\(\int \frac {(-a+x) (-3 a b+(a+2 b) x) (-b^3+3 b^2 x-3 b x^2+x^3)}{x (x (-a+x) (-b+x)^2)^{3/4} (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3)} \, dx\) [2075]

Optimal result

Integrand size = 93, antiderivative size = 150 \[ \int \frac {(-a+x) (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x \left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx=-\frac {4 \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{x}-2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )+2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(4*(a - x)*(b - x)^2*AppellF1[-3/4, -1/4, -3/2, 1/4, x/a, x/b])/((-((a - x)*(b - x)^2*x))^(3/4)*(1 - x/a)^(1/4
)*Sqrt[1 - x/b]) - (4*(5*a + b)*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/2)*Defer[Subst][Defer[Int][((-a + x^4)^(1/4
)*(-b + x^4)^(3/2))/(-(b^2*x^4) + 2*b*x^8 + (-1 + d)*x^12 + a*(b - x^4)^2), x], x, x^(1/4)])/(-((a - x)*(b - x
)^2*x))^(3/4) + (12*(a + 2*b)*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/2)*Defer[Subst][Defer[Int][(x^4*(-a + x^4)^(1
/4)*(-b + x^4)^(3/2))/(-(b^2*x^4) + 2*b*x^8 + (-1 + d)*x^12 + a*(b - x^4)^2), x], x, x^(1/4)])/(b*(-((a - x)*(
b - x)^2*x))^(3/4)) - (12*(1 - d)*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/2)*Defer[Subst][Defer[Int][(x^8*(-a + x^4
)^(1/4)*(-b + x^4)^(3/2))/(-(b^2*x^4) + 2*b*x^8 + (-1 + d)*x^12 + a*(b - x^4)^2), x], x, x^(1/4)])/(b*(-((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]{-a+x} (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x^{7/4} (-b+x)^{3/2} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\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 \frac {\sqrt [4]{-a+x} (-3 a b+(a+2 b) x) \left (b^2-2 b x+x^2\right )}{x^{7/4} \sqrt {-b+x} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\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 \frac {\sqrt [4]{-a+x} (-b+x)^{3/2} (-3 a b+(a+2 b) x)}{x^{7/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{3/4}} \\ & = \frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2} \left (-3 a b+(a+2 b) x^4\right )}{x^4 \left (a b^2-b (2 a+b) x^4+(a+2 b) x^8+(-1+d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4}} \\ & = \frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \text {Subst}\left (\int \left (-\frac {3 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{b x^4}+\frac {\sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2} \left (-b (5 a+b)+3 (a+2 b) x^4-3 (1-d) x^8\right )}{b \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4}} \\ & = \frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2} \left (-b (5 a+b)+3 (a+2 b) x^4-3 (1-d) x^8\right )}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}} \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (12 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{x^4} \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4}} \\ & = \frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \text {Subst}\left (\int \left (\frac {(-5 a-b) b \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}}+\frac {3 (a+2 b) x^4 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}}+\frac {3 (-1+d) x^8 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}}\right ) \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (12 x^{3/4} (-a+x) (-b+x)^{3/2}\right ) \text {Subst}\left (\int \frac {\left (-b+x^4\right )^{3/2} \sqrt [4]{1-\frac {x^4}{a}}}{x^4} \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4} \sqrt [4]{1-\frac {x}{a}}} \\ & = -\frac {\left (4 (5 a+b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left (12 (a+2 b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}} \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (12 (1-d) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \text {Subst}\left (\int \frac {x^8 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}} \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left (12 x^{3/4} (-a+x) (-b+x)^2\right ) \text {Subst}\left (\int \frac {\sqrt [4]{1-\frac {x^4}{a}} \left (1-\frac {x^4}{b}\right )^{3/2}}{x^4} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \sqrt [4]{1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}} \\ & = \frac {4 (a-x) (b-x)^2 \operatorname {AppellF1}\left (-\frac {3}{4},-\frac {1}{4},-\frac {3}{2},\frac {1}{4},\frac {x}{a},\frac {x}{b}\right )}{\left (-\left ((a-x) (b-x)^2 x\right )\right )^{3/4} \sqrt [4]{1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}}-\frac {\left (4 (5 a+b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{-b^2 x^4+2 b x^8+(-1+d) x^{12}+a \left (b-x^4\right )^2} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left (12 (a+2 b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{-b^2 x^4+2 b x^8+(-1+d) x^{12}+a \left (b-x^4\right )^2} \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (12 (1-d) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \text {Subst}\left (\int \frac {x^8 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{-b^2 x^4+2 b x^8+(-1+d) x^{12}+a \left (b-x^4\right )^2} \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.73

method	result	size
pseudoelliptic	\(\frac {x \left (\ln \left (\frac {d^{\frac {1}{4}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}{-d^{\frac {1}{4}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}{x \,d^{\frac {1}{4}}}\right )\right ) d^{\frac {1}{4}}-4 \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}{x}\)	\(109\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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