\(\int \frac {x^3 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3)} \, dx\) [2250]

Optimal result

Integrand size = 85, antiderivative size = 168 \[ \int \frac {x^3 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx=\frac {4 \left (-a b^2 x+2 a b x^2+b^2 x^2-a x^3-2 b x^3+x^4\right )^{3/4}}{(-a+x) (-b+x)^2}+2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{x}\right )-2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{x}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(-4*(a + 2*b)*x^2*(1 - x/a)^(1/4)*Hypergeometric2F1[5/4, 7/4, 11/4, -(((a - b)*x)/(a*(b - x)))])/(7*a*b*(1 - d
)*(-((a - x)*(b - x)^2*x))^(1/4)*(1 - x/b)^(5/4)) + (16*(a^2*d + 4*b^2*d + a*b*(3 + d))*x*Gamma[7/4]*(11*b*(7*
b - 4*x)*(a - x)*Hypergeometric2F1[1, 5/4, 11/4, ((a - b)*x)/(b*(a - x))] + 20*(a - b)*(b - x)*x*Hypergeometri
c2F1[2, 9/4, 15/4, ((a - b)*x)/(b*(a - x))]))/(693*b^3*(1 - d)^2*(a - x)^2*(-((a - x)*(b - x)^2*x))^(1/4)*Gamm
a[3/4]) - (4*a*b^2*d*(a^2*d + 4*b^2*d + a*b*(3 + d))*x^(1/4)*(-a + x)^(1/4)*Sqrt[-b + x]*Defer[Subst][Defer[In
t][x^2/((-a + x^4)^(5/4)*(-b + x^4)^(3/2)*(a*b^2*d - 2*a*b*(1 + b/(2*a))*d*x^4 + a*(1 + (2*b)/a)*d*x^8 + (1 -
d)*x^12)), x], x, x^(1/4)])/((1 - d)^2*(-((a - x)*(b - x)^2*x))^(1/4)) + (4*b*d*(2*a^3*d + 4*b^3*d + a^2*b*(7
+ 2*d) + a*b^2*(5 + 7*d))*x^(1/4)*(-a + x)^(1/4)*Sqrt[-b + x]*Defer[Subst][Defer[Int][x^6/((-a + x^4)^(5/4)*(-
b + x^4)^(3/2)*(a*b^2*d - 2*a*b*(1 + b/(2*a))*d*x^4 + a*(1 + (2*b)/a)*d*x^8 + (1 - d)*x^12)), x], x, x^(1/4)])
/((1 - d)^2*(-((a - x)*(b - x)^2*x))^(1/4)) - (4*(a + 2*b)*d*(a*b*(5 - d) + a^2*d + b^2*(1 + 3*d))*x^(1/4)*(-a
+ x)^(1/4)*Sqrt[-b + x]*Defer[Subst][Defer[Int][x^10/((-a + x^4)^(5/4)*(-b + x^4)^(3/2)*(a*b^2*d - 2*a*b*(1 +
b/(2*a))*d*x^4 + a*(1 + (2*b)/a)*d*x^8 + (1 - d)*x^12)), x], x, x^(1/4)])/((1 - d)^2*(-((a - x)*(b - x)^2*x))
^(1/4))

