Integrand size = 22, antiderivative size = 50 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{-x^3+x^4}}{x^8} \, dx=\frac {4 \sqrt [4]{-x^3+x^4} \left (13923-663 x-21255 x^2+615 x^3+820 x^4+1312 x^5+5248 x^6\right )}{348075 x^7} \] Output:
4/348075*(x^4-x^3)^(1/4)*(5248*x^6+1312*x^5+820*x^4+615*x^3-21255*x^2-663* x+13923)/x^7
Time = 1.44 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.76 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{-x^3+x^4}}{x^8} \, dx=\frac {4 \left ((-1+x) x^3\right )^{9/4} \left (13923+27183 x+19188 x^2+11808 x^3+5248 x^4\right )}{348075 x^{13}} \] Input:
Integrate[((-1 + x^2)*(-x^3 + x^4)^(1/4))/x^8,x]
Output:
(4*((-1 + x)*x^3)^(9/4)*(13923 + 27183*x + 19188*x^2 + 11808*x^3 + 5248*x^ 4))/(348075*x^13)
Leaf count is larger than twice the leaf count of optimal. \(121\) vs. \(2(50)=100\).
Time = 0.33 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.42, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2449, 2009}
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.
\(\displaystyle \int \frac {\left (x^2-1\right ) \sqrt [4]{x^4-x^3}}{x^8} \, dx\) |
\(\Big \downarrow \) 2449 |
\(\displaystyle \int \left (\frac {\sqrt [4]{x^4-x^3}}{x^6}-\frac {\sqrt [4]{x^4-x^3}}{x^8}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 \left (x^4-x^3\right )^{5/4}}{25 x^{10}}-\frac {16 \left (x^4-x^3\right )^{5/4}}{105 x^9}+\frac {164 \left (x^4-x^3\right )^{5/4}}{1785 x^8}+\frac {656 \left (x^4-x^3\right )^{5/4}}{7735 x^7}+\frac {5248 \left (x^4-x^3\right )^{5/4}}{69615 x^6}+\frac {20992 \left (x^4-x^3\right )^{5/4}}{348075 x^5}\) |
Input:
Int[((-1 + x^2)*(-x^3 + x^4)^(1/4))/x^8,x]
Output:
(-4*(-x^3 + x^4)^(5/4))/(25*x^10) - (16*(-x^3 + x^4)^(5/4))/(105*x^9) + (1 64*(-x^3 + x^4)^(5/4))/(1785*x^8) + (656*(-x^3 + x^4)^(5/4))/(7735*x^7) + (5248*(-x^3 + x^4)^(5/4))/(69615*x^6) + (20992*(-x^3 + x^4)^(5/4))/(348075 *x^5)
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_S ymbol] :> 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]) && !Integer Q[p] && NeQ[n, j]
Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.80
method | result | size |
pseudoelliptic | \(\frac {4 \left (-1+x \right )^{2} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (5248 x^{4}+11808 x^{3}+19188 x^{2}+27183 x +13923\right )}{348075 x^{7}}\) | \(40\) |
gosper | \(\frac {4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \left (-1+x \right )^{2} \left (5248 x^{4}+11808 x^{3}+19188 x^{2}+27183 x +13923\right )}{348075 x^{7}}\) | \(42\) |
trager | \(\frac {4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \left (5248 x^{6}+1312 x^{5}+820 x^{4}+615 x^{3}-21255 x^{2}-663 x +13923\right )}{348075 x^{7}}\) | \(47\) |
orering | \(\frac {4 \left (5248 x^{4}+11808 x^{3}+19188 x^{2}+27183 x +13923\right ) \left (-1+x \right ) \left (x^{2}-1\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{348075 x^{7} \left (1+x \right )}\) | \(50\) |
risch | \(\frac {4 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (5248 x^{7}-3936 x^{6}-492 x^{5}-205 x^{4}-21870 x^{3}+20592 x^{2}+14586 x -13923\right )}{348075 x^{7} \left (-1+x \right )}\) | \(55\) |
meijerg | \(\frac {4 \operatorname {signum}\left (-1+x \right )^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {25}{4}, -\frac {1}{4}\right ], \left [-\frac {21}{4}\right ], x\right )}{25 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {25}{4}}}-\frac {4 \operatorname {signum}\left (-1+x \right )^{\frac {1}{4}} \left (-\frac {128}{195} x^{4}-\frac {32}{195} x^{3}-\frac {4}{39} x^{2}-\frac {1}{13} x +1\right ) \left (1-x \right )^{\frac {1}{4}}}{17 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {17}{4}}}\) | \(74\) |
