Integrand size = 28, antiderivative size = 38 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=\frac {4 \left (-1+x^5\right )^{3/4} \left (7+11 x^4-14 x^5-11 x^9+7 x^{10}\right )}{77 x^{11}} \]
Time = 1.88 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=\frac {4 \left (-1+x^5\right )^{7/4} \left (-7-11 x^4+7 x^5\right )}{77 x^{11}} \]
Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(38)=76\).
Time = 0.47 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.34, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2364, 27, 2374, 9, 27, 2374, 27, 951}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (x^5-1\right )^{3/4} \left (x^5+4\right ) \left (x^5-x^4-1\right )}{x^{12}} \, dx\) |
\(\Big \downarrow \) 2364 |
\(\displaystyle \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (x^5-1\right )^{3/4}-\frac {15}{4} \int \frac {-154 x^{10}+77 x^9-77 x^5+88 x^4+56}{154 x^7 \sqrt [4]{x^5-1}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (x^5-1\right )^{3/4}-\frac {15}{616} \int \frac {-154 x^{10}+77 x^9-77 x^5+88 x^4+56}{x^7 \sqrt [4]{x^5-1}}dx\) |
\(\Big \downarrow \) 2374 |
\(\displaystyle \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (x^5-1\right )^{3/4}-\frac {15}{616} \left (\frac {1}{12} \int \frac {12 \left (-154 x^9+77 x^8-56 x^4+88 x^3\right )}{x^6 \sqrt [4]{x^5-1}}dx+\frac {28 \left (x^5-1\right )^{3/4}}{3 x^6}\right )\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (x^5-1\right )^{3/4}-\frac {15}{616} \left (\frac {1}{12} \int \frac {12 \left (-154 x^6+77 x^5-56 x+88\right )}{x^3 \sqrt [4]{x^5-1}}dx+\frac {28 \left (x^5-1\right )^{3/4}}{3 x^6}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (x^5-1\right )^{3/4}-\frac {15}{616} \left (\int \frac {-154 x^6+77 x^5-56 x+88}{x^3 \sqrt [4]{x^5-1}}dx+\frac {28 \left (x^5-1\right )^{3/4}}{3 x^6}\right )\) |
\(\Big \downarrow \) 2374 |
\(\displaystyle \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (x^5-1\right )^{3/4}-\frac {15}{616} \left (\frac {1}{4} \int -\frac {56 \left (11 x^5+4\right )}{x^2 \sqrt [4]{x^5-1}}dx+\frac {28 \left (x^5-1\right )^{3/4}}{3 x^6}+\frac {44 \left (x^5-1\right )^{3/4}}{x^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (x^5-1\right )^{3/4}-\frac {15}{616} \left (-14 \int \frac {11 x^5+4}{x^2 \sqrt [4]{x^5-1}}dx+\frac {28 \left (x^5-1\right )^{3/4}}{3 x^6}+\frac {44 \left (x^5-1\right )^{3/4}}{x^2}\right )\) |
\(\Big \downarrow \) 951 |
\(\displaystyle \frac {1}{154} \left (\frac {56}{x^{11}}+\frac {88}{x^7}-\frac {77}{x^6}+\frac {77}{x^2}-\frac {154}{x}\right ) \left (x^5-1\right )^{3/4}-\frac {15}{616} \left (-\frac {56 \left (x^5-1\right )^{3/4}}{x}+\frac {28 \left (x^5-1\right )^{3/4}}{3 x^6}+\frac {44 \left (x^5-1\right )^{3/4}}{x^2}\right )\) |
((56/x^11 + 88/x^7 - 77/x^6 + 77/x^2 - 154/x)*(-1 + x^5)^(3/4))/154 - (15* ((28*(-1 + x^5)^(3/4))/(3*x^6) + (44*(-1 + x^5)^(3/4))/x^2 - (56*(-1 + x^5 )^(3/4))/x))/616
3.5.85.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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] - Simp[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]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wit h[{Pq0 = Coeff[Pq, x, 0]}, Simp[Pq0*(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c *(m + 1))), x] + Simp[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]]
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 | \(-\frac {\operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {3}{4}, -\frac {1}{5}\right ], \left [\frac {4}{5}\right ], x^{5}\right )}{{\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x}+\frac {\operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {3}{4}, -\frac {2}{5}\right ], \left [\frac {3}{5}\right ], x^{5}\right )}{2 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x^{2}}-\frac {\operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {6}{5}, -\frac {3}{4}\right ], \left [-\frac {1}{5}\right ], x^{5}\right )}{2 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x^{6}}+\frac {4 \operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {7}{5}, -\frac {3}{4}\right ], \left [-\frac {2}{5}\right ], x^{5}\right )}{7 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x^{7}}+\frac {4 \operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {11}{5}, -\frac {3}{4}\right ], \left [-\frac {6}{5}\right ], x^{5}\right )}{11 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x^{11}}\) | \(162\) |
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=\frac {4 \, {\left (7 \, x^{10} - 11 \, x^{9} - 14 \, x^{5} + 11 \, x^{4} + 7\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \]
Result contains complex when optimal does not.
Time = 3.18 (sec) , antiderivative size = 199, normalized size of antiderivative = 5.24 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=- \frac {e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {1}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{5} \\ \frac {4}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x \Gamma \left (\frac {4}{5}\right )} + \frac {e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {2}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {2}{5} \\ \frac {3}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x^{2} \Gamma \left (\frac {3}{5}\right )} - \frac {3 e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {6}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {6}{5}, - \frac {3}{4} \\ - \frac {1}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x^{6} \Gamma \left (- \frac {1}{5}\right )} + \frac {4 e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{5}, - \frac {3}{4} \\ - \frac {2}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x^{7} \Gamma \left (- \frac {2}{5}\right )} + \frac {4 e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {11}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{5}, - \frac {3}{4} \\ - \frac {6}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x^{11} \Gamma \left (- \frac {6}{5}\right )} \]
-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*ga mma(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))
Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=\frac {4 \, {\left (7 \, x^{10} - 11 \, x^{9} - 14 \, x^{5} + 11 \, x^{4} + 7\right )} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \]
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
\[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=\int { \frac {{\left (x^{5} - x^{4} - 1\right )} {\left (x^{5} + 4\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}}}{x^{12}} \,d x } \]
Time = 5.42 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.61 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right ) \left (-1-x^4+x^5\right )}{x^{12}} \, dx=\frac {4\,{\left (x^5-1\right )}^{3/4}}{11\,x}-\frac {4\,{\left (x^5-1\right )}^{3/4}}{7\,x^2}-\frac {8\,{\left (x^5-1\right )}^{3/4}}{11\,x^6}+\frac {4\,{\left (x^5-1\right )}^{3/4}}{7\,x^7}+\frac {4\,{\left (x^5-1\right )}^{3/4}}{11\,x^{11}} \]