\(\int \frac {-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4}{\sqrt [4]{x (-a+x) (-b+x)^3} (a b^3 d-b^2 (3 a+b) d x+3 b (a+b) d x^2-(1+a d+3 b d) x^3+d x^4)} \, dx\) [2033]

Optimal result

Integrand size = 99, antiderivative size = 145 \[ \int \frac {-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4}{\sqrt [4]{x (-a+x) (-b+x)^3} \left (a b^3 d-b^2 (3 a+b) d x+3 b (a+b) d x^2-(1+a d+3 b d) x^3+d x^4\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a b^3 x+\left (-3 a b^2-b^3\right ) x^2+\left (3 a b+3 b^2\right ) x^3+(-a-3 b) x^4+x^5}}{x}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b^3 x+\left (-3 a b^2-b^3\right ) x^2+\left (3 a b+3 b^2\right ) x^3+(-a-3 b) x^4+x^5}}{x}\right )}{d^{3/4}} \]

[Out]

2*arctan(d^(1/4)*(a*b^3*x+(-3*a*b^2-b^3)*x^2+(3*a*b+3*b^2)*x^3+(-a-3*b)*x^4+x^5)^(1/4)/x)/d^(3/4)-2*arctanh(d^
(1/4)*(a*b^3*x+(-3*a*b^2-b^3)*x^2+(3*a*b+3*b^2)*x^3+(-a-3*b)*x^4+x^5)^(1/4)/x)/d^(3/4)

Rubi [F]

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

[In]

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

[Out]

(12*a*b*x^(1/4)*(-a + x)^(1/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x^2*(-b + x^4)^(5/4))/((-a + x^4)^(1/4)
*(-(a*b^3*d) + 3*a*b^2*(1 + b/(3*a))*d*x^4 - 3*a*b*(1 + b/a)*d*x^8 + (1 + (a + 3*b)*d)*x^12 - d*x^16)), x], x,
x^(1/4)])/((a - x)*(b - x)^3*x)^(1/4) + (8*b*x^(1/4)*(-a + x)^(1/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x
^6*(-b + x^4)^(5/4))/((-a + x^4)^(1/4)*(a*b^3*d - 3*a*b^2*(1 + b/(3*a))*d*x^4 + 3*a*b*(1 + b/a)*d*x^8 - (1 + (
a + 3*b)*d)*x^12 + d*x^16)), x], x, x^(1/4)])/((a - x)*(b - x)^3*x)^(1/4) + (4*x^(1/4)*(-a + x)^(1/4)*(-b + x)
^(3/4)*Defer[Subst][Defer[Int][(x^10*(-b + x^4)^(5/4))/((-a + x^4)^(1/4)*(a*b^3*d - 3*a*b^2*(1 + b/(3*a))*d*x^
4 + 3*a*b*(1 + b/a)*d*x^8 - (1 + (a + 3*b)*d)*x^12 + d*x^16)), x], x, x^(1/4)])/((a - x)*(b - x)^3*x)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \int \frac {-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4}{\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4} \left (a b^3 d-b^2 (3 a+b) d x+3 b (a+b) d x^2-(1+a d+3 b d) x^3+d x^4\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^3}} \\ & = \frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \int \frac {\sqrt [4]{-b+x} \left (3 a b^2+\left (-3 a b-2 b^2\right ) x+b x^2+x^3\right )}{\sqrt [4]{x} \sqrt [4]{-a+x} \left (a b^3 d-b^2 (3 a+b) d x+3 b (a+b) d x^2-(1+a d+3 b d) x^3+d x^4\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^3}} \\ & = \frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \int \frac {(-b+x)^{5/4} \left (-3 a b+2 b x+x^2\right )}{\sqrt [4]{x} \sqrt [4]{-a+x} \left (a b^3 d-b^2 (3 a+b) d x+3 b (a+b) d x^2-(1+a d+3 b d) x^3+d x^4\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^3}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-b+x^4\right )^{5/4} \left (-3 a b+2 b x^4+x^8\right )}{\sqrt [4]{-a+x^4} \left (a b^3 d-b^2 (3 a+b) d x^4+3 b (a+b) d x^8-(1+a d+3 b d) x^{12}+d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^3}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \left (\frac {3 a b x^2 \left (-b+x^4\right )^{5/4}}{\sqrt [4]{-a+x^4} \left (-a b^3 d+3 a b^2 \left (1+\frac {b}{3 a}\right ) d x^4-3 a b \left (1+\frac {b}{a}\right ) d x^8+(1+(a+3 b) d) x^{12}-d x^{16}\right )}+\frac {2 b x^6 \left (-b+x^4\right )^{5/4}}{\sqrt [4]{-a+x^4} \left (a b^3 d-3 a b^2 \left (1+\frac {b}{3 a}\right ) d x^4+3 a b \left (1+\frac {b}{a}\right ) d x^8-(1+(a+3 b) d) x^{12}+d x^{16}\right )}+\frac {x^{10} \left (-b+x^4\right )^{5/4}}{\sqrt [4]{-a+x^4} \left (a b^3 d-3 a b^2 \left (1+\frac {b}{3 a}\right ) d x^4+3 a b \left (1+\frac {b}{a}\right ) d x^8-(1+(a+3 b) d) x^{12}+d x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^3}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^{10} \left (-b+x^4\right )^{5/4}}{\sqrt [4]{-a+x^4} \left (a b^3 d-3 a b^2 \left (1+\frac {b}{3 a}\right ) d x^4+3 a b \left (1+\frac {b}{a}\right ) d x^8-(1+(a+3 b) d) x^{12}+d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^3}}+\frac {\left (8 b \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^6 \left (-b+x^4\right )^{5/4}}{\sqrt [4]{-a+x^4} \left (a b^3 d-3 a b^2 \left (1+\frac {b}{3 a}\right ) d x^4+3 a b \left (1+\frac {b}{a}\right ) d x^8-(1+(a+3 b) d) x^{12}+d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^3}}+\frac {\left (12 a b \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-b+x^4\right )^{5/4}}{\sqrt [4]{-a+x^4} \left (-a b^3 d+3 a b^2 \left (1+\frac {b}{3 a}\right ) d x^4-3 a b \left (1+\frac {b}{a}\right ) d x^8+(1+(a+3 b) d) x^{12}-d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.42 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.46 \[ \int \frac {-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4}{\sqrt [4]{x (-a+x) (-b+x)^3} \left (a b^3 d-b^2 (3 a+b) d x+3 b (a+b) d x^2-(1+a d+3 b d) x^3+d x^4\right )} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x) (-b+x)^3}}{x}\right )-\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x) (-b+x)^3}}{x}\right )\right )}{d^{3/4}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.74 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.65

method	result	size
pseudoelliptic	\(\frac {2 \arctan \left (\frac {\left (x \left (a -x \right ) \left (b -x \right )^{3}\right )^{\frac {1}{4}}}{x \left (\frac {1}{d}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x \left (\frac {1}{d}\right )^{\frac {1}{4}}+\left (x \left (a -x \right ) \left (b -x \right )^{3}\right )^{\frac {1}{4}}}{-x \left (\frac {1}{d}\right )^{\frac {1}{4}}+\left (x \left (a -x \right ) \left (b -x \right )^{3}\right )^{\frac {1}{4}}}\right )}{\left (\frac {1}{d}\right )^{\frac {1}{4}} d}\)	\(94\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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