\(\int \frac {2 a b x+(-3 a+b) x^2}{\sqrt [4]{x^2 (-a+x) (-b+x)} (a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3)} \, dx\) [1283]

Optimal result

Integrand size = 65, antiderivative size = 93 \[ \int \frac {2 a b x+(-3 a+b) x^2}{\sqrt [4]{x^2 (-a+x) (-b+x)} \left (a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right )}{d^{3/4}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(211\) vs. \(2(93)=186\).

Time = 33.04 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.27 \[ \int \frac {2 a b x+(-3 a+b) x^2}{\sqrt [4]{x^2 (-a+x) (-b+x)} \left (a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3\right )} \, dx=-\frac {x \sqrt [4]{\frac {-b+x}{a-x}} \left (\arctan \left (\frac {a-x \left (1+\sqrt {d} \sqrt {\frac {-b+x}{a-x}}\right )}{\sqrt {2} \sqrt [4]{d} \sqrt {\frac {x}{a-x}} (-a+x) \sqrt [4]{\frac {-b+x}{a-x}}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{d} x \sqrt [4]{\frac {-b+x}{a-x}}}{\sqrt {\frac {x}{2 a-2 x}} \left (a+x \left (-1+\sqrt {d} \sqrt {\frac {-b+x}{a-x}}\right )\right )}\right )\right )}{d^{3/4} \sqrt {\frac {x}{2 a-2 x}} \sqrt [4]{x^2 (-a+x) (-b+x)}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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