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

Optimal. Leaf size=20 \[ \frac {4 \left (x^4-x^3\right )^{5/4}}{5 x^5} \]

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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 )^{5/4}}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^3 + x^4)^(1/4)/x^3,x]

[Out]

(4*(-x^3 + x^4)^(5/4))/(5*x^5)

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

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.90 \begin {gather*} \frac {4 \left ((x-1) x^3\right )^{5/4}}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^3 + x^4)^(1/4)/x^3,x]

[Out]

(4*((-1 + x)*x^3)^(5/4))/(5*x^5)

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IntegrateAlgebraic [A]  time = 0.18, size = 20, normalized size = 1.00 \begin {gather*} \frac {4 \left (-x^3+x^4\right )^{5/4}}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-x^3 + x^4)^(5/4))/(5*x^5)

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fricas [A]  time = 0.45, size = 19, normalized size = 0.95 \begin {gather*} \frac {4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x - 1\right )}}{5 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*(x^4 - x^3)^(1/4)*(x - 1)/x^2

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giac [A]  time = 0.25, size = 11, normalized size = 0.55 \begin {gather*} -\frac {4}{5} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/5*(-1/x + 1)^(5/4)

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maple [A]  time = 0.06, size = 20, normalized size = 1.00

method	result	size
gosper	\(\frac {4 \left (-1+x \right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{5 x^{2}}\)	\(20\)
trager	\(\frac {4 \left (-1+x \right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{5 x^{2}}\)	\(20\)
meijerg	\(-\frac {4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} \left (1-x \right )^{\frac {5}{4}}}{5 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {5}{4}}}\)	\(27\)
risch	\(\frac {4 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x^{2}-2 x +1\right )}{5 x^{2} \left (-1+x \right )}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5/x^2*(-1+x)*(x^4-x^3)^(1/4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 - x^3)^(1/4)/x^3, x)

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mupad [B]  time = 0.17, size = 33, normalized size = 1.65 \begin {gather*} \frac {4\,x\,{\left (x^4-x^3\right )}^{1/4}-4\,{\left (x^4-x^3\right )}^{1/4}}{5\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 - x^3)^(1/4)/x^3,x)

[Out]

(4*x*(x^4 - x^3)^(1/4) - 4*(x^4 - x^3)^(1/4))/(5*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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