3.9.32 \(\int x^4 \sqrt [4]{-x+x^4} \, dx\) [832]

Optimal. Leaf size=63 \[ \frac {1}{24} \sqrt [4]{-x+x^4} \left (-x^2+4 x^5\right )+\frac {1}{16} \text {ArcTan}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right )-\frac {1}{16} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right ) \]

[Out]

1/24*(x^4-x)^(1/4)*(4*x^5-x^2)+1/16*arctan(x/(x^4-x)^(1/4))-1/16*arctanh(x/(x^4-x)^(1/4))

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(127\) vs. \(2(63)=126\).
time = 0.08, antiderivative size = 127, normalized size of antiderivative = 2.02, number of steps used = 9, number of rules used = 9, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2046, 2049, 2057, 335, 281, 338, 304, 209, 212} \begin {gather*} \frac {\left (x^3-1\right )^{3/4} x^{3/4} \text {ArcTan}\left (\frac {x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{16 \left (x^4-x\right )^{3/4}}+\frac {1}{6} \sqrt [4]{x^4-x} x^5-\frac {1}{24} \sqrt [4]{x^4-x} x^2-\frac {\left (x^3-1\right )^{3/4} x^{3/4} \tanh ^{-1}\left (\frac {x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{16 \left (x^4-x\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(-x + x^4)^(1/4),x]

[Out]

-1/24*(x^2*(-x + x^4)^(1/4)) + (x^5*(-x + x^4)^(1/4))/6 + (x^(3/4)*(-1 + x^3)^(3/4)*ArcTan[x^(3/4)/(-1 + x^3)^
(1/4)])/(16*(-x + x^4)^(3/4)) - (x^(3/4)*(-1 + x^3)^(3/4)*ArcTanh[x^(3/4)/(-1 + x^3)^(1/4)])/(16*(-x + x^4)^(3
/4))

Rule 209

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

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 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

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 2049

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 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rubi steps

\begin {align*} \int x^4 \sqrt [4]{-x+x^4} \, dx &=\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {1}{8} \int \frac {x^5}{\left (-x+x^4\right )^{3/4}} \, dx\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {3}{32} \int \frac {x^2}{\left (-x+x^4\right )^{3/4}} \, dx\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (3 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \int \frac {x^{5/4}}{\left (-1+x^3\right )^{3/4}} \, dx}{32 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (3 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^8}{\left (-1+x^{12}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{8 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,x^{3/4}\right )}{8 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{8 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{16 \left (-x+x^4\right )^{3/4}}+\frac {\left (x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{16 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}+\frac {x^{3/4} \left (-1+x^3\right )^{3/4} \tan ^{-1}\left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{16 \left (-x+x^4\right )^{3/4}}-\frac {x^{3/4} \left (-1+x^3\right )^{3/4} \tanh ^{-1}\left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{16 \left (-x+x^4\right )^{3/4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.03, size = 56, normalized size = 0.89 \begin {gather*} \frac {x^2 \sqrt [4]{x \left (-1+x^3\right )} \left (-\left (1-x^3\right )^{5/4}+\, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};x^3\right )\right )}{6 \sqrt [4]{1-x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(-x + x^4)^(1/4),x]

[Out]

(x^2*(x*(-1 + x^3))^(1/4)*(-(1 - x^3)^(5/4) + Hypergeometric2F1[-1/4, 3/4, 7/4, x^3]))/(6*(1 - x^3)^(1/4))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 4.25, size = 33, normalized size = 0.52

method	result	size
meijerg	\(\frac {4 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {21}{4}} \hypergeom \left (\left [-\frac {1}{4}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], x^{3}\right )}{21 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{4}}}\)	\(33\)
trager	\(\frac {x^{2} \left (4 x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{24}+\frac {\ln \left (-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x \sqrt {x^{4}-x}-2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+2 x^{3}-1\right )}{32}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{32}\)	\(142\)
risch	\(\frac {x^{2} \left (4 x^{3}-1\right ) \left (x \left (x^{3}-1\right )\right )^{\frac {1}{4}}}{24}+\frac {\left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 x^{9}-2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-5 x^{6}+4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}-1}{\left (x^{2}+x +1\right )^{2} \left (-1+x \right )^{2}}\right )}{32}-\frac {\ln \left (\frac {2 x^{9}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}-5 x^{6}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}-4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}-1}{\left (x^{2}+x +1\right )^{2} \left (-1+x \right )^{2}}\right )}{32}\right ) \left (x \left (x^{3}-1\right )\right )^{\frac {1}{4}} \left (x^{3} \left (x^{3}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{3}-1\right )}\)	\(450\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/21*signum(x^3-1)^(1/4)/(-signum(x^3-1))^(1/4)*x^(21/4)*hypergeom([-1/4,7/4],[11/4],x^3)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 - x)^(1/4)*x^4, x)

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Fricas [A]
time = 1.04, size = 99, normalized size = 1.57 \begin {gather*} \frac {1}{24} \, {\left (4 \, x^{5} - x^{2}\right )} {\left (x^{4} - x\right )}^{\frac {1}{4}} - \frac {1}{32} \, \arctan \left (2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}}\right ) + \frac {1}{32} \, \log \left (2 \, x^{3} - 2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} - x} x - 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(4*x^5 - x^2)*(x^4 - x)^(1/4) - 1/32*arctan(2*(x^4 - x)^(1/4)*x^2 + 2*(x^4 - x)^(3/4)) + 1/32*log(2*x^3 -
2*(x^4 - x)^(1/4)*x^2 + 2*sqrt(x^4 - x)*x - 2*(x^4 - x)^(3/4) - 1)

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

[Out]

Integral(x**4*(x*(x - 1)*(x**2 + x + 1))**(1/4), x)

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Giac [A]
time = 0.40, size = 68, normalized size = 1.08 \begin {gather*} -\frac {1}{24} \, {\left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right )} x^{6} + \frac {1}{16} \, \arctan \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{32} \, \log \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{32} \, \log \left ({\left | {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*((-1/x^3 + 1)^(5/4) + 3*(-1/x^3 + 1)^(1/4))*x^6 + 1/16*arctan((-1/x^3 + 1)^(1/4)) + 1/32*log((-1/x^3 + 1
)^(1/4) + 1) - 1/32*log(abs((-1/x^3 + 1)^(1/4) - 1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^4\,{\left (x^4-x\right )}^{1/4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(x^4 - x)^(1/4),x)

[Out]

int(x^4*(x^4 - x)^(1/4), x)

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