\(\int \frac {(a^2-2 a x+x^2) (-a b+2 (a-b) x+x^2)}{(x (-a+x) (-b+x))^{3/4} (-a^3 d+(b+3 a^2 d) x-(1+3 a d) x^2+d x^3)} \, dx\) [1316]

Optimal result

Integrand size = 79, antiderivative size = 95 \[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b+2 (a-b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \left (a b x+(-a-b) x^2+x^3\right )^{3/4}}{x (-b+x)}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \left (a b x+(-a-b) x^2+x^3\right )^{3/4}}{x (-b+x)}\right )}{d^{3/4}} \]

[Out]

2*arctan(d^(1/4)*(a*b*x+(-a-b)*x^2+x^3)^(3/4)/x/(-b+x))/d^(3/4)-2*arctanh(d^(1/4)*(a*b*x+(-a-b)*x^2+x^3)^(3/4)
/x/(-b+x))/d^(3/4)

Rubi [F]

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

[In]

Int[((a^2 - 2*a*x + x^2)*(-(a*b) + 2*(a - b)*x + x^2))/((x*(-a + x)*(-b + x))^(3/4)*(-(a^3*d) + (b + 3*a^2*d)*
x - (1 + 3*a*d)*x^2 + d*x^3)),x]

[Out]

(4*a*b*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(-a + x^4)^(5/4)/((-b + x^4)^(3/4)*(a^3*d
- b*(1 + (3*a^2*d)/b)*x^4 + (1 + 3*a*d)*x^8 - d*x^12)), x], x, x^(1/4)])/((a - x)*(b - x)*x)^(3/4) - (8*(a -
b)*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x^4*(-a + x^4)^(5/4))/((-b + x^4)^(3/4)*(a^3
*d - b*(1 + (3*a^2*d)/b)*x^4 + (1 + 3*a*d)*x^8 - d*x^12)), x], x, x^(1/4)])/((a - x)*(b - x)*x)^(3/4) + (4*x^(
3/4)*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x^8*(-a + x^4)^(5/4))/((-b + x^4)^(3/4)*(-(a^3*d)
+ b*(1 + (3*a^2*d)/b)*x^4 - (1 + 3*a*d)*x^8 + d*x^12)), x], x, x^(1/4)])/((a - x)*(b - x)*x)^(3/4)

Rubi steps \begin{align*} \text {integral}& = \int \frac {(-a+x)^2 \left (-a b+2 (a-b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx \\ & = \frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \int \frac {(-a+x)^{5/4} \left (-a b+2 (a-b) x+x^2\right )}{x^{3/4} (-b+x)^{3/4} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx}{(x (-a+x) (-b+x))^{3/4}} \\ & = \frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {\left (-a+x^4\right )^{5/4} \left (-a b+2 (a-b) x^4+x^8\right )}{\left (-b+x^4\right )^{3/4} \left (-a^3 d+\left (b+3 a^2 d\right ) x^4-(1+3 a d) x^8+d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(x (-a+x) (-b+x))^{3/4}} \\ & = \frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \left (\frac {a b \left (-a+x^4\right )^{5/4}}{\left (-b+x^4\right )^{3/4} \left (a^3 d-b \left (1+\frac {3 a^2 d}{b}\right ) x^4+(1+3 a d) x^8-d x^{12}\right )}+\frac {2 (-a+b) x^4 \left (-a+x^4\right )^{5/4}}{\left (-b+x^4\right )^{3/4} \left (a^3 d-b \left (1+\frac {3 a^2 d}{b}\right ) x^4+(1+3 a d) x^8-d x^{12}\right )}+\frac {x^8 \left (-a+x^4\right )^{5/4}}{\left (-b+x^4\right )^{3/4} \left (-a^3 d+b \left (1+\frac {3 a^2 d}{b}\right ) x^4-(1+3 a d) x^8+d x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{(x (-a+x) (-b+x))^{3/4}} \\ & = \frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^8 \left (-a+x^4\right )^{5/4}}{\left (-b+x^4\right )^{3/4} \left (-a^3 d+b \left (1+\frac {3 a^2 d}{b}\right ) x^4-(1+3 a d) x^8+d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(x (-a+x) (-b+x))^{3/4}}-\frac {\left (8 (a-b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^4 \left (-a+x^4\right )^{5/4}}{\left (-b+x^4\right )^{3/4} \left (a^3 d-b \left (1+\frac {3 a^2 d}{b}\right ) x^4+(1+3 a d) x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(x (-a+x) (-b+x))^{3/4}}+\frac {\left (4 a b x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {\left (-a+x^4\right )^{5/4}}{\left (-b+x^4\right )^{3/4} \left (a^3 d-b \left (1+\frac {3 a^2 d}{b}\right ) x^4+(1+3 a d) x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(x (-a+x) (-b+x))^{3/4}} \\ \end{align*}

Mathematica [F]

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

[In]

Integrate[((a^2 - 2*a*x + x^2)*(-(a*b) + 2*(a - b)*x + x^2))/((x*(-a + x)*(-b + x))^(3/4)*(-(a^3*d) + (b + 3*a
^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]

[Out]

Integrate[((a^2 - 2*a*x + x^2)*(-(a*b) + 2*(a - b)*x + x^2))/((x*(-a + x)*(-b + x))^(3/4)*(-(a^3*d) + (b + 3*a
^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)), x]

Maple [F]

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

[In]

int((a^2-2*a*x+x^2)*(-a*b+2*(a-b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-d*a^3+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x^3),x)

[Out]

int((a^2-2*a*x+x^2)*(-a*b+2*(a-b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-d*a^3+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x^3),x)

Fricas [F(-1)]

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

[In]

integrate((a^2-2*a*x+x^2)*(-a*b+2*(a-b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x
^3),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

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

[In]

integrate((a**2-2*a*x+x**2)*(-a*b+2*(a-b)*x+x**2)/(x*(-a+x)*(-b+x))**(3/4)/(-a**3*d+(3*a**2*d+b)*x-(3*a*d+1)*x
**2+d*x**3),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((a^2-2*a*x+x^2)*(-a*b+2*(a-b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x
^3),x, algorithm="maxima")

[Out]

integrate((a^2 - 2*a*x + x^2)*(a*b - 2*(a - b)*x - x^2)/((a^3*d - d*x^3 + (3*a*d + 1)*x^2 - (3*a^2*d + b)*x)*(
(a - x)*(b - x)*x)^(3/4)), x)

Giac [F]

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

[In]

integrate((a^2-2*a*x+x^2)*(-a*b+2*(a-b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x
^3),x, algorithm="giac")

[Out]

integrate((a^2 - 2*a*x + x^2)*(a*b - 2*(a - b)*x - x^2)/((a^3*d - d*x^3 + (3*a*d + 1)*x^2 - (3*a^2*d + b)*x)*(
(a - x)*(b - x)*x)^(3/4)), x)

Mupad [F(-1)]

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

[In]

int(((a^2 - 2*a*x + x^2)*(2*x*(a - b) - a*b + x^2))/((x*(a - x)*(b - x))^(3/4)*(x*(b + 3*a^2*d) - a^3*d + d*x^
3 - x^2*(3*a*d + 1))),x)

[Out]

int(((a^2 - 2*a*x + x^2)*(2*x*(a - b) - a*b + x^2))/((x*(a - x)*(b - x))^(3/4)*(x*(b + 3*a^2*d) - a^3*d + d*x^
3 - x^2*(3*a*d + 1))), x)