3.7.35 \(\int x^6 \sqrt {-x+x^4} \, dx\)

Optimal. Leaf size=50 \[ \frac {1}{72} \sqrt {x^4-x} \left (8 x^7-2 x^4-3 x\right )-\frac {1}{24} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 73, normalized size of antiderivative = 1.46, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2021, 2024, 2029, 206} \begin {gather*} -\frac {1}{36} \sqrt {x^4-x} x^4-\frac {1}{24} \sqrt {x^4-x} x+\frac {1}{9} \sqrt {x^4-x} x^7-\frac {1}{24} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6*Sqrt[-x + x^4],x]

[Out]

-1/24*(x*Sqrt[-x + x^4]) - (x^4*Sqrt[-x + x^4])/36 + (x^7*Sqrt[-x + x^4])/9 - ArcTanh[x^2/Sqrt[-x + x^4]]/24

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 2024

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n +
1)*(a*x^j + b*x^n)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^(n - j)*(m + j*p - n + j + 1))/(b*(m + n*p + 1)
), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j
, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 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^6 \sqrt {-x+x^4} \, dx &=\frac {1}{9} x^7 \sqrt {-x+x^4}-\frac {1}{6} \int \frac {x^7}{\sqrt {-x+x^4}} \, dx\\ &=-\frac {1}{36} x^4 \sqrt {-x+x^4}+\frac {1}{9} x^7 \sqrt {-x+x^4}-\frac {1}{8} \int \frac {x^4}{\sqrt {-x+x^4}} \, dx\\ &=-\frac {1}{24} x \sqrt {-x+x^4}-\frac {1}{36} x^4 \sqrt {-x+x^4}+\frac {1}{9} x^7 \sqrt {-x+x^4}-\frac {1}{16} \int \frac {x}{\sqrt {-x+x^4}} \, dx\\ &=-\frac {1}{24} x \sqrt {-x+x^4}-\frac {1}{36} x^4 \sqrt {-x+x^4}+\frac {1}{9} x^7 \sqrt {-x+x^4}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {-x+x^4}}\right )\\ &=-\frac {1}{24} x \sqrt {-x+x^4}-\frac {1}{36} x^4 \sqrt {-x+x^4}+\frac {1}{9} x^7 \sqrt {-x+x^4}-\frac {1}{24} \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 58, normalized size = 1.16 \begin {gather*} \frac {\sqrt {x \left (x^3-1\right )} \left (\frac {3 \sin ^{-1}\left (x^{3/2}\right )}{\sqrt {1-x^3}}+\left (8 x^6-2 x^3-3\right ) x^{3/2}\right )}{72 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6*Sqrt[-x + x^4],x]

[Out]

(Sqrt[x*(-1 + x^3)]*(x^(3/2)*(-3 - 2*x^3 + 8*x^6) + (3*ArcSin[x^(3/2)])/Sqrt[1 - x^3]))/(72*Sqrt[x])

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IntegrateAlgebraic [A]  time = 0.43, size = 50, normalized size = 1.00 \begin {gather*} \frac {1}{72} \sqrt {-x+x^4} \left (-3 x-2 x^4+8 x^7\right )-\frac {1}{24} \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^6*Sqrt[-x + x^4],x]

[Out]

(Sqrt[-x + x^4]*(-3*x - 2*x^4 + 8*x^7))/72 - ArcTanh[x^2/Sqrt[-x + x^4]]/24

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fricas [A]  time = 0.54, size = 48, normalized size = 0.96 \begin {gather*} \frac {1}{72} \, {\left (8 \, x^{7} - 2 \, x^{4} - 3 \, x\right )} \sqrt {x^{4} - x} + \frac {1}{48} \, \log \left (2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^4-x)^(1/2),x, algorithm="fricas")

[Out]

1/72*(8*x^7 - 2*x^4 - 3*x)*sqrt(x^4 - x) + 1/48*log(2*x^3 - 2*sqrt(x^4 - x)*x - 1)

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giac [A]  time = 0.20, size = 56, normalized size = 1.12 \begin {gather*} \frac {1}{72} \, {\left (2 \, {\left (4 \, x^{3} - 1\right )} x^{3} - 3\right )} \sqrt {x^{4} - x} x - \frac {1}{48} \, \log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) + \frac {1}{48} \, \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^4-x)^(1/2),x, algorithm="giac")

[Out]

1/72*(2*(4*x^3 - 1)*x^3 - 3)*sqrt(x^4 - x)*x - 1/48*log(sqrt(-1/x^3 + 1) + 1) + 1/48*log(abs(sqrt(-1/x^3 + 1)
- 1))

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maple [A]  time = 0.61, size = 48, normalized size = 0.96

method	result	size
trager	\(\frac {x \left (8 x^{6}-2 x^{3}-3\right ) \sqrt {x^{4}-x}}{72}+\frac {\ln \left (2 x^{3}-2 x \sqrt {x^{4}-x}-1\right )}{48}\)	\(48\)
meijerg	\(\frac {i \sqrt {\mathrm {signum}\left (x^{3}-1\right )}\, \left (\frac {i \sqrt {\pi }\, x^{\frac {3}{2}} \left (-40 x^{6}+10 x^{3}+15\right ) \sqrt {-x^{3}+1}}{60}-\frac {i \sqrt {\pi }\, \arcsin \left (x^{\frac {3}{2}}\right )}{4}\right )}{6 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{3}-1\right )}}\)	\(66\)
risch	\(\frac {x^{2} \left (8 x^{6}-2 x^{3}-3\right ) \left (x^{3}-1\right )}{72 \sqrt {x \left (x^{3}-1\right )}}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{8 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(320\)
default	\(\frac {x^{7} \sqrt {x^{4}-x}}{9}-\frac {x^{4} \sqrt {x^{4}-x}}{36}-\frac {x \sqrt {x^{4}-x}}{24}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{8 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(329\)
elliptic	\(\frac {x^{7} \sqrt {x^{4}-x}}{9}-\frac {x^{4} \sqrt {x^{4}-x}}{36}-\frac {x \sqrt {x^{4}-x}}{24}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{8 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(329\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x^4-x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/72*x*(8*x^6-2*x^3-3)*(x^4-x)^(1/2)+1/48*ln(2*x^3-2*x*(x^4-x)^(1/2)-1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{4} - x} x^{6}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^4-x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 - x)*x^6, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^6\,\sqrt {x^4-x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x^4 - x)^(1/2),x)

[Out]

int(x^6*(x^4 - x)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{6} \sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(x**4-x)**(1/2),x)

[Out]

Integral(x**6*sqrt(x*(x - 1)*(x**2 + x + 1)), x)

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