\(\int \frac {(a-5 b+4 x) (-a^3+3 a^2 x-3 a x^2+x^3)}{((-a+x) (-b+x))^{2/3} (b-a^5 d-(1-5 a^4 d) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5)} \, dx\) [2895]

Optimal result

Integrand size = 96, antiderivative size = 317 \[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a^2 \sqrt [3]{d}-2 \sqrt {3} a \sqrt [3]{d} x+\sqrt {3} \sqrt [3]{d} x^2}{a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2+2 \sqrt [3]{a b+(-a-b) x+x^2}}\right )}{d^{2/3}}+\frac {\log \left (a^3 \sqrt [3]{d}-2 a^2 \sqrt [3]{d} x+a \sqrt [3]{d} x^2-a \sqrt [3]{a b+(-a-b) x+x^2}\right )}{d^{2/3}}-\frac {\log \left (a^6 d^{2/3}-4 a^5 d^{2/3} x+6 a^4 d^{2/3} x^2-4 a^3 d^{2/3} x^3+a^2 d^{2/3} x^4+a^2 \left (a b+(-a-b) x+x^2\right )^{2/3}+\sqrt [3]{a b+(-a-b) x+x^2} \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \sqrt [3]{d} x^2\right )\right )}{2 d^{2/3}} \]

[Out]

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)

Rubi [F]

\[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx \]

[In]

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]

[Out]

a^3*(a - 5*b)*Defer[Int][1/((a*b - (a + b)*x + x^2)^(2/3)*(-(b*(1 - (a^5*d)/b)) + (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] + a^2*(a + 15*b)*Defer[Int][x/((a*b - (a + b)*x + x^2)^(2/3)*(-(b*
(1 - (a^5*d)/b)) + (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] + (11*a + 5*b)*Defe
r[Int][x^3/((a*b - (a + b)*x + x^2)^(2/3)*(-(b*(1 - (a^5*d)/b)) + (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*a*(3*a + 5*b)*Defer[Int][x^2/((a*b - (a + b)*x + x^2)^(2/3)*(b*(1 - (a^5*d)/b)
- (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] + 4*Defer[Int][x^4/((a*b - (a + b)*
x + x^2)^(2/3)*(b*(1 - (a^5*d)/b) - (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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{\left (a b+(-a-b) x+x^2\right )^{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 )}{\left (a b-(a+b) x+x^2\right )^{2/3} \left (b \left (1-\frac {a^5 d}{b}\right )-\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) (a-x)^3}{((a-x) (b-x))^{2/3} \left (b \left (1-\frac {a^5 d}{b}\right )-\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) (a-x)^3}{\left (a b+(-a-b) x+x^2\right )^{2/3} \left (b \left (1-\frac {a^5 d}{b}\right )-\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) (a-x)^3}{\left (a b-(a+b) x+x^2\right )^{2/3} \left (b \left (1-\frac {a^5 d}{b}\right )-\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 \left (\frac {a^4 \left (1-\frac {5 b}{a}\right )}{\left (a b-(a+b) x+x^2\right )^{2/3} \left (-b \left (1-\frac {a^5 d}{b}\right )+\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 )}+\frac {a^3 \left (1+\frac {15 b}{a}\right ) x}{\left (a b-(a+b) x+x^2\right )^{2/3} \left (-b \left (1-\frac {a^5 d}{b}\right )+\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 )}+\frac {11 a \left (1+\frac {5 b}{11 a}\right ) x^3}{\left (a b-(a+b) x+x^2\right )^{2/3} \left (-b \left (1-\frac {a^5 d}{b}\right )+\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 )}+\frac {9 a^2 \left (1+\frac {5 b}{3 a}\right ) x^2}{\left (a b-(a+b) x+x^2\right )^{2/3} \left (b \left (1-\frac {a^5 d}{b}\right )-\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 )}+\frac {4 x^4}{\left (a b-(a+b) x+x^2\right )^{2/3} \left (b \left (1-\frac {a^5 d}{b}\right )-\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 )}\right ) \, dx \\ & = 4 \int \frac {x^4}{\left (a b-(a+b) x+x^2\right )^{2/3} \left (b \left (1-\frac {a^5 d}{b}\right )-\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+\left (a^3 (a-5 b)\right ) \int \frac {1}{\left (a b-(a+b) x+x^2\right )^{2/3} \left (-b \left (1-\frac {a^5 d}{b}\right )+\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+(3 a (3 a+5 b)) \int \frac {x^2}{\left (a b-(a+b) x+x^2\right )^{2/3} \left (b \left (1-\frac {a^5 d}{b}\right )-\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+(11 a+5 b) \int \frac {x^3}{\left (a b-(a+b) x+x^2\right )^{2/3} \left (-b \left (1-\frac {a^5 d}{b}\right )+\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+\left (a^2 (a+15 b)\right ) \int \frac {x}{\left (a b-(a+b) x+x^2\right )^{2/3} \left (-b \left (1-\frac {a^5 d}{b}\right )+\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 \\ \end{align*}

Mathematica [F]

\[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx \]

[In]

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]

[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 - (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]

Maple [F]

\[\int \frac {\left (a -5 b +4 x \right ) \left (-a^{3}+3 a^{2} x -3 a \,x^{2}+x^{3}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (b -a^{5} d -\left (-5 a^{4} d +1\right ) x -10 a^{3} d \,x^{2}+10 a^{2} d \,x^{3}-5 a d \,x^{4}+d \,x^{5}\right )}d x\]

[In]

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)

[Out]

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)

Fricas [F(-1)]

Timed out. \[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\text {Timed out} \]

[In]

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, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\int \frac {\left (- a + x\right )^{3} \left (a - 5 b + 4 x\right )}{\left (\left (- a + x\right ) \left (- b + x\right )\right )^{\frac {2}{3}} \left (- a^{5} d + 5 a^{4} d x - 10 a^{3} d x^{2} + 10 a^{2} d x^{3} - 5 a d x^{4} + b + d x^{5} - x\right )}\, dx \]

[In]

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)

[Out]

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)

Maxima [F]

\[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\int { \frac {{\left (a^{3} - 3 \, a^{2} x + 3 \, a x^{2} - x^{3}\right )} {\left (a - 5 \, b + 4 \, x\right )}}{{\left (a^{5} d + 10 \, a^{3} d x^{2} - 10 \, a^{2} d x^{3} + 5 \, a d x^{4} - d x^{5} - {\left (5 \, a^{4} d - 1\right )} x - b\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {2}{3}}} \,d x } \]

[In]

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, algorithm="maxima")

[Out]

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)

Giac [F]

\[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\int { \frac {{\left (a^{3} - 3 \, a^{2} x + 3 \, a x^{2} - x^{3}\right )} {\left (a - 5 \, b + 4 \, x\right )}}{{\left (a^{5} d + 10 \, a^{3} d x^{2} - 10 \, a^{2} d x^{3} + 5 \, a d x^{4} - d x^{5} - {\left (5 \, a^{4} d - 1\right )} x - b\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {2}{3}}} \,d x } \]

[In]

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, algorithm="giac")

[Out]

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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx=\int -\frac {\left (a-5\,b+4\,x\right )\,\left (a^3-3\,a^2\,x+3\,a\,x^2-x^3\right )}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (b-a^5\,d+d\,x^5+x\,\left (5\,a^4\,d-1\right )+10\,a^2\,d\,x^3-10\,a^3\,d\,x^2-5\,a\,d\,x^4\right )} \,d x \]

[In]

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)

[Out]

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)