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {x^{11/4} (-3 a b+(a+2 b) x)}{(-a+x)^{5/4} (-b+x)^{3/2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^2}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^{14} \left (-3 a b+(a+2 b) x^4\right )}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (-a b^2 d+b (2 a+b) d x^4-(a+2 b) d x^8+(-1+d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^2}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \left (\frac {\left (a^2 d+4 b^2 d+a b (3+d)\right ) x^2}{(1-d)^2 \left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2}}-\frac {(a+2 b) x^6}{(1-d) \left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2}}+\frac {x^2 \left (a b^2 d \left (a^2 d+4 b^2 d+a b (3+d)\right )-b d \left (2 a^3 d+4 b^3 d+a^2 b (7+2 d)+a b^2 (5+7 d)\right ) x^4+(a+2 b) d \left (a b (5-d)+a^2 d+b^2 (1+3 d)\right ) x^8\right )}{(-1+d)^2 \left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (-a b^2 d+b (2 a+b) d x^4-(a+2 b) d x^8+(-1+d) x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^2}} \\ & = -\frac {\left (4 (a+2 b) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^6}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2}} \, dx,x,\sqrt [4]{x}\right )}{(1-d) \sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2 \left (a b^2 d \left (a^2 d+4 b^2 d+a b (3+d)\right )-b d \left (2 a^3 d+4 b^3 d+a^2 b (7+2 d)+a b^2 (5+7 d)\right ) x^4+(a+2 b) d \left (a b (5-d)+a^2 d+b^2 (1+3 d)\right ) x^8\right )}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (-a b^2 d+b (2 a+b) d x^4-(a+2 b) d x^8+(-1+d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (4 \left (a^2 d+4 b^2 d+a b (3+d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2}} \, dx,x,\sqrt [4]{x}\right )}{(1-d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \left (\frac {a b^2 d \left (-a^2 d-4 b^2 d-a b (3+d)\right ) x^2}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )}+\frac {b d \left (2 a^3 d+4 b^3 d+a^2 b (7+2 d)+a b^2 (5+7 d)\right ) x^6}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )}+\frac {(a+2 b) d \left (-a b (5-d)-a^2 d-b^2 (1+3 d)\right ) x^{10}}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (4 (a+2 b) \sqrt [4]{x} \sqrt {-b+x} \sqrt [4]{1-\frac {x}{a}}\right ) \text {Subst}\left (\int \frac {x^6}{\left (-b+x^4\right )^{3/2} \left (1-\frac {x^4}{a}\right )^{5/4}} \, dx,x,\sqrt [4]{x}\right )}{a (1-d) \sqrt [4]{x (-a+x) (-b+x)^2}}-\frac {\left (4 \left (a^2 d+4 b^2 d+a b (3+d)\right ) \sqrt [4]{x} \sqrt {-b+x} \sqrt [4]{1-\frac {x}{a}}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-b+x^4\right )^{3/2} \left (1-\frac {x^4}{a}\right )^{5/4}} \, dx,x,\sqrt [4]{x}\right )}{a (1-d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}} \\ & = -\frac {\left (4 a b^2 d \left (a^2 d+4 b^2 d+a b (3+d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}-\frac {\left (4 (a+2 b) d \left (a b (5-d)+a^2 d+b^2 (1+3 d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^{10}}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (4 b d \left (2 a^3 d+4 b^3 d+a^2 b (7+2 d)+a b^2 (5+7 d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^6}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}-\frac {\left (4 (a+2 b) \sqrt [4]{x} \sqrt [4]{1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \text {Subst}\left (\int \frac {x^6}{\left (1-\frac {x^4}{a}\right )^{5/4} \left (1-\frac {x^4}{b}\right )^{3/2}} \, dx,x,\sqrt [4]{x}\right )}{a b (1-d) \sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (4 \left (a^2 d+4 b^2 d+a b (3+d)\right ) \sqrt [4]{x} \sqrt [4]{1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1-\frac {x^4}{a}\right )^{5/4} \left (1-\frac {x^4}{b}\right )^{3/2}} \, dx,x,\sqrt [4]{x}\right )}{a b (1-d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}} \\ & = -\frac {4 (a+2 b) x^2 \sqrt [4]{1-\frac {x}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {7}{4},\frac {11}{4},\frac {\left (\frac {1}{a}-\frac {1}{b}\right ) b x}{b-x}\right )}{7 a b (1-d) \sqrt [4]{-\left ((a-x) (b-x)^2 x\right )} \left (1-\frac {x}{b}\right )^{5/4}}+\frac {16 \left (a^2 d+4 b^2 d+a b (3+d)\right ) x \operatorname {Gamma}\left (\frac {7}{4}\right ) \left (11 b (7 b-4 x) (a-x) \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4},\frac {11}{4},\frac {(a-b) x}{b (a-x)}\right )+20 (a-b) (b-x) x \operatorname {Hypergeometric2F1}\left (2,\frac {9}{4},\frac {15}{4},\frac {(a-b) x}{b (a-x)}\right )\right )}{693 b^3 (1-d)^2 (a-x)^2 \sqrt [4]{-\left ((a-x) (b-x)^2 x\right )} \operatorname {Gamma}\left (\frac {3}{4}\right )}-\frac {\left (4 a b^2 d \left (a^2 d+4 b^2 d+a b (3+d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}-\frac {\left (4 (a+2 b) d \left (a b (5-d)+a^2 d+b^2 (1+3 d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^{10}}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (4 b d \left (2 a^3 d+4 b^3 d+a^2 b (7+2 d)+a b^2 (5+7 d)\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^6}{\left (-a+x^4\right )^{5/4} \left (-b+x^4\right )^{3/2} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{(-1+d)^2 \sqrt [4]{x (-a+x) (-b+x)^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 41.66 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx=\frac {4 \sqrt {\frac {b-x}{a-x}} x+2 \sqrt [4]{d} \sqrt [4]{\frac {x}{-a+x}} (-b+x) \arctan \left (\frac {\sqrt [4]{d} \sqrt {\frac {b-x}{a-x}}}{\left (\frac {x}{-a+x}\right )^{3/4}}\right )-2 \sqrt [4]{d} \sqrt [4]{\frac {x}{-a+x}} (-b+x) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt {\frac {b-x}{a-x}}}{\left (\frac {x}{-a+x}\right )^{3/4}}\right )}{\sqrt {\frac {b-x}{a-x}} \sqrt [4]{(b-x)^2 x (-a+x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.91

method	result	size
pseudoelliptic	\(-\frac {\ln \left (\frac {x \left (\frac {1}{d}\right )^{\frac {1}{4}}+\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}{-x \left (\frac {1}{d}\right )^{\frac {1}{4}}+\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}\right ) \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}-4 x \left (\frac {1}{d}\right )^{\frac {1}{4}}-2 \arctan \left (\frac {\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}{x \left (\frac {1}{d}\right )^{\frac {1}{4}}}\right ) \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}{\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}} \left (\frac {1}{d}\right )^{\frac {1}{4}}}\)	\(153\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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