\(\int \frac {1}{x^4 \sqrt [4]{-x^3+x^4}} \, dx\) [437]

Optimal result

Integrand size = 17, antiderivative size = 35 \[ \int \frac {1}{x^4 \sqrt [4]{-x^3+x^4}} \, dx=\frac {4 \left (77+84 x+96 x^2+128 x^3\right ) \left (-x^3+x^4\right )^{3/4}}{1155 x^6} \]

[Out]

4/1155*(128*x^3+96*x^2+84*x+77)*(x^4-x^3)^(3/4)/x^6

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(35)=70\).

Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.31, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2041, 2039} \[ \int \frac {1}{x^4 \sqrt [4]{-x^3+x^4}} \, dx=\frac {512 \left (x^4-x^3\right )^{3/4}}{1155 x^3}+\frac {128 \left (x^4-x^3\right )^{3/4}}{385 x^4}+\frac {4 \left (x^4-x^3\right )^{3/4}}{15 x^6}+\frac {16 \left (x^4-x^3\right )^{3/4}}{55 x^5} \]

[In]

Int[1/(x^4*(-x^3 + x^4)^(1/4)),x]

[Out]

(4*(-x^3 + x^4)^(3/4))/(15*x^6) + (16*(-x^3 + x^4)^(3/4))/(55*x^5) + (128*(-x^3 + x^4)^(3/4))/(385*x^4) + (512
*(-x^3 + x^4)^(3/4))/(1155*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}}{15 x^6}+\frac {4}{5} \int \frac {1}{x^3 \sqrt [4]{-x^3+x^4}} \, dx \\ & = \frac {4 \left (-x^3+x^4\right )^{3/4}}{15 x^6}+\frac {16 \left (-x^3+x^4\right )^{3/4}}{55 x^5}+\frac {32}{55} \int \frac {1}{x^2 \sqrt [4]{-x^3+x^4}} \, dx \\ & = \frac {4 \left (-x^3+x^4\right )^{3/4}}{15 x^6}+\frac {16 \left (-x^3+x^4\right )^{3/4}}{55 x^5}+\frac {128 \left (-x^3+x^4\right )^{3/4}}{385 x^4}+\frac {128}{385} \int \frac {1}{x \sqrt [4]{-x^3+x^4}} \, dx \\ & = \frac {4 \left (-x^3+x^4\right )^{3/4}}{15 x^6}+\frac {16 \left (-x^3+x^4\right )^{3/4}}{55 x^5}+\frac {128 \left (-x^3+x^4\right )^{3/4}}{385 x^4}+\frac {512 \left (-x^3+x^4\right )^{3/4}}{1155 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^4 \sqrt [4]{-x^3+x^4}} \, dx=\frac {4 \left ((-1+x) x^3\right )^{3/4} \left (77+84 x+96 x^2+128 x^3\right )}{1155 x^6} \]

[In]

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

[Out]

(4*((-1 + x)*x^3)^(3/4)*(77 + 84*x + 96*x^2 + 128*x^3))/(1155*x^6)

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {4 \left (128 x^{3}+96 x^{2}+84 x +77\right ) \left (x^{3} \left (x -1\right )\right )^{\frac {3}{4}}}{1155 x^{6}}\)	\(30\)
trager	\(\frac {4 \left (128 x^{3}+96 x^{2}+84 x +77\right ) \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{1155 x^{6}}\)	\(32\)
gosper	\(\frac {4 \left (x -1\right ) \left (128 x^{3}+96 x^{2}+84 x +77\right )}{1155 x^{3} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}\)	\(35\)
risch	\(\frac {-\frac {4}{165} x -\frac {4}{15}-\frac {16}{385} x^{2}-\frac {128}{1155} x^{3}+\frac {512}{1155} x^{4}}{x^{3} \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}\)	\(35\)
meijerg	\(-\frac {4 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{4}} \left (\frac {128}{77} x^{3}+\frac {96}{77} x^{2}+\frac {12}{11} x +1\right ) \left (1-x \right )^{\frac {3}{4}}}{15 \operatorname {signum}\left (x -1\right )^{\frac {1}{4}} x^{\frac {15}{4}}}\)	\(42\)

[In]

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

[Out]

4/1155*(128*x^3+96*x^2+84*x+77)*(x^3*(x-1))^(3/4)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^4 \sqrt [4]{-x^3+x^4}} \, dx=\frac {4 \, {\left (x^{4} - x^{3}\right )}^{\frac {3}{4}} {\left (128 \, x^{3} + 96 \, x^{2} + 84 \, x + 77\right )}}{1155 \, x^{6}} \]

[In]

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

[Out]

4/1155*(x^4 - x^3)^(3/4)*(128*x^3 + 96*x^2 + 84*x + 77)/x^6

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int \frac {1}{x^4 \sqrt [4]{-x^3+x^4}} \, dx=\frac {4}{15} \, {\left (\frac {1}{x} - 1\right )}^{3} {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{4}} + \frac {12}{11} \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{4}} - \frac {12}{7} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {7}{4}} + \frac {4}{3} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{4}} \]

[In]

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

[Out]

4/15*(1/x - 1)^3*(-1/x + 1)^(3/4) + 12/11*(1/x - 1)^2*(-1/x + 1)^(3/4) - 12/7*(-1/x + 1)^(7/4) + 4/3*(-1/x + 1
)^(3/4)

Mupad [B] (verification not implemented)

Time = 5.47 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.86 \[ \int \frac {1}{x^4 \sqrt [4]{-x^3+x^4}} \, dx=\frac {512\,{\left (x^4-x^3\right )}^{3/4}}{1155\,x^3}+\frac {128\,{\left (x^4-x^3\right )}^{3/4}}{385\,x^4}+\frac {16\,{\left (x^4-x^3\right )}^{3/4}}{55\,x^5}+\frac {4\,{\left (x^4-x^3\right )}^{3/4}}{15\,x^6} \]

[In]

int(1/(x^4*(x^4 - x^3)^(1/4)),x)

[Out]

(512*(x^4 - x^3)^(3/4))/(1155*x^3) + (128*(x^4 - x^3)^(3/4))/(385*x^4) + (16*(x^4 - x^3)^(3/4))/(55*x^5) + (4*
(x^4 - x^3)^(3/4))/(15*x^6)