Integrand size = 96, antiderivative size = 317 \[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a^2 \sqrt [3]{d}-2 \sqrt {3} a \sqrt [3]{d} x+\sqrt {3} \sqrt [3]{d} x^2}{a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2+2 \sqrt [3]{a b+(-a-b) x+x^2}}\right )}{d^{2/3}}+\frac {\log \left (a^3 \sqrt [3]{d}-2 a^2 \sqrt [3]{d} x+a \sqrt [3]{d} x^2-a \sqrt [3]{a b+(-a-b) x+x^2}\right )}{d^{2/3}}-\frac {\log \left (a^6 d^{2/3}-4 a^5 d^{2/3} x+6 a^4 d^{2/3} x^2-4 a^3 d^{2/3} x^3+a^2 d^{2/3} x^4+a^2 \left (a b+(-a-b) x+x^2\right )^{2/3}+\sqrt [3]{a b+(-a-b) x+x^2} \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \sqrt [3]{d} x^2\right )\right )}{2 d^{2/3}} \]
3^(1/2)*arctan((3^(1/2)*a^2*d^(1/3)-2*3^(1/2)*a*d^(1/3)*x+3^(1/2)*d^(1/3)* x^2)/(a^2*d^(1/3)-2*a*d^(1/3)*x+d^(1/3)*x^2+2*(a*b+(-a-b)*x+x^2)^(1/3)))/d ^(2/3)+ln(a^3*d^(1/3)-2*a^2*d^(1/3)*x+a*d^(1/3)*x^2-a*(a*b+(-a-b)*x+x^2)^( 1/3))/d^(2/3)-1/2*ln(a^6*d^(2/3)-4*a^5*d^(2/3)*x+6*a^4*d^(2/3)*x^2-4*a^3*d ^(2/3)*x^3+a^2*d^(2/3)*x^4+a^2*(a*b+(-a-b)*x+x^2)^(2/3)+(a*b+(-a-b)*x+x^2) ^(1/3)*(a^4*d^(1/3)-2*a^3*d^(1/3)*x+a^2*d^(1/3)*x^2))/d^(2/3)
\[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx \]
Integrate[((a - 5*b + 4*x)*(-a^3 + 3*a^2*x - 3*a*x^2 + x^3))/(((-a + x)*(- b + x))^(2/3)*(b - a^5*d - (1 - 5*a^4*d)*x - 10*a^3*d*x^2 + 10*a^2*d*x^3 - 5*a*d*x^4 + d*x^5)),x]
Integrate[((a - 5*b + 4*x)*(-a^3 + 3*a^2*x - 3*a*x^2 + x^3))/(((-a + x)*(- b + x))^(2/3)*(b - a^5*d - (1 - 5*a^4*d)*x - 10*a^3*d*x^2 + 10*a^2*d*x^3 - 5*a*d*x^4 + d*x^5)), x]
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.
| \(\displaystyle \int \frac {\left (-a^3+3 a^2 x-3 a x^2+x^3\right ) (a-5 b+4 x)}{((x-a) (x-b))^{2/3} \left (a^5 (-d)-x \left (1-5 a^4 d\right )-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+b+d x^5\right )} \, dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {(x-a)^3 (a-5 b+4 x)}{((x-a) (x-b))^{2/3} \left (a^5 (-d)-x \left (1-5 a^4 d\right )-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+b+d x^5\right )}dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int \frac {(x-a)^3 (a-5 b+4 x)}{\left (x (-a-b)+a b+x^2\right )^{2/3} \left (a^5 (-d)-x \left (1-5 a^4 d\right )-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+b+d x^5\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {(a-x)^3 (-a+5 b-4 x)}{\left (-x (a+b)+a b+x^2\right )^{2/3} \left (b \left (1-\frac {a^5 d}{b}\right )-x \left (1-5 a^4 d\right )-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {a^4 \left (1-\frac {5 b}{a}\right )}{\left (-x (a+b)+a b+x^2\right )^{2/3} \left (-b \left (1-\frac {a^5 d}{b}\right )+x \left (1-5 a^4 d\right )+10 a^3 d x^2-10 a^2 d x^3+5 a d x^4-d x^5\right )}+\frac {a^3 x \left (\frac {15 b}{a}+1\right )}{\left (-x (a+b)+a b+x^2\right )^{2/3} \left (-b \left (1-\frac {a^5 d}{b}\right )+x \left (1-5 a^4 d\right )+10 a^3 d x^2-10 a^2 d x^3+5 a d x^4-d x^5\right )}+\frac {9 a^2 x^2 \left (\frac {5 b}{3 a}+1\right )}{\left (-x (a+b)+a b+x^2\right )^{2/3} \left (b \left (1-\frac {a^5 d}{b}\right )-x \left (1-5 a^4 d\right )-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )}+\frac {11 a x^3 \left (\frac {5 b}{11 a}+1\right )}{\left (-x (a+b)+a b+x^2\right )^{2/3} \left (-b \left (1-\frac {a^5 d}{b}\right )+x \left (1-5 a^4 d\right )+10 a^3 d x^2-10 a^2 d x^3+5 a d x^4-d x^5\right )}+\frac {4 x^4}{\left (-x (a+b)+a b+x^2\right )^{2/3} \left (b \left (1-\frac {a^5 d}{b}\right )-x \left (1-5 a^4 d\right )-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^3 (a-5 b) \int \frac {1}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (-d x^5+5 a d x^4-10 a^2 d x^3+10 a^3 d x^2+\left (1-5 a^4 d\right ) x-b \left (1-\frac {a^5 d}{b}\right )\right )}dx+a^2 (a+15 b) \int \frac {x}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (-d x^5+5 a d x^4-10 a^2 d x^3+10 a^3 d x^2+\left (1-5 a^4 d\right ) x-b \left (1-\frac {a^5 d}{b}\right )\right )}dx+3 a (3 a+5 b) \int \frac {x^2}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (d x^5-5 a d x^4+10 a^2 d x^3-10 a^3 d x^2-\left (1-5 a^4 d\right ) x+b \left (1-\frac {a^5 d}{b}\right )\right )}dx+(11 a+5 b) \int \frac {x^3}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (-d x^5+5 a d x^4-10 a^2 d x^3+10 a^3 d x^2+\left (1-5 a^4 d\right ) x-b \left (1-\frac {a^5 d}{b}\right )\right )}dx+4 \int \frac {x^4}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (d x^5-5 a d x^4+10 a^2 d x^3-10 a^3 d x^2-\left (1-5 a^4 d\right ) x+b \left (1-\frac {a^5 d}{b}\right )\right )}dx\) |
Int[((a - 5*b + 4*x)*(-a^3 + 3*a^2*x - 3*a*x^2 + x^3))/(((-a + x)*(-b + x) )^(2/3)*(b - a^5*d - (1 - 5*a^4*d)*x - 10*a^3*d*x^2 + 10*a^2*d*x^3 - 5*a*d *x^4 + d*x^5)),x]
3.29.95.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) , x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F reeQ[{a, b, c, d, e, n, p}, x]
\[\int \frac {\left (a -5 b +4 x \right ) \left (-a^{3}+3 a^{2} x -3 a \,x^{2}+x^{3}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (b -a^{5} d -\left (-5 a^{4} d +1\right ) x -10 a^{3} d \,x^{2}+10 a^{2} d \,x^{3}-5 a d \,x^{4}+d \,x^{5}\right )}d x\]
int((a-5*b+4*x)*(-a^3+3*a^2*x-3*a*x^2+x^3)/((-a+x)*(-b+x))^(2/3)/(b-a^5*d- (-5*a^4*d+1)*x-10*a^3*d*x^2+10*a^2*d*x^3-5*a*d*x^4+d*x^5),x)
int((a-5*b+4*x)*(-a^3+3*a^2*x-3*a*x^2+x^3)/((-a+x)*(-b+x))^(2/3)/(b-a^5*d- (-5*a^4*d+1)*x-10*a^3*d*x^2+10*a^2*d*x^3-5*a*d*x^4+d*x^5),x)
Timed out. \[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\text {Timed out} \]
