\(\int \frac {1}{(-1+x) \sqrt [4]{-x^3+x^4}} \, dx\) [225]

Optimal result

Integrand size = 19, antiderivative size = 23 \[ \int \frac {1}{(-1+x) \sqrt [4]{-x^3+x^4}} \, dx=-\frac {4 \left (-x^3+x^4\right )^{3/4}}{(-1+x) x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2081, 37} \[ \int \frac {1}{(-1+x) \sqrt [4]{-x^3+x^4}} \, dx=-\frac {4 x}{\sqrt [4]{x^4-x^3}} \]

[In]

Int[1/((-1 + x)*(-x^3 + x^4)^(1/4)),x]

[Out]

(-4*x)/(-x^3 + x^4)^(1/4)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{-1+x} x^{3/4}\right ) \int \frac {1}{(-1+x)^{5/4} x^{3/4}} \, dx}{\sqrt [4]{-x^3+x^4}} \\ & = -\frac {4 x}{\sqrt [4]{-x^3+x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(-1+x) \sqrt [4]{-x^3+x^4}} \, dx=-\frac {4 x}{\sqrt [4]{(-1+x) x^3}} \]

[In]

Integrate[1/((-1 + x)*(-x^3 + x^4)^(1/4)),x]

[Out]

(-4*x)/((-1 + x)*x^3)^(1/4)

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57

method	result	size
risch	\(-\frac {4 x}{\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}\)	\(13\)
pseudoelliptic	\(-\frac {4 x}{\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}\)	\(13\)
gosper	\(-\frac {4 x}{\left (x^{4}-x^{3}\right )^{\frac {1}{4}}}\)	\(15\)
trager	\(-\frac {4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{\left (x -1\right ) x^{2}}\)	\(22\)
meijerg	\(-\frac {4 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{4}} x^{\frac {1}{4}}}{\operatorname {signum}\left (x -1\right )^{\frac {1}{4}} \left (1-x \right )^{\frac {1}{4}}}\)	\(27\)

[In]

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

[Out]

-4*x/(x^3*(x-1))^(1/4)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(-1+x) \sqrt [4]{-x^3+x^4}} \, dx=-\frac {4 \, {\left (x^{4} - x^{3}\right )}^{\frac {3}{4}}}{x^{3} - x^{2}} \]

[In]

integrate(1/(-1+x)/(x^4-x^3)^(1/4),x, algorithm="fricas")

[Out]

-4*(x^4 - x^3)^(3/4)/(x^3 - x^2)

Sympy [F]

\[ \int \frac {1}{(-1+x) \sqrt [4]{-x^3+x^4}} \, dx=\int \frac {1}{\sqrt [4]{x^{3} \left (x - 1\right )} \left (x - 1\right )}\, dx \]

[In]

integrate(1/(-1+x)/(x**4-x**3)**(1/4),x)

[Out]

Integral(1/((x**3*(x - 1))**(1/4)*(x - 1)), x)

Maxima [F]

\[ \int \frac {1}{(-1+x) \sqrt [4]{-x^3+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x - 1\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/((x^4 - x^3)^(1/4)*(x - 1)), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48 \[ \int \frac {1}{(-1+x) \sqrt [4]{-x^3+x^4}} \, dx=-\frac {4}{{\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}} \]

[In]

integrate(1/(-1+x)/(x^4-x^3)^(1/4),x, algorithm="giac")

[Out]

-4/(-1/x + 1)^(1/4)

Mupad [B] (verification not implemented)

Time = 5.59 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(-1+x) \sqrt [4]{-x^3+x^4}} \, dx=-\frac {4\,{\left (x^4-x^3\right )}^{3/4}}{x^2\,\left (x-1\right )} \]

[In]

int(1/((x^4 - x^3)^(1/4)*(x - 1)),x)

[Out]

-(4*(x^4 - x^3)^(3/4))/(x^2*(x - 1))