\(\int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} (-a^2+(2 a-b^3 d) x+(-1+3 b^2 d) x^2-3 b d x^3+d x^4)} \, dx\) [2553]

Optimal result

Integrand size = 108, antiderivative size = 215 \[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a^2 b^3 x+\left (3 a^2 b^2+2 a b^3\right ) x^2+\left (-3 a^2 b-6 a b^2-b^3\right ) x^3+\left (a^2+6 a b+3 b^2\right ) x^4+(-2 a-3 b) x^5+x^6}}{a-x}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a^2 b^3 x+\left (3 a^2 b^2+2 a b^3\right ) x^2+\left (-3 a^2 b-6 a b^2-b^3\right ) x^3+\left (a^2+6 a b+3 b^2\right ) x^4+(-2 a-3 b) x^5+x^6}}{a-x}\right )}{d^{3/4}} \]

[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)/(a-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)/(a-x))/d^(3/4)

Rubi [F]

\[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=\int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx \]

[In]

Int[(a*b^3 - (6*a - b)*b^2*x + 9*a*b*x^2 - (4*a + 3*b)*x^3 + 2*x^4)/((x*(-a + x)^2*(-b + x)^3)^(1/4)*(-a^2 + (
2*a - b^3*d)*x + (-1 + 3*b^2*d)*x^2 - 3*b*d*x^3 + d*x^4)),x]

[Out]

(-4*a*b*x^(1/4)*Sqrt[-a + x]*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x^2*(-b + x^4)^(5/4))/(Sqrt[-a + x^4]*(a^
2 - 2*a*(1 - (b^3*d)/(2*a))*x^4 + (1 - 3*b^2*d)*x^8 + 3*b*d*x^12 - d*x^16)), x], x, x^(1/4)])/(-((a - x)^2*(b
- x)^3*x))^(1/4) + (4*(4*a - b)*x^(1/4)*Sqrt[-a + x]*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x^6*(-b + x^4)^(5
/4))/(Sqrt[-a + x^4]*(a^2 - 2*a*(1 - (b^3*d)/(2*a))*x^4 + (1 - 3*b^2*d)*x^8 + 3*b*d*x^12 - d*x^16)), 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^1
0*(-b + x^4)^(5/4))/(Sqrt[-a + x^4]*(a^2 - 2*a*(1 - (b^3*d)/(2*a))*x^4 + (1 - 3*b^2*d)*x^8 + 3*b*d*x^12 - d*x^
16)), 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 b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\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 [4]{-b+x} \left (-a b^2+\left (5 a b-b^2\right ) x+(-4 a-b) x^2+2 x^3\right )}{\sqrt [4]{x} \sqrt {-a+x} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\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 [4]{-b+x^4} \left (-a b^2+\left (5 a b-b^2\right ) x^4+(-4 a-b) x^8+2 x^{12}\right )}{\sqrt {-a+x^4} \left (-a^2+\left (2 a-b^3 d\right ) x^4+\left (-1+3 b^2 d\right ) x^8-3 b d x^{12}+d x^{16}\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 \frac {x^2 \left (b-x^4\right ) \sqrt [4]{-b+x^4} \left (a b-(4 a-b) x^4+2 x^8\right )}{\sqrt {-a+x^4} \left (a^2-\left (2 a-b^3 d\right ) x^4-\left (-1+3 b^2 d\right ) x^8+3 b d x^{12}-d x^{16}\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 \frac {x^2 \left (-b+x^4\right )^{5/4} \left (a b-(4 a-b) x^4+2 x^8\right )}{\sqrt {-a+x^4} \left (a^2-\left (2 a-b^3 d\right ) x^4-\left (-1+3 b^2 d\right ) x^8+3 b d x^{12}-d x^{16}\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 {a b x^2 \left (-b+x^4\right )^{5/4}}{\sqrt {-a+x^4} \left (a^2-2 a \left (1-\frac {b^3 d}{2 a}\right ) x^4+\left (1-3 b^2 d\right ) x^8+3 b d x^{12}-d x^{16}\right )}+\frac {(-4 a+b) x^6 \left (-b+x^4\right )^{5/4}}{\sqrt {-a+x^4} \left (a^2-2 a \left (1-\frac {b^3 d}{2 a}\right ) x^4+\left (1-3 b^2 d\right ) x^8+3 b d x^{12}-d x^{16}\right )}+\frac {2 x^{10} \left (-b+x^4\right )^{5/4}}{\sqrt {-a+x^4} \left (a^2-2 a \left (1-\frac {b^3 d}{2 a}\right ) x^4+\left (1-3 b^2 d\right ) x^8+3 b d x^{12}-d x^{16}\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} \left (-b+x^4\right )^{5/4}}{\sqrt {-a+x^4} \left (a^2-2 a \left (1-\frac {b^3 d}{2 a}\right ) x^4+\left (1-3 b^2 d\right ) x^8+3 b d x^{12}-d x^{16}\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 \left (-b+x^4\right )^{5/4}}{\sqrt {-a+x^4} \left (a^2-2 a \left (1-\frac {b^3 d}{2 a}\right ) x^4+\left (1-3 b^2 d\right ) x^8+3 b d x^{12}-d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}}-\frac {\left (4 (-4 a+b) \sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^6 \left (-b+x^4\right )^{5/4}}{\sqrt {-a+x^4} \left (a^2-2 a \left (1-\frac {b^3 d}{2 a}\right ) x^4+\left (1-3 b^2 d\right ) x^8+3 b d x^{12}-d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 15.48 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.35 \[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=-\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x)^2 (-b+x)^3}}{a-x}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{(a-x)^2 x (-b+x)^3}}{-a+x}\right )\right )}{d^{3/4}} \]

