{5}-1\right )}{5 \sqrt {x \left (x^{5}-1\right )}}-\frac {\sqrt {-\mathrm {signum}\left (x^{5}-1\right )}\, \arcsin \left (x^{\frac {5}{2}}\right )}{5 \sqrt {\mathrm {signum}\left (x^{5}-1\right )}}\)	\(45\)
meijerg	\(\frac {i \sqrt {\mathrm {signum}\left (x^{5}-1\right )}\, \left (-2 i \sqrt {\pi }\, x^{\frac {5}{2}} \sqrt {-x^{5}+1}-2 i \sqrt {\pi }\, \arcsin \left (x^{\frac {5}{2}}\right )\right )}{10 \sqrt {-\mathrm {signum}\left (x^{5}-1\right )}\, \sqrt {\pi }}\)	\(54\)

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^2*(x^6-x)^(1/2)-1/10*ln(2*x^5+2*x^2*(x^6-x)^(1/2)-1)

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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)

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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^6 - x)^(1/2),x)

[Out]

int(x*(x^6 - x)^(1/2), x)

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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)

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