\(\int \frac {(-1+x^5)^{3/4} (4+x^5) (-1-x^4+x^5)}{x^{12}} \, dx\) [485]

Optimal result

Integrand size = 28, antiderivative size = 38 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=\frac {4 \left (-1+x^5\right )^{3/4} \left (7+11 x^4-14 x^5-11 x^9+7 x^{10}\right )}{77 x^{11}} \]

[Out]

4/77*(x^5-1)^(3/4)*(7*x^10-11*x^9-14*x^5+11*x^4+7)/x^11

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(88\) vs. \(2(38)=76\).

Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.32, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1834, 1839, 1849, 1600, 460} \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=\frac {15 \left (x^5-1\right )^{3/4}}{11 x}-\frac {5 \left (x^5-1\right )^{3/4}}{22 x^6}-\frac {15 \left (x^5-1\right )^{3/4}}{14 x^2}+\frac {1}{154} \left (x^5-1\right )^{3/4} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \]

[In]

Int[((-1 + x^5)^(3/4)*(4 + x^5)*(-1 - x^4 + x^5))/x^12,x]

[Out]

((56/x^11 + 88/x^7 - 77/x^6 + 77/x^2 - 154/x)*(-1 + x^5)^(3/4))/154 - (5*(-1 + x^5)^(3/4))/(22*x^6) - (15*(-1
+ x^5)^(3/4))/(14*x^2) + (15*(-1 + x^5)^(3/4))/(11*x)

Rule 460

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[
a*d*(m + 1) - b*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]

Rule 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1834

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 1839

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{u = IntHide[x^m*Pq, x]}, Simp[u*(a +
b*x^n)^p, x] - Dist[b*n*p, Int[x^(m + n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x], x]] /; FreeQ[{a
, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1, 0]

Rule 1849

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{Pq0 = Coeff[Pq, x, 0]}, Simp[Pq0
*(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(2*a*c*(m + 1)), Int[(c*x)^(m + 1)*ExpandToSum
[2*a*(m + 1)*((Pq - Pq0)/x) - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b*x^n)^p, x], x] /; NeQ[Pq0, 0]]
/; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (-1+x^5\right )^{3/4}-\frac {15}{4} \int \frac {\frac {4}{11}+\frac {4 x^4}{7}-\frac {x^5}{2}+\frac {x^9}{2}-x^{10}}{x^7 \sqrt [4]{-1+x^5}} \, dx \\ & = \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (-1+x^5\right )^{3/4}-\frac {5 \left (-1+x^5\right )^{3/4}}{22 x^6}-\frac {5}{16} \int \frac {\frac {48 x^3}{7}-\frac {48 x^4}{11}+6 x^8-12 x^9}{x^6 \sqrt [4]{-1+x^5}} \, dx \\ & = \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (-1+x^5\right )^{3/4}-\frac {5 \left (-1+x^5\right )^{3/4}}{22 x^6}-\frac {5}{16} \int \frac {\frac {48 x^2}{7}-\frac {48 x^3}{11}+6 x^7-12 x^8}{x^5 \sqrt [4]{-1+x^5}} \, dx \\ & = \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (-1+x^5\right )^{3/4}-\frac {5 \left (-1+x^5\right )^{3/4}}{22 x^6}-\frac {5}{16} \int \frac {\frac {48 x}{7}-\frac {48 x^2}{11}+6 x^6-12 x^7}{x^4 \sqrt [4]{-1+x^5}} \, dx \\ & = \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (-1+x^5\right )^{3/4}-\frac {5 \left (-1+x^5\right )^{3/4}}{22 x^6}-\frac {5}{16} \int \frac {\frac {48}{7}-\frac {48 x}{11}+6 x^5-12 x^6}{x^3 \sqrt [4]{-1+x^5}} \, dx \\ & = \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (-1+x^5\right )^{3/4}-\frac {5 \left (-1+x^5\right )^{3/4}}{22 x^6}-\frac {15 \left (-1+x^5\right )^{3/4}}{14 x^2}-\frac {5}{64} \int \frac {-\frac {192}{11}-48 x^5}{x^2 \sqrt [4]{-1+x^5}} \, dx \\ & = \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (-1+x^5\right )^{3/4}-\frac {5 \left (-1+x^5\right )^{3/4}}{22 x^6}-\frac {15 \left (-1+x^5\right )^{3/4}}{14 x^2}+\frac {15 \left (-1+x^5\right )^{3/4}}{11 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.88 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=\frac {4 \left (-1+x^5\right )^{7/4} \left (-7-11 x^4+7 x^5\right )}{77 x^{11}} \]

[In]

Integrate[((-1 + x^5)^(3/4)*(4 + x^5)*(-1 - x^4 + x^5))/x^12,x]

[Out]

(4*(-1 + x^5)^(7/4)*(-7 - 11*x^4 + 7*x^5))/(77*x^11)