integrate((a-5*b+4*x)*(-a^3+3*a^2*x-3*a*x^2+x^3)/((-a+x)*(-b+x))^(2/3)/(b- a^5*d-(-5*a^4*d+1)*x-10*a^3*d*x^2+10*a^2*d*x^3-5*a*d*x^4+d*x^5),x, algorit hm="fricas")
\[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\int \frac {\left (- a + x\right )^{3} \left (a - 5 b + 4 x\right )}{\left (\left (- a + x\right ) \left (- b + x\right )\right )^{\frac {2}{3}} \left (- a^{5} d + 5 a^{4} d x - 10 a^{3} d x^{2} + 10 a^{2} d x^{3} - 5 a d x^{4} + b + d x^{5} - x\right )}\, dx \]
integrate((a-5*b+4*x)*(-a**3+3*a**2*x-3*a*x**2+x**3)/((-a+x)*(-b+x))**(2/3 )/(b-a**5*d-(-5*a**4*d+1)*x-10*a**3*d*x**2+10*a**2*d*x**3-5*a*d*x**4+d*x** 5),x)
Integral((-a + x)**3*(a - 5*b + 4*x)/(((-a + x)*(-b + x))**(2/3)*(-a**5*d + 5*a**4*d*x - 10*a**3*d*x**2 + 10*a**2*d*x**3 - 5*a*d*x**4 + b + d*x**5 - x)), x)
\[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\int { \frac {{\left (a^{3} - 3 \, a^{2} x + 3 \, a x^{2} - x^{3}\right )} {\left (a - 5 \, b + 4 \, x\right )}}{{\left (a^{5} d + 10 \, a^{3} d x^{2} - 10 \, a^{2} d x^{3} + 5 \, a d x^{4} - d x^{5} - {\left (5 \, a^{4} d - 1\right )} x - b\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {2}{3}}} \,d x } \]
integrate((a-5*b+4*x)*(-a^3+3*a^2*x-3*a*x^2+x^3)/((-a+x)*(-b+x))^(2/3)/(b- a^5*d-(-5*a^4*d+1)*x-10*a^3*d*x^2+10*a^2*d*x^3-5*a*d*x^4+d*x^5),x, algorit hm="maxima")
integrate((a^3 - 3*a^2*x + 3*a*x^2 - x^3)*(a - 5*b + 4*x)/((a^5*d + 10*a^3 *d*x^2 - 10*a^2*d*x^3 + 5*a*d*x^4 - d*x^5 - (5*a^4*d - 1)*x - b)*((a - x)* (b - x))^(2/3)), x)
\[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\int { \frac {{\left (a^{3} - 3 \, a^{2} x + 3 \, a x^{2} - x^{3}\right )} {\left (a - 5 \, b + 4 \, x\right )}}{{\left (a^{5} d + 10 \, a^{3} d x^{2} - 10 \, a^{2} d x^{3} + 5 \, a d x^{4} - d x^{5} - {\left (5 \, a^{4} d - 1\right )} x - b\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {2}{3}}} \,d x } \]
integrate((a-5*b+4*x)*(-a^3+3*a^2*x-3*a*x^2+x^3)/((-a+x)*(-b+x))^(2/3)/(b- a^5*d-(-5*a^4*d+1)*x-10*a^3*d*x^2+10*a^2*d*x^3-5*a*d*x^4+d*x^5),x, algorit hm="giac")
integrate((a^3 - 3*a^2*x + 3*a*x^2 - x^3)*(a - 5*b + 4*x)/((a^5*d + 10*a^3 *d*x^2 - 10*a^2*d*x^3 + 5*a*d*x^4 - d*x^5 - (5*a^4*d - 1)*x - b)*((a - x)* (b - x))^(2/3)), x)
Timed out. \[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\int -\frac {\left (a-5\,b+4\,x\right )\,\left (a^3-3\,a^2\,x+3\,a\,x^2-x^3\right )}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (b-a^5\,d+d\,x^5+x\,\left (5\,a^4\,d-1\right )+10\,a^2\,d\,x^3-10\,a^3\,d\,x^2-5\,a\,d\,x^4\right )} \,d x \]
int(-((a - 5*b + 4*x)*(3*a*x^2 - 3*a^2*x + a^3 - x^3))/(((a - x)*(b - x))^ (2/3)*(b - a^5*d + d*x^5 + x*(5*a^4*d - 1) + 10*a^2*d*x^3 - 10*a^3*d*x^2 - 5*a*d*x^4)),x)