\(\int \frac {1}{x^2 \sqrt [4]{-x^3+x^4}} \, dx\) [268]

Optimal result

Integrand size = 17, antiderivative size = 25 \[ \int \frac {1}{x^2 \sqrt [4]{-x^3+x^4}} \, dx=\frac {4 (3+4 x) \left (-x^3+x^4\right )^{3/4}}{21 x^4} \]

[Out]

4/21*(3+4*x)*(x^4-x^3)^(3/4)/x^4

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2041, 2039} \[ \int \frac {1}{x^2 \sqrt [4]{-x^3+x^4}} \, dx=\frac {16 \left (x^4-x^3\right )^{3/4}}{21 x^3}+\frac {4 \left (x^4-x^3\right )^{3/4}}{7 x^4} \]

[In]

Int[1/(x^2*(-x^3 + x^4)^(1/4)),x]

[Out]

(4*(-x^3 + x^4)^(3/4))/(7*x^4) + (16*(-x^3 + x^4)^(3/4))/(21*x^3)

Rule 2039

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] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2041

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*(m + j*p + 1))), x] - Dist[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps \begin{align*} \text {integral}& = \frac {4 \left (-x^3+x^4\right )^{3/4}}{7 x^4}+\frac {4}{7} \int \frac {1}{x \sqrt [4]{-x^3+x^4}} \, dx \\ & = \frac {4 \left (-x^3+x^4\right )^{3/4}}{7 x^4}+\frac {16 \left (-x^3+x^4\right )^{3/4}}{21 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 \sqrt [4]{-x^3+x^4}} \, dx=\frac {4 \left ((-1+x) x^3\right )^{3/4} (3+4 x)}{21 x^4} \]

[In]

Integrate[1/(x^2*(-x^3 + x^4)^(1/4)),x]

[Out]

(4*((-1 + x)*x^3)^(3/4)*(3 + 4*x))/(21*x^4)

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
pseudoelliptic	\(\frac {4 \left (4 x +3\right ) \left (x^{3} \left (x -1\right )\right )^{\frac {3}{4}}}{21 x^{4}}\)	\(20\)
trager	\(\frac {4 \left (4 x +3\right ) \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{21 x^{4}}\)	\(22\)
gosper	\(\frac {4 \left (x -1\right ) \left (4 x +3\right )}{21 x \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}\)	\(25\)
risch	\(\frac {-\frac {4}{21} x -\frac {4}{7}+\frac {16}{21} x^{2}}{x \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}\)	\(25\)
meijerg	\(-\frac {4 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{4}} \left (1+\frac {4 x}{3}\right ) \left (1-x \right )^{\frac {3}{4}}}{7 \operatorname {signum}\left (x -1\right )^{\frac {1}{4}} x^{\frac {7}{4}}}\)	\(32\)

[In]

int(1/x^2/(x^4-x^3)^(1/4),x,method=_RETURNVERBOSE)

[Out]

4/21*(4*x+3)*(x^3*(x-1))^(3/4)/x^4

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^2 \sqrt [4]{-x^3+x^4}} \, dx=\frac {4 \, {\left (x^{4} - x^{3}\right )}^{\frac {3}{4}} {\left (4 \, x + 3\right )}}{21 \, x^{4}} \]

[In]

integrate(1/x^2/(x^4-x^3)^(1/4),x, algorithm="fricas")

[Out]

4/21*(x^4 - x^3)^(3/4)*(4*x + 3)/x^4

Sympy [F]

\[ \int \frac {1}{x^2 \sqrt [4]{-x^3+x^4}} \, dx=\int \frac {1}{x^{2} \sqrt [4]{x^{3} \left (x - 1\right )}}\, dx \]

[In]

integrate(1/x**2/(x**4-x**3)**(1/4),x)

[Out]

Integral(1/(x**2*(x**3*(x - 1))**(1/4)), x)

Maxima [F]

\[ \int \frac {1}{x^2 \sqrt [4]{-x^3+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(x^4-x^3)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((x^4 - x^3)^(1/4)*x^2), x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 \sqrt [4]{-x^3+x^4}} \, dx=-\frac {4}{7} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {7}{4}} + \frac {4}{3} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{4}} \]

[In]

integrate(1/x^2/(x^4-x^3)^(1/4),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^2 \sqrt [4]{-x^3+x^4}} \, dx=\frac {16\,x\,{\left (x^4-x^3\right )}^{3/4}+12\,{\left (x^4-x^3\right )}^{3/4}}{21\,x^4} \]

[In]

int(1/(x^2*(x^4 - x^3)^(1/4)),x)

[Out]

(16*x*(x^4 - x^3)^(3/4) + 12*(x^4 - x^3)^(3/4))/(21*x^4)