\(\int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx\) [133]

Optimal result

Integrand size = 20, antiderivative size = 18 \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \left (-x+x^4\right )^{7/4}}{21 x^7} \]

[Out]

4/21*(x^4-x)^(7/4)/x^7

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(18)=36\).

Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2077, 2041, 2039} \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \left (x^4-x\right )^{3/4}}{21 x^3}-\frac {4 \left (x^4-x\right )^{3/4}}{21 x^6} \]

[In]

Int[(-1 + x^3)/(x^6*(-x + x^4)^(1/4)),x]

[Out]

(-4*(-x + x^4)^(3/4))/(21*x^6) + (4*(-x + 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])

Rule 2077

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(c*x)
^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !In
tegerQ[p] && NeQ[n, j]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{x^6 \sqrt [4]{-x+x^4}}+\frac {1}{x^3 \sqrt [4]{-x+x^4}}\right ) \, dx \\ & = -\int \frac {1}{x^6 \sqrt [4]{-x+x^4}} \, dx+\int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx \\ & = -\frac {4 \left (-x+x^4\right )^{3/4}}{21 x^6}+\frac {4 \left (-x+x^4\right )^{3/4}}{9 x^3}-\frac {4}{7} \int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx \\ & = -\frac {4 \left (-x+x^4\right )^{3/4}}{21 x^6}+\frac {4 \left (-x+x^4\right )^{3/4}}{21 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 9.54 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \left (x \left (-1+x^3\right )\right )^{7/4}}{21 x^7} \]

[In]

Integrate[(-1 + x^3)/(x^6*(-x + x^4)^(1/4)),x]

[Out]

(4*(x*(-1 + x^3))^(7/4))/(21*x^7)

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11

method	result	size
trager	\(\frac {4 \left (x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {3}{4}}}{21 x^{6}}\)	\(20\)
pseudoelliptic	\(\frac {4 \left (x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {3}{4}}}{21 x^{6}}\)	\(20\)
risch	\(\frac {\frac {4}{21} x^{6}-\frac {8}{21} x^{3}+\frac {4}{21}}{x^{5} {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}}}\)	\(25\)
gosper	\(\frac {4 \left (x -1\right ) \left (x^{2}+x +1\right ) \left (x^{3}-1\right )}{21 x^{5} \left (x^{4}-x \right )^{\frac {1}{4}}}\)	\(29\)
meijerg	\(\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (1+\frac {4 x^{3}}{3}\right ) \left (-x^{3}+1\right )^{\frac {3}{4}}}{21 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {21}{4}}}-\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (-x^{3}+1\right )^{\frac {3}{4}}}{9 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {9}{4}}}\)	\(73\)

[In]

int((x^3-1)/x^6/(x^4-x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

4/21*(x^3-1)*(x^4-x)^(3/4)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} {\left (x^{3} - 1\right )}}{21 \, x^{6}} \]

[In]

integrate((x^3-1)/x^6/(x^4-x)^(1/4),x, algorithm="fricas")

[Out]

4/21*(x^4 - x)^(3/4)*(x^3 - 1)/x^6

Sympy [F]

\[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x^{2} + x + 1\right )}{x^{6} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}\, dx \]

[In]

integrate((x**3-1)/x**6/(x**4-x)**(1/4),x)

[Out]

Integral((x - 1)*(x**2 + x + 1)/(x**6*(x*(x - 1)*(x**2 + x + 1))**(1/4)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (14) = 28\).

Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.22 \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \, {\left (x^{4} - x\right )}}{9 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {13}{4}}} - \frac {4 \, {\left (4 \, x^{7} - x^{4} - 3 \, x\right )}}{63 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {25}{4}}} \]

[In]

integrate((x^3-1)/x^6/(x^4-x)^(1/4),x, algorithm="maxima")

[Out]

4/9*(x^4 - x)/((x^2 + x + 1)^(1/4)*(x - 1)^(1/4)*x^(13/4)) - 4/63*(4*x^7 - x^4 - 3*x)/((x^2 + x + 1)^(1/4)*(x
- 1)^(1/4)*x^(25/4))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4}{21} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {7}{4}} \]

[In]

integrate((x^3-1)/x^6/(x^4-x)^(1/4),x, algorithm="giac")

[Out]

4/21*(-1/x^3 + 1)^(7/4)

Mupad [B] (verification not implemented)

Time = 5.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=-\frac {4\,{\left (x^4-x\right )}^{3/4}-4\,x^3\,{\left (x^4-x\right )}^{3/4}}{21\,x^6} \]

[In]

int((x^3 - 1)/(x^6*(x^4 - x)^(1/4)),x)

[Out]

-(4*(x^4 - x)^(3/4) - 4*x^3*(x^4 - x)^(3/4))/(21*x^6)