\(\int \frac {-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} (b+(-1+a^2 d) x-2 a d x^2+d x^3)} \, dx\) [2551]

Optimal result

Integrand size = 75, antiderivative size = 215 \[ \int \frac {-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (b+\left (-1+a^2 d\right ) x-2 a d x^2+d x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a^2 b^3 x+\left (3 a^2 b^2+2 a b^3\right ) x^2+\left (-3 a^2 b-6 a b^2-b^3\right ) x^3+\left (a^2+6 a b+3 b^2\right ) x^4+(-2 a-3 b) x^5+x^6}}{b-x}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a^2 b^3 x+\left (3 a^2 b^2+2 a b^3\right ) x^2+\left (-3 a^2 b-6 a b^2-b^3\right ) x^3+\left (a^2+6 a b+3 b^2\right ) x^4+(-2 a-3 b) x^5+x^6}}{b-x}\right )}{d^{3/4}} \]

[Out]

-2*arctan(d^(1/4)*(-a^2*b^3*x+(3*a^2*b^2+2*a*b^3)*x^2+(-3*a^2*b-6*a*b^2-b^3)*x^3+(a^2+6*a*b+3*b^2)*x^4+(-2*a-3
*b)*x^5+x^6)^(1/4)/(b-x))/d^(3/4)+2*arctanh(d^(1/4)*(-a^2*b^3*x+(3*a^2*b^2+2*a*b^3)*x^2+(-3*a^2*b-6*a*b^2-b^3)
*x^3+(a^2+6*a*b+3*b^2)*x^4+(-2*a-3*b)*x^5+x^6)^(1/4)/(b-x))/d^(3/4)

Rubi [F]

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

[In]

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

[Out]

(12*b*x^(1/4)*Sqrt[-a + x]*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x^6*Sqrt[-a + x^4])/((-b + x^4)^(3/4)*(-b +
(1 - a^2*d)*x^4 + 2*a*d*x^8 - d*x^12)), x], x, x^(1/4)])/(-((a - x)^2*(b - x)^3*x))^(1/4) + (4*a*b*x^(1/4)*Sq
rt[-a + x]*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x^2*Sqrt[-a + x^4])/((-b + x^4)^(3/4)*(b - (1 - a^2*d)*x^4
- 2*a*d*x^8 + d*x^12)), x], x, x^(1/4)])/(-((a - x)^2*(b - x)^3*x))^(1/4) + (8*x^(1/4)*Sqrt[-a + x]*(-b + x)^(
3/4)*Defer[Subst][Defer[Int][(x^10*Sqrt[-a + x^4])/((-b + x^4)^(3/4)*(b - (1 - a^2*d)*x^4 - 2*a*d*x^8 + d*x^12
)), x], x, x^(1/4)])/(-((a - x)^2*(b - x)^3*x))^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \int \frac {-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{\sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4} \left (b+\left (-1+a^2 d\right ) x-2 a d x^2+d x^3\right )} \, dx}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}} \\ & = \frac {\left (\sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \int \frac {\sqrt {-a+x} \left (a b-3 b x+2 x^2\right )}{\sqrt [4]{x} (-b+x)^{3/4} \left (b+\left (-1+a^2 d\right ) x-2 a d x^2+d x^3\right )} \, dx}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {-a+x^4} \left (a b-3 b x^4+2 x^8\right )}{\left (-b+x^4\right )^{3/4} \left (b+\left (-1+a^2 d\right ) x^4-2 a d x^8+d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \left (\frac {3 b x^6 \sqrt {-a+x^4}}{\left (-b+x^4\right )^{3/4} \left (-b+\left (1-a^2 d\right ) x^4+2 a d x^8-d x^{12}\right )}+\frac {a b x^2 \sqrt {-a+x^4}}{\left (-b+x^4\right )^{3/4} \left (b-\left (1-a^2 d\right ) x^4-2 a d x^8+d x^{12}\right )}+\frac {2 x^{10} \sqrt {-a+x^4}}{\left (-b+x^4\right )^{3/4} \left (b-\left (1-a^2 d\right ) x^4-2 a d x^8+d x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}} \\ & = \frac {\left (8 \sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^{10} \sqrt {-a+x^4}}{\left (-b+x^4\right )^{3/4} \left (b-\left (1-a^2 d\right ) x^4-2 a d x^8+d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}}+\frac {\left (12 b \sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt {-a+x^4}}{\left (-b+x^4\right )^{3/4} \left (-b+\left (1-a^2 d\right ) x^4+2 a d x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}}+\frac {\left (4 a b \sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {-a+x^4}}{\left (-b+x^4\right )^{3/4} \left (b-\left (1-a^2 d\right ) x^4-2 a d x^8+d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.58 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.35 \[ \int \frac {-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (b+\left (-1+a^2 d\right ) x-2 a d x^2+d x^3\right )} \, dx=-\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x)^2 (-b+x)^3}}{b-x}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{(a-x)^2 x (-b+x)^3}}{-b+x}\right )\right )}{d^{3/4}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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