3.6.80 \(\int x^3 \sqrt {-x+x^4} \, dx\)

Optimal. Leaf size=45 \[ \frac {1}{12} \sqrt {x^4-x} \left (2 x^4-x\right )-\frac {1}{12} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 55, normalized size of antiderivative = 1.22, number of steps used = 4, 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}{6} \sqrt {x^4-x} x^4-\frac {1}{12} \sqrt {x^4-x} x-\frac {1}{12} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[-x + x^4],x]

[Out]

-1/12*(x*Sqrt[-x + x^4]) + (x^4*Sqrt[-x + x^4])/6 - ArcTanh[x^2/Sqrt[-x + x^4]]/12

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

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 62, normalized size = 1.38 \begin {gather*} \frac {\sqrt {x \left (x^3-1\right )} \left (\sin ^{-1}\left (x^{3/2}\right )+\sqrt {1-x^3} \left (2 x^3-1\right ) x^{3/2}\right )}{12 \sqrt {x} \sqrt {1-x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[-x + x^4],x]

[Out]

(Sqrt[x*(-1 + x^3)]*(x^(3/2)*Sqrt[1 - x^3]*(-1 + 2*x^3) + ArcSin[x^(3/2)]))/(12*Sqrt[x]*Sqrt[1 - x^3])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.38, size = 45, normalized size = 1.00 \begin {gather*} \frac {1}{12} \sqrt {-x+x^4} \left (-x+2 x^4\right )-\frac {1}{12} \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^3*Sqrt[-x + x^4],x]

[Out]

(Sqrt[-x + x^4]*(-x + 2*x^4))/12 - ArcTanh[x^2/Sqrt[-x + x^4]]/12

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 43, normalized size = 0.96 \begin {gather*} \frac {1}{12} \, {\left (2 \, x^{4} - x\right )} \sqrt {x^{4} - x} + \frac {1}{24} \, \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^3*(x^4-x)^(1/2),x, algorithm="fricas")

[Out]

1/12*(2*x^4 - x)*sqrt(x^4 - x) + 1/24*log(2*x^3 - 2*sqrt(x^4 - x)*x - 1)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 49, normalized size = 1.09 \begin {gather*} \frac {1}{12} \, \sqrt {x^{4} - x} {\left (2 \, x^{3} - 1\right )} x - \frac {1}{24} \, \log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) + \frac {1}{24} \, \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^3*(x^4-x)^(1/2),x, algorithm="giac")

[Out]

1/12*sqrt(x^4 - x)*(2*x^3 - 1)*x - 1/24*log(sqrt(-1/x^3 + 1) + 1) + 1/24*log(abs(sqrt(-1/x^3 + 1) - 1))

________________________________________________________________________________________

maple [A]  time = 0.29, size = 43, normalized size = 0.96

method	result	size
trager	\(\frac {x \left (2 x^{3}-1\right ) \sqrt {x^{4}-x}}{12}+\frac {\ln \left (2 x^{3}-2 x \sqrt {x^{4}-x}-1\right )}{24}\)	\(43\)
meijerg	\(-\frac {i \sqrt {\mathrm {signum}\left (x^{3}-1\right )}\, \left (-\frac {i \sqrt {\pi }\, x^{\frac {3}{2}} \left (-6 x^{3}+3\right ) \sqrt {-x^{3}+1}}{6}+\frac {i \sqrt {\pi }\, \arcsin \left (x^{\frac {3}{2}}\right )}{2}\right )}{6 \sqrt {-\mathrm {signum}\left (x^{3}-1\right )}\, \sqrt {\pi }}\)	\(61\)
default	\(\frac {x^{4} \sqrt {x^{4}-x}}{6}-\frac {x \sqrt {x^{4}-x}}{12}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{4 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(315\)
risch	\(\frac {x^{2} \left (2 x^{3}-1\right ) \left (x^{3}-1\right )}{12 \sqrt {x \left (x^{3}-1\right )}}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{4 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(315\)
elliptic	\(\frac {x^{4} \sqrt {x^{4}-x}}{6}-\frac {x \sqrt {x^{4}-x}}{12}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{4 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(315\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(x^4-x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/12*x*(2*x^3-1)*(x^4-x)^(1/2)+1/24*ln(2*x^3-2*x*(x^4-x)^(1/2)-1)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{4} - x} x^{3}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^4-x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 - x)*x^3, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^3\,\sqrt {x^4-x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(x^4 - x)^(1/2),x)

[Out]

int(x^3*(x^4 - x)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \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**3*(x**4-x)**(1/2),x)

[Out]

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

________________________________________________________________________________________