\(\int \frac {-2 a b x^2+(a+b) x^3}{(x^2 (-a+x) (-b+x))^{3/4} (-a b+(a+b) x+(-1+d) x^2)} \, dx\) [1076]

Optimal result

Integrand size = 53, antiderivative size = 81 \[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{3/4}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.56 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00

method	result	size
pseudoelliptic	\(\frac {\ln \left (\frac {d^{\frac {1}{4}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{-d^{\frac {1}{4}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{x \,d^{\frac {1}{4}}}\right )}{d^{\frac {3}{4}}}\)	\(81\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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