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

Optimal. Leaf size=20 \[ \frac {4 \left (x^4-x^3\right )^{3/4}}{3 x^3} \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2014} \begin {gather*} \frac {4 \left (x^4-x^3\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(-x^3 + x^4)^(1/4)),x]

[Out]

(4*(-x^3 + x^4)^(3/4))/(3*x^3)

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.90 \begin {gather*} \frac {4 (x-1)}{3 \sqrt [4]{(x-1) x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(-x^3 + x^4)^(1/4)),x]

[Out]

(4*(-1 + x))/(3*((-1 + x)*x^3)^(1/4))

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IntegrateAlgebraic [A]  time = 0.20, size = 20, normalized size = 1.00 \begin {gather*} \frac {4 \left (-x^3+x^4\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-x^3 + x^4)^(3/4))/(3*x^3)

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fricas [A]  time = 0.43, size = 16, normalized size = 0.80 \begin {gather*} \frac {4 \, {\left (x^{4} - x^{3}\right )}^{\frac {3}{4}}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(x^4 - x^3)^(3/4)/x^3

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giac [A]  time = 0.67, size = 11, normalized size = 0.55 \begin {gather*} -\frac {4}{3} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(-1/x + 1)^(3/4)

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maple [A]  time = 0.08, size = 15, normalized size = 0.75

method	result	size
risch	\(\frac {-\frac {4}{3}+\frac {4 x}{3}}{\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}\)	\(15\)
gosper	\(\frac {-\frac {4}{3}+\frac {4 x}{3}}{\left (x^{4}-x^{3}\right )^{\frac {1}{4}}}\)	\(17\)
trager	\(\frac {4 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(17\)
meijerg	\(-\frac {4 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} \left (1-x \right )^{\frac {3}{4}}}{3 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} x^{\frac {3}{4}}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(-1+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}} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((x^4 - x^3)^(1/4)*x), x)

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mupad [B]  time = 0.30, size = 16, normalized size = 0.80 \begin {gather*} \frac {4\,{\left (x^4-x^3\right )}^{3/4}}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^4 - x^3)^(1/4)),x)

[Out]

(4*(x^4 - x^3)^(3/4))/(3*x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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