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

Optimal. Leaf size=30 \[ -\frac {4 (7 x-1) \left (x^4-x^3\right )^{3/4}}{3 (x-1) x^3} \]

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Rubi [A]  time = 0.16, antiderivative size = 36, normalized size of antiderivative = 1.20, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2056, 78, 37} \begin {gather*} \frac {4}{3 \sqrt [4]{x^4-x^3}}-\frac {28 x}{3 \sqrt [4]{x^4-x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4/(3*(-x^3 + x^4)^(1/4)) - (28*x)/(3*(-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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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+x}{(-1+x) x \sqrt [4]{-x^3+x^4}} \, dx &=\frac {\left (\sqrt [4]{-1+x} x^{3/4}\right ) \int \frac {1+x}{(-1+x)^{5/4} x^{7/4}} \, dx}{\sqrt [4]{-x^3+x^4}}\\ &=\frac {4}{3 \sqrt [4]{-x^3+x^4}}+\frac {\left (7 \sqrt [4]{-1+x} x^{3/4}\right ) \int \frac {1}{(-1+x)^{5/4} x^{3/4}} \, dx}{3 \sqrt [4]{-x^3+x^4}}\\ &=\frac {4}{3 \sqrt [4]{-x^3+x^4}}-\frac {28 x}{3 \sqrt [4]{-x^3+x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.67 \begin {gather*} -\frac {4 (7 x-1)}{3 \sqrt [4]{(x-1) x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(-1 + 7*x))/(3*((-1 + x)*x^3)^(1/4))

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IntegrateAlgebraic [A]  time = 0.22, size = 30, normalized size = 1.00 \begin {gather*} -\frac {4 (-1+7 x) \left (-x^3+x^4\right )^{3/4}}{3 (-1+x) x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(-1 + 7*x)*(-x^3 + x^4)^(3/4))/(3*(-1 + x)*x^3)

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fricas [A]  time = 0.47, size = 18, normalized size = 0.60 \begin {gather*} -\frac {4 \, {\left (7 \, x - 1\right )}}{3 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(7*x - 1)/(x^4 - x^3)^(1/4)

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giac [A]  time = 0.77, size = 23, normalized size = 0.77 \begin {gather*} \frac {4}{3} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{4}} + \frac {8}{{\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(-1/x + 1)^(3/4) + 8/(-1/x + 1)^(1/4)

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maple [A]  time = 0.10, size = 17, normalized size = 0.57

method	result	size
risch	\(-\frac {4 \left (-1+7 x \right )}{3 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}\)	\(17\)
gosper	\(-\frac {4 \left (-1+7 x \right )}{3 \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}\)	\(19\)
trager	\(-\frac {4 \left (-1+7 x \right ) \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{3 \left (-1+x \right ) x^{3}}\)	\(27\)
meijerg	\(\frac {4 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} \left (1-4 x \right )}{3 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} \left (1-x \right )^{\frac {1}{4}} x^{\frac {3}{4}}}-\frac {4 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {1}{4}}}{\mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} \left (1-x \right )^{\frac {1}{4}}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(-1+7*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 {x + 1}{{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x - 1\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.23, size = 18, normalized size = 0.60 \begin {gather*} -\frac {28\,x-4}{3\,{\left (x^4-x^3\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(28*x - 4)/(3*(x^4 - x^3)^(1/4))

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

[Out]

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

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