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66

method	result	size
pseudoelliptic	\(\frac {4 \left (x^{5}-1\right )^{\frac {7}{4}} \left (7 x^{5}-11 x^{4}-7\right )}{77 x^{11}}\)	\(25\)
trager	\(\frac {4 \left (x^{5}-1\right )^{\frac {3}{4}} \left (7 x^{10}-11 x^{9}-14 x^{5}+11 x^{4}+7\right )}{77 x^{11}}\)	\(35\)
gosper	\(\frac {4 \left (x^{5}-1\right )^{\frac {3}{4}} \left (7 x^{5}-11 x^{4}-7\right ) \left (x -1\right ) \left (x^{4}+x^{3}+x^{2}+x +1\right )}{77 x^{11}}\)	\(40\)
risch	\(\frac {\frac {4}{11} x^{15}-\frac {12}{11} x^{10}+\frac {12}{11} x^{5}-\frac {4}{11}-\frac {4}{7} x^{14}+\frac {8}{7} x^{9}-\frac {4}{7} x^{4}}{x^{11} \left (x^{5}-1\right )^{\frac {1}{4}}}\)	\(45\)
meijerg	\(-\frac {\operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {3}{4}, -\frac {1}{5}\right ], \left [\frac {4}{5}\right ], x^{5}\right )}{{\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x}+\frac {\operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {3}{4}, -\frac {2}{5}\right ], \left [\frac {3}{5}\right ], x^{5}\right )}{2 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x^{2}}-\frac {\operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {6}{5}, -\frac {3}{4}\right ], \left [-\frac {1}{5}\right ], x^{5}\right )}{2 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x^{6}}+\frac {4 \operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {7}{5}, -\frac {3}{4}\right ], \left [-\frac {2}{5}\right ], x^{5}\right )}{7 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x^{7}}+\frac {4 \operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {11}{5}, -\frac {3}{4}\right ], \left [-\frac {6}{5}\right ], x^{5}\right )}{11 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x^{11}}\)	\(162\)

[In]

int((x^5-1)^(3/4)*(x^5+4)*(x^5-x^4-1)/x^12,x,method=_RETURNVERBOSE)

[Out]

4/77*(x^5-1)^(7/4)*(7*x^5-11*x^4-7)/x^11

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=\frac {4 \, {\left (7 \, x^{10} - 11 \, x^{9} - 14 \, x^{5} + 11 \, x^{4} + 7\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \]

[In]

integrate((x^5-1)^(3/4)*(x^5+4)*(x^5-x^4-1)/x^12,x, algorithm="fricas")

[Out]

4/77*(7*x^10 - 11*x^9 - 14*x^5 + 11*x^4 + 7)*(x^5 - 1)^(3/4)/x^11

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.18 (sec) , antiderivative size = 199, normalized size of antiderivative = 5.24 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=- \frac {e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {1}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{5} \\ \frac {4}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x \Gamma \left (\frac {4}{5}\right )} + \frac {e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {2}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {2}{5} \\ \frac {3}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x^{2} \Gamma \left (\frac {3}{5}\right )} - \frac {3 e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {6}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {6}{5}, - \frac {3}{4} \\ - \frac {1}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x^{6} \Gamma \left (- \frac {1}{5}\right )} + \frac {4 e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{5}, - \frac {3}{4} \\ - \frac {2}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x^{7} \Gamma \left (- \frac {2}{5}\right )} + \frac {4 e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {11}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{5}, - \frac {3}{4} \\ - \frac {6}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x^{11} \Gamma \left (- \frac {6}{5}\right )} \]

[In]

integrate((x**5-1)**(3/4)*(x**5+4)*(x**5-x**4-1)/x**12,x)

[Out]

-exp(-I*pi/4)*gamma(-1/5)*hyper((-3/4, -1/5), (4/5,), x**5)/(5*x*gamma(4/5)) + exp(-I*pi/4)*gamma(-2/5)*hyper(
(-3/4, -2/5), (3/5,), x**5)/(5*x**2*gamma(3/5)) - 3*exp(-I*pi/4)*gamma(-6/5)*hyper((-6/5, -3/4), (-1/5,), x**5
)/(5*x**6*gamma(-1/5)) + 4*exp(-I*pi/4)*gamma(-7/5)*hyper((-7/5, -3/4), (-2/5,), x**5)/(5*x**7*gamma(-2/5)) +
4*exp(-I*pi/4)*gamma(-11/5)*hyper((-11/5, -3/4), (-6/5,), x**5)/(5*x**11*gamma(-6/5))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=\frac {4 \, {\left (7 \, x^{10} - 11 \, x^{9} - 14 \, x^{5} + 11 \, x^{4} + 7\right )} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \]

[In]

integrate((x^5-1)^(3/4)*(x^5+4)*(x^5-x^4-1)/x^12,x, algorithm="maxima")

[Out]

4/77*(7*x^10 - 11*x^9 - 14*x^5 + 11*x^4 + 7)*(x^4 + x^3 + x^2 + x + 1)^(3/4)*(x - 1)^(3/4)/x^11

Giac [F]

\[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=\int { \frac {{\left (x^{5} - x^{4} - 1\right )} {\left (x^{5} + 4\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}}}{x^{12}} \,d x } \]

[In]

integrate((x^5-1)^(3/4)*(x^5+4)*(x^5-x^4-1)/x^12,x, algorithm="giac")

[Out]

integrate((x^5 - x^4 - 1)*(x^5 + 4)*(x^5 - 1)^(3/4)/x^12, x)

Mupad [B] (verification not implemented)

Time = 5.42 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.61 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=\frac {4\,{\left (x^5-1\right )}^{3/4}}{11\,x}-\frac {4\,{\left (x^5-1\right )}^{3/4}}{7\,x^2}-\frac {8\,{\left (x^5-1\right )}^{3/4}}{11\,x^6}+\frac {4\,{\left (x^5-1\right )}^{3/4}}{7\,x^7}+\frac {4\,{\left (x^5-1\right )}^{3/4}}{11\,x^{11}} \]

[In]

int(-((x^5 - 1)^(3/4)*(x^5 + 4)*(x^4 - x^5 + 1))/x^12,x)

[Out]

(4*(x^5 - 1)^(3/4))/(11*x) - (4*(x^5 - 1)^(3/4))/(7*x^2) - (8*(x^5 - 1)^(3/4))/(11*x^6) + (4*(x^5 - 1)^(3/4))/
(7*x^7) + (4*(x^5 - 1)^(3/4))/(11*x^11)