Integrand size = 26, antiderivative size = 26 \[ \int \frac {\left (-4+x^5\right ) \left (1+x^4+x^5\right )}{x^6 \left (1+x^5\right )^{3/4}} \, dx=\frac {4 \sqrt [4]{1+x^5} \left (1+5 x^4+x^5\right )}{5 x^5} \]
Time = 1.51 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+x^5\right ) \left (1+x^4+x^5\right )}{x^6 \left (1+x^5\right )^{3/4}} \, dx=\frac {4 \sqrt [4]{1+x^5} \left (1+5 x^4+x^5\right )}{5 x^5} \]
Time = 0.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2374, 9, 27, 2374, 27, 793}
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.
\(\displaystyle \int \frac {\left (x^5-4\right ) \left (x^5+x^4+1\right )}{x^6 \left (x^5+1\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 2374 |
\(\displaystyle \frac {4 \sqrt [4]{x^5+1}}{5 x^5}-\frac {1}{10} \int \frac {10 \left (-x^9-x^8+4 x^3\right )}{x^5 \left (x^5+1\right )^{3/4}}dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \frac {4 \sqrt [4]{x^5+1}}{5 x^5}-\frac {1}{10} \int \frac {10 \left (-x^6-x^5+4\right )}{x^2 \left (x^5+1\right )^{3/4}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 \sqrt [4]{x^5+1}}{5 x^5}-\int \frac {-x^6-x^5+4}{x^2 \left (x^5+1\right )^{3/4}}dx\) |
\(\Big \downarrow \) 2374 |
\(\displaystyle \frac {1}{2} \int \frac {2 x^4}{\left (x^5+1\right )^{3/4}}dx+\frac {4 \sqrt [4]{x^5+1}}{x}+\frac {4 \sqrt [4]{x^5+1}}{5 x^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {x^4}{\left (x^5+1\right )^{3/4}}dx+\frac {4 \sqrt [4]{x^5+1}}{x}+\frac {4 \sqrt [4]{x^5+1}}{5 x^5}\) |
\(\Big \downarrow \) 793 |
\(\displaystyle \frac {4 \sqrt [4]{x^5+1}}{x}+\frac {4 \sqrt [4]{x^5+1}}{5 x^5}+\frac {4}{5} \sqrt [4]{x^5+1}\) |
3.3.86.3.1 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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n) ^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
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 = 1.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
trager | \(\frac {4 \left (x^{5}+1\right )^{\frac {1}{4}} \left (x^{5}+5 x^{4}+1\right )}{5 x^{5}}\) | \(23\) |
pseudoelliptic | \(\frac {4 \left (x^{5}+1\right )^{\frac {1}{4}} \left (x^{5}+5 x^{4}+1\right )}{5 x^{5}}\) | \(23\) |
risch | \(\frac {\frac {8}{5} x^{5}+\frac {4}{5}+4 x^{9}+4 x^{4}+\frac {4}{5} x^{10}}{\left (x^{5}+1\right )^{\frac {3}{4}} x^{5}}\) | \(33\) |
gosper | \(\frac {4 \left (x^{5}+5 x^{4}+1\right ) \left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}{5 \left (x^{5}+1\right )^{\frac {3}{4}} x^{5}}\) | \(42\) |
meijerg | \(\frac {x^{5} \operatorname {hypergeom}\left (\left [\frac {3}{4}, 1\right ], \left [2\right ], -x^{5}\right )}{5}+\frac {x^{4} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {4}{5}\right ], \left [\frac {9}{5}\right ], -x^{5}\right )}{4}-\frac {3 \left (-\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], -x^{5}\right )}{4}+\left (-3 \ln \left (2\right )+\frac {\pi }{2}+5 \ln \left (x \right )\right ) \Gamma \left (\frac {3}{4}\right )\right )}{5 \Gamma \left (\frac {3}{4}\right )}-\frac {4 \left (\frac {21 \Gamma \left (\frac {3}{4}\right ) x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {11}{4}\right ], \left [2, 3\right ], -x^{5}\right )}{32}-\frac {3 \left (\frac {1}{3}-3 \ln \left (2\right )+\frac {\pi }{2}+5 \ln \left (x \right )\right ) \Gamma \left (\frac {3}{4}\right )}{4}-\frac {\Gamma \left (\frac {3}{4}\right )}{x^{5}}\right )}{5 \Gamma \left (\frac {3}{4}\right )}+\frac {4 \operatorname {hypergeom}\left (\left [-\frac {1}{5}, \frac {3}{4}\right ], \left [\frac {4}{5}\right ], -x^{5}\right )}{x}\) | \(143\) |
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-4+x^5\right ) \left (1+x^4+x^5\right )}{x^6 \left (1+x^5\right )^{3/4}} \, dx=\frac {4 \, {\left (x^{5} + 5 \, x^{4} + 1\right )} {\left (x^{5} + 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \]
Result contains complex when optimal does not.