[In]

Integrate[(a*b^3 - (6*a - b)*b^2*x + 9*a*b*x^2 - (4*a + 3*b)*x^3 + 2*x^4)/((x*(-a + x)^2*(-b + x)^3)^(1/4)*(-a
^2 + (2*a - b^3*d)*x + (-1 + 3*b^2*d)*x^2 - 3*b*d*x^3 + d*x^4)),x]

[Out]

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

Maple [F]

\[\int \frac {a \,b^{3}-\left (6 a -b \right ) b^{2} x +9 a b \,x^{2}-\left (4 a +3 b \right ) x^{3}+2 x^{4}}{\left (x \left (-a +x \right )^{2} \left (-b +x \right )^{3}\right )^{\frac {1}{4}} \left (-a^{2}+\left (-b^{3} d +2 a \right ) x +\left (3 b^{2} d -1\right ) x^{2}-3 b d \,x^{3}+d \,x^{4}\right )}d x\]

[In]

int((a*b^3-(6*a-b)*b^2*x+9*a*b*x^2-(4*a+3*b)*x^3+2*x^4)/(x*(-a+x)^2*(-b+x)^3)^(1/4)/(-a^2+(-b^3*d+2*a)*x+(3*b^
2*d-1)*x^2-3*b*d*x^3+d*x^4),x)

[Out]

int((a*b^3-(6*a-b)*b^2*x+9*a*b*x^2-(4*a+3*b)*x^3+2*x^4)/(x*(-a+x)^2*(-b+x)^3)^(1/4)/(-a^2+(-b^3*d+2*a)*x+(3*b^
2*d-1)*x^2-3*b*d*x^3+d*x^4),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate((a*b^3-(6*a-b)*b^2*x+9*a*b*x^2-(4*a+3*b)*x^3+2*x^4)/(x*(-a+x)^2*(-b+x)^3)^(1/4)/(-a^2+(-b^3*d+2*a)*x
+(3*b^2*d-1)*x^2-3*b*d*x^3+d*x^4),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate((a*b**3-(6*a-b)*b**2*x+9*a*b*x**2-(4*a+3*b)*x**3+2*x**4)/(x*(-a+x)**2*(-b+x)**3)**(1/4)/(-a**2+(-b**
3*d+2*a)*x+(3*b**2*d-1)*x**2-3*b*d*x**3+d*x**4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=\int { -\frac {a b^{3} - {\left (6 \, a - b\right )} b^{2} x + 9 \, a b x^{2} - {\left (4 \, a + 3 \, b\right )} x^{3} + 2 \, x^{4}}{\left (-{\left (a - x\right )}^{2} {\left (b - x\right )}^{3} x\right )^{\frac {1}{4}} {\left (3 \, b d x^{3} - d x^{4} - {\left (3 \, b^{2} d - 1\right )} x^{2} + a^{2} + {\left (b^{3} d - 2 \, a\right )} x\right )}} \,d x } \]

