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

Optimal. Leaf size=23 \[ -\frac {4 \left (x^4-x^3\right )^{3/4}}{(x-1) x^2} \]

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2056, 37} \begin {gather*} -\frac {4 x}{\sqrt [4]{x^4-x^3}} \end {gather*}

Antiderivative was successfully verified.

[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 2056

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

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.61 \begin {gather*} -\frac {4 x}{\sqrt [4]{(x-1) x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*x)/((-1 + x)*x^3)^(1/4)

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IntegrateAlgebraic [A]  time = 0.21, size = 23, normalized size = 1.00 \begin {gather*} -\frac {4 \left (-x^3+x^4\right )^{3/4}}{(-1+x) x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 24, normalized size = 1.04 \begin {gather*} -\frac {4 \, {\left (x^{4} - x^{3}\right )}^{\frac {3}{4}}}{x^{3} - x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.32, size = 11, normalized size = 0.48 \begin {gather*} \frac {4}{{\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/(-1/x + 1)^(1/4)

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maple [A]  time = 0.08, size = 13, normalized size = 0.57

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.17, size = 21, normalized size = 0.91 \begin {gather*} -\frac {4\,{\left (x^4-x^3\right )}^{3/4}}{x^2\,\left (x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x^4 - x^3)^(1/4)*(x - 1)),x)

[Out]

-(4*(x^4 - x^3)^(3/4))/(x^2*(x - 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{3} \left (x - 1\right )} \left (x - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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