\(\int \frac {-2 a b x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} (-a b d+(a+b) d x+(1-d) x^2)} \, dx\) [1835]

Optimal result

Integrand size = 71, antiderivative size = 125 \[ \int \frac {-2 a b x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x+(1-d) x^2\right )} \, dx=-\frac {4 \left (a b x^2+(-a-b) x^3+x^4\right )^{3/4}}{x (-a+x) (-b+x)}+2 \sqrt [4]{d} \arctan \left (\frac {x}{\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )+2 \sqrt [4]{d} \text {arctanh}\left (\frac {x}{\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 67.10 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.37 \[ \int \frac {-2 a b x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x+(1-d) x^2\right )} \, dx=-\frac {2 x \left (2 \sqrt {\frac {x}{-a+x}}+\sqrt [4]{d} \sqrt [4]{\frac {b-x}{a-x}} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b-x}{a-x}}}{\sqrt {\frac {x}{-a+x}}}\right )-\sqrt [4]{d} \sqrt [4]{\frac {b-x}{a-x}} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b-x}{a-x}}}{\sqrt {\frac {x}{-a+x}}}\right )\right )}{\sqrt {\frac {x}{-a+x}} \sqrt [4]{x^2 (-a+x) (-b+x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.08

method	result	size
pseudoelliptic	\(-\frac {2 \left (2 x \left (\frac {1}{d}\right )^{\frac {1}{4}}+\frac {\left (2 \arctan \left (\frac {\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{x \left (\frac {1}{d}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x \left (\frac {1}{d}\right )^{\frac {1}{4}}+\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{-x \left (\frac {1}{d}\right )^{\frac {1}{4}}+\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}\right )\right ) \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{2}\right )}{\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}} \left (\frac {1}{d}\right )^{\frac {1}{4}}}\)	\(135\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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