[In]

integrate((a*b^3-(6*a-b)*b^2*x+9*a*b*x^2-(4*a+3*b)*x^3+2*x^4)/(x*(-a+x)^2*(-b+x)^3)^(1/4)/(-a^2+(-b^3*d+2*a)*x
+(3*b^2*d-1)*x^2-3*b*d*x^3+d*x^4),x, algorithm="maxima")

[Out]

-integrate((a*b^3 - (6*a - b)*b^2*x + 9*a*b*x^2 - (4*a + 3*b)*x^3 + 2*x^4)/((-(a - x)^2*(b - x)^3*x)^(1/4)*(3*
b*d*x^3 - d*x^4 - (3*b^2*d - 1)*x^2 + a^2 + (b^3*d - 2*a)*x)), x)

Giac [F]

\[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=\int { -\frac {a b^{3} - {\left (6 \, a - b\right )} b^{2} x + 9 \, a b x^{2} - {\left (4 \, a + 3 \, b\right )} x^{3} + 2 \, x^{4}}{\left (-{\left (a - x\right )}^{2} {\left (b - x\right )}^{3} x\right )^{\frac {1}{4}} {\left (3 \, b d x^{3} - d x^{4} - {\left (3 \, b^{2} d - 1\right )} x^{2} + a^{2} + {\left (b^{3} d - 2 \, a\right )} x\right )}} \,d x } \]

[In]

integrate((a*b^3-(6*a-b)*b^2*x+9*a*b*x^2-(4*a+3*b)*x^3+2*x^4)/(x*(-a+x)^2*(-b+x)^3)^(1/4)/(-a^2+(-b^3*d+2*a)*x
+(3*b^2*d-1)*x^2-3*b*d*x^3+d*x^4),x, algorithm="giac")

[Out]

integrate(-(a*b^3 - (6*a - b)*b^2*x + 9*a*b*x^2 - (4*a + 3*b)*x^3 + 2*x^4)/((-(a - x)^2*(b - x)^3*x)^(1/4)*(3*
b*d*x^3 - d*x^4 - (3*b^2*d - 1)*x^2 + a^2 + (b^3*d - 2*a)*x)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=\int \frac {a\,b^3-x^3\,\left (4\,a+3\,b\right )+2\,x^4-b^2\,x\,\left (6\,a-b\right )+9\,a\,b\,x^2}{{\left (-x\,{\left (a-x\right )}^2\,{\left (b-x\right )}^3\right )}^{1/4}\,\left (x^2\,\left (3\,b^2\,d-1\right )+x\,\left (2\,a-b^3\,d\right )+d\,x^4-a^2-3\,b\,d\,x^3\right )} \,d x \]

[In]

int((a*b^3 - x^3*(4*a + 3*b) + 2*x^4 - b^2*x*(6*a - b) + 9*a*b*x^2)/((-x*(a - x)^2*(b - x)^3)^(1/4)*(x^2*(3*b^
2*d - 1) + x*(2*a - b^3*d) + d*x^4 - a^2 - 3*b*d*x^3)),x)

[Out]

int((a*b^3 - x^3*(4*a + 3*b) + 2*x^4 - b^2*x*(6*a - b) + 9*a*b*x^2)/((-x*(a - x)^2*(b - x)^3)^(1/4)*(x^2*(3*b^
2*d - 1) + x*(2*a - b^3*d) + d*x^4 - a^2 - 3*b*d*x^3)), x)