Time = 2.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 5.50 \[ \int \frac {\left (-4+x^5\right ) \left (1+x^4+x^5\right )}{x^6 \left (1+x^5\right )^{3/4}} \, dx=\frac {x^{4} \Gamma \left (\frac {4}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {4}{5} \\ \frac {9}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 \Gamma \left (\frac {9}{5}\right )} + \frac {4 \sqrt [4]{x^{5} + 1}}{5} - \frac {4 \Gamma \left (- \frac {1}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{5}, \frac {3}{4} \\ \frac {4}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x \Gamma \left (\frac {4}{5}\right )} + \frac {3 \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{5}}} \right )}}{5 x^{\frac {15}{4}} \Gamma \left (\frac {7}{4}\right )} + \frac {4 \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{5}}} \right )}}{5 x^{\frac {35}{4}} \Gamma \left (\frac {11}{4}\right )} \]
x**4*gamma(4/5)*hyper((3/4, 4/5), (9/5,), x**5*exp_polar(I*pi))/(5*gamma(9 /5)) + 4*(x**5 + 1)**(1/4)/5 - 4*gamma(-1/5)*hyper((-1/5, 3/4), (4/5,), x* *5*exp_polar(I*pi))/(5*x*gamma(4/5)) + 3*gamma(3/4)*hyper((3/4, 3/4), (7/4 ,), exp_polar(I*pi)/x**5)/(5*x**(15/4)*gamma(7/4)) + 4*gamma(7/4)*hyper((3 /4, 7/4), (11/4,), exp_polar(I*pi)/x**5)/(5*x**(35/4)*gamma(11/4))
\[ \int \frac {\left (-4+x^5\right ) \left (1+x^4+x^5\right )}{x^6 \left (1+x^5\right )^{3/4}} \, dx=\int { \frac {{\left (x^{5} + x^{4} + 1\right )} {\left (x^{5} - 4\right )}}{{\left (x^{5} + 1\right )}^{\frac {3}{4}} x^{6}} \,d x } \]
4/5*(x^5 + 1)^(1/4)/x^5 - 6/5*arctan((x^5 + 1)^(1/4)) + integrate((x^6 + x ^5 - 3*x - 4)*(x^4 - x^3 + x^2 - x + 1)^(1/4)*(x + 1)^(1/4)/(x^7 + x^2), x ) - 3/5*log((x^5 + 1)^(1/4) + 1) + 3/5*log((x^5 + 1)^(1/4) - 1)
\[ \int \frac {\left (-4+x^5\right ) \left (1+x^4+x^5\right )}{x^6 \left (1+x^5\right )^{3/4}} \, dx=\int { \frac {{\left (x^{5} + x^{4} + 1\right )} {\left (x^{5} - 4\right )}}{{\left (x^{5} + 1\right )}^{\frac {3}{4}} x^{6}} \,d x } \]
Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-4+x^5\right ) \left (1+x^4+x^5\right )}{x^6 \left (1+x^5\right )^{3/4}} \, dx=\frac {4\,{\left (x^5+1\right )}^{5/4}+20\,x^4\,{\left (x^5+1\right )}^{1/4}}{5\,x^5} \]