\(\int \frac {x (-a+x) (a b+(a-2 b) x)}{(x (-a+x) (-b+x)^2)^{3/4} (b^2 d+(a-2 b d) x+(-1+d) x^2)} \, dx\) [2347]

Optimal result

Integrand size = 59, antiderivative size = 186 \[ \int \frac {x (-a+x) (a b+(a-2 b) x)}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (b^2 d+(a-2 b d) x+(-1+d) x^2\right )} \, dx=\frac {4 \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b-x}+2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{3/4}}{(b-x) x (-a+x)}\right )-2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{3/4}}{(b-x) x (-a+x)}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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