\(\int \frac {(a^2-2 a x+x^2) (-2 a b x+(3 a-b) x^2)}{(x^2 (-a+x) (-b+x))^{3/4} (a^3 d-3 a^2 d x+(-b+3 a d) x^2+(1-d) x^3)} \, dx\) [1375]

Optimal result

Integrand size = 83, antiderivative size = 99 \[ \int \frac {\left (a^2-2 a x+x^2\right ) \left (-2 a b x+(3 a-b) x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (a^3 d-3 a^2 d x+(-b+3 a d) x^2+(1-d) x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{3/4}}{x^2 (-b+x)}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{3/4}}{x^2 (-b+x)}\right )}{d^{3/4}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(4*a*b*x^(3/2)*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(-a + x^2)^(5/4)/((-b + x^2)^(3/4)*(-(a^3
*d) + 3*a^2*d*x^2 + b*(1 - (3*a*d)/b)*x^4 - (1 - d)*x^6)), x], x, Sqrt[x]])/((a - x)*(b - x)*x^2)^(3/4) + (2*(
3*a - b)*x^(3/2)*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x^2*(-a + x^2)^(5/4))/((-b + x^2)^(3/4
)*(a^3*d - 3*a^2*d*x^2 - b*(1 - (3*a*d)/b)*x^4 + (1 - d)*x^6)), x], x, Sqrt[x]])/((a - x)*(b - x)*x^2)^(3/4)

Rubi steps \begin{align*} \text {integral}& = \int \frac {(-a+x)^2 \left (-2 a b x+(3 a-b) x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (a^3 d-3 a^2 d x+(-b+3 a d) x^2+(1-d) x^3\right )} \, dx \\ & = \int \frac {x (-a+x)^2 (-2 a b+(3 a-b) x)}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (a^3 d-3 a^2 d x+(-b+3 a d) x^2+(1-d) x^3\right )} \, dx \\ & = \frac {\left (x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \int \frac {(-a+x)^{5/4} (-2 a b+(3 a-b) x)}{\sqrt {x} (-b+x)^{3/4} \left (a^3 d-3 a^2 d x+(-b+3 a d) x^2+(1-d) x^3\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}} \\ & = \frac {\left (2 x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {\left (-a+x^2\right )^{5/4} \left (-2 a b+(3 a-b) x^2\right )}{\left (-b+x^2\right )^{3/4} \left (a^3 d-3 a^2 d x^2+(-b+3 a d) x^4+(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}} \\ & = \frac {\left (2 x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \left (\frac {2 a b \left (-a+x^2\right )^{5/4}}{\left (-b+x^2\right )^{3/4} \left (-a^3 d+3 a^2 d x^2+b \left (1-\frac {3 a d}{b}\right ) x^4-(1-d) x^6\right )}+\frac {(3 a-b) x^2 \left (-a+x^2\right )^{5/4}}{\left (-b+x^2\right )^{3/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}} \\ & = \frac {\left (2 (3 a-b) x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-a+x^2\right )^{5/4}}{\left (-b+x^2\right )^{3/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}}+\frac {\left (4 a b x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {\left (-a+x^2\right )^{5/4}}{\left (-b+x^2\right )^{3/4} \left (-a^3 d+3 a^2 d x^2+b \left (1-\frac {3 a d}{b}\right ) x^4-(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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