\(\int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} (-a b d+(a+b) d x-d x^2+x^3)} \, dx\) [1073]

Optimal result

Integrand size = 54, antiderivative size = 81 \[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a b x+(-a-b) x^2+x^3}}{x}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b x+(-a-b) x^2+x^3}}{x}\right )}{d^{3/4}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 11.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.77 \[ \int \frac {3 a b-2 (a+b) x+x^2}{\sqrt [4]{x (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^3\right )} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x) (-b+x)}}{x}\right )-\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x) (-b+x)}}{x}\right )\right )}{d^{3/4}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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