Input:
int((x^2-1)*(x^4-x^3)^(1/4)/x^8,x,method=_RETURNVERBOSE)
Output:
4/348075/x^7*(-1+x)^2*(x^3*(-1+x))^(1/4)*(5248*x^4+11808*x^3+19188*x^2+271 83*x+13923)
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{-x^3+x^4}}{x^8} \, dx=\frac {4 \, {\left (5248 \, x^{6} + 1312 \, x^{5} + 820 \, x^{4} + 615 \, x^{3} - 21255 \, x^{2} - 663 \, x + 13923\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{348075 \, x^{7}} \] Input:
integrate((x^2-1)*(x^4-x^3)^(1/4)/x^8,x, algorithm="fricas")
Output:
4/348075*(5248*x^6 + 1312*x^5 + 820*x^4 + 615*x^3 - 21255*x^2 - 663*x + 13 923)*(x^4 - x^3)^(1/4)/x^7
\[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{-x^3+x^4}}{x^8} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x - 1\right ) \left (x + 1\right )}{x^{8}}\, dx \] Input:
integrate((x**2-1)*(x**4-x**3)**(1/4)/x**8,x)
Output:
Integral((x**3*(x - 1))**(1/4)*(x - 1)*(x + 1)/x**8, x)
\[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{-x^3+x^4}}{x^8} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{x^{8}} \,d x } \] Input:
integrate((x^2-1)*(x^4-x^3)^(1/4)/x^8,x, algorithm="maxima")
Output:
integrate((x^4 - x^3)^(1/4)*(x^2 - 1)/x^8, x)
Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.82 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{-x^3+x^4}}{x^8} \, dx=\frac {4}{25} \, {\left (\frac {1}{x} - 1\right )}^{6} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \frac {20}{21} \, {\left (\frac {1}{x} - 1\right )}^{5} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \frac {36}{17} \, {\left (\frac {1}{x} - 1\right )}^{4} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \frac {28}{13} \, {\left (\frac {1}{x} - 1\right )}^{3} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \frac {8}{9} \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \] Input:
integrate((x^2-1)*(x^4-x^3)^(1/4)/x^8,x, algorithm="giac")
Output:
4/25*(1/x - 1)^6*(-1/x + 1)^(1/4) + 20/21*(1/x - 1)^5*(-1/x + 1)^(1/4) + 3 6/17*(1/x - 1)^4*(-1/x + 1)^(1/4) + 28/13*(1/x - 1)^3*(-1/x + 1)^(1/4) + 8 /9*(1/x - 1)^2*(-1/x + 1)^(1/4)
Time = 8.71 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.26 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{-x^3+x^4}}{x^8} \, dx=\frac {20992\,{\left (x^4-x^3\right )}^{1/4}}{348075\,x}+\frac {5248\,{\left (x^4-x^3\right )}^{1/4}}{348075\,x^2}+\frac {656\,{\left (x^4-x^3\right )}^{1/4}}{69615\,x^3}+\frac {164\,{\left (x^4-x^3\right )}^{1/4}}{23205\,x^4}-\frac {436\,{\left (x^4-x^3\right )}^{1/4}}{1785\,x^5}-\frac {4\,{\left (x^4-x^3\right )}^{1/4}}{525\,x^6}+\frac {4\,{\left (x^4-x^3\right )}^{1/4}}{25\,x^7} \] Input:
int(((x^2 - 1)*(x^4 - x^3)^(1/4))/x^8,x)
Output:
(20992*(x^4 - x^3)^(1/4))/(348075*x) + (5248*(x^4 - x^3)^(1/4))/(348075*x^ 2) + (656*(x^4 - x^3)^(1/4))/(69615*x^3) + (164*(x^4 - x^3)^(1/4))/(23205* x^4) - (436*(x^4 - x^3)^(1/4))/(1785*x^5) - (4*(x^4 - x^3)^(1/4))/(525*x^6 ) + (4*(x^4 - x^3)^(1/4))/(25*x^7)
Time = 0.15 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{-x^3+x^4}}{x^8} \, dx=\frac {4 \left (x -1\right )^{\frac {1}{4}} \left (5248 x^{6}+1312 x^{5}+820 x^{4}+615 x^{3}-21255 x^{2}-663 x +13923\right )}{348075 x^{\frac {25}{4}}} \] Input:
int((x^2-1)*(x^4-x^3)^(1/4)/x^8,x)
Output:
(4*x**(3/4)*(x - 1)**(1/4)*(5248*x**6 + 1312*x**5 + 820*x**4 + 615*x**3 - 21255*x**2 - 663*x + 13923))/(348075*x**7)