\(\int \frac {g+h x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}} \, dx\) [146]

Optimal result

Integrand size = 29, antiderivative size = 582 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}} \, dx=-\frac {d (3 b d e g-b c (2 f g+e h)-a (d f g+2 d e h-3 c f h)) \sqrt {e+f x}}{2 (b c-a d)^2 (b e-a f) (d e-c f) (c+d x)^2}-\frac {(b g-a h) \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}+\frac {d \left (a^2 d f (3 d f g-4 d e h+c f h)-b^2 \left (12 d^2 e^2 g-c d e (19 f g+4 e h)+c^2 f (4 f g+7 e h)\right )+a b \left (11 c^2 f^2 h+d^2 e (5 f g+8 e h)-c d f (11 f g+13 e h)\right )\right ) \sqrt {e+f x}}{4 (b c-a d)^3 (b e-a f) (d e-c f)^2 (c+d x)}+\frac {b^{3/2} \left (5 a^2 d f h+b^2 (6 d e g+c f g-2 c e h)-a b (7 d f g+4 d e h-c f h)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d)^4 (b e-a f)^{3/2}}-\frac {\sqrt {d} \left (a^2 d^2 f (3 d f g-4 d e h+c f h)-2 a b d \left (5 c^2 f^2 h+c d f (7 f g-16 e h)-4 d^2 e (f g-2 e h)\right )+b^2 \left (24 d^3 e^2 g-15 c^3 f^2 h-8 c d^2 e (7 f g+e h)+5 c^2 d f (7 f g+4 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 (b c-a d)^4 (d e-c f)^{5/2}} \] Output:

-1/2*d*(3*b*d*e*g-b*c*(e*h+2*f*g)-a*(-3*c*f*h+2*d*e*h+d*f*g))*(f*x+e)^(1/2 
)/(-a*d+b*c)^2/(-a*f+b*e)/(-c*f+d*e)/(d*x+c)^2-(-a*h+b*g)*(f*x+e)^(1/2)/(- 
a*d+b*c)/(-a*f+b*e)/(b*x+a)/(d*x+c)^2+1/4*d*(a^2*d*f*(c*f*h-4*d*e*h+3*d*f* 
g)-b^2*(12*d^2*e^2*g-c*d*e*(4*e*h+19*f*g)+c^2*f*(7*e*h+4*f*g))+a*b*(11*c^2 
*f^2*h+d^2*e*(8*e*h+5*f*g)-c*d*f*(13*e*h+11*f*g)))*(f*x+e)^(1/2)/(-a*d+b*c 
)^3/(-a*f+b*e)/(-c*f+d*e)^2/(d*x+c)+b^(3/2)*(5*a^2*d*f*h+b^2*(-2*c*e*h+c*f 
*g+6*d*e*g)-a*b*(-c*f*h+4*d*e*h+7*d*f*g))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(- 
a*f+b*e)^(1/2))/(-a*d+b*c)^4/(-a*f+b*e)^(3/2)-1/4*d^(1/2)*(a^2*d^2*f*(c*f* 
h-4*d*e*h+3*d*f*g)-2*a*b*d*(5*c^2*f^2*h+c*d*f*(-16*e*h+7*f*g)-4*d^2*e*(-2* 
e*h+f*g))+b^2*(24*d^3*e^2*g-15*c^3*f^2*h-8*c*d^2*e*(e*h+7*f*g)+5*c^2*d*f*( 
4*e*h+7*f*g)))*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/(-a*d+b*c)^ 
4/(-c*f+d*e)^(5/2)
 

Mathematica [A] (verified)

Time = 12.23 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.25 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}} \, dx=\frac {\frac {(b c-a d) \sqrt {e+f x} \left (a^3 d^2 f \left (-c^2 f h+c d (5 f g-2 e h+f h x)+d^2 (3 f g x-2 e (g+2 h x))\right )+a^2 b d \left (9 c^3 f^2 h+c^2 d f (-13 f g-5 e h+6 f h x)+c d^2 \left (2 e^2 h+e f (5 g-7 h x)+f^2 x (-6 g+h x)\right )+d^3 \left (3 f^2 g x^2+e f x (3 g-4 h x)+2 e^2 (g+2 h x)\right )\right )-b^3 \left (4 c^4 f^2 g+12 d^4 e^2 g x^2+c d^3 e x (18 e g-19 f g x-4 e h x)+c^3 d f (-8 e g+8 f g x+9 e h x)+c^2 d^2 \left (4 f^2 g x^2+e^2 (4 g-6 h x)+e f x (-29 g+7 h x)\right )\right )+a b^2 \left (4 c^4 f^2 h+17 c^3 d f h (-e+f x)+d^4 e x (-6 e g+5 f g x+8 e h x)+c d^3 \left (-11 f^2 g x^2+e f x (16 g-13 h x)-2 e^2 (5 g-7 h x)\right )+c^2 d^2 \left (10 e^2 h+e f (13 g-28 h x)+f^2 x (-13 g+11 h x)\right )\right )\right )}{(b e-a f) (d e-c f)^2 (a+b x) (c+d x)^2}+\frac {4 b^{3/2} \left (5 a^2 d f h+b^2 (6 d e g+c f g-2 c e h)+a b (-7 d f g-4 d e h+c f h)\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{(-b e+a f)^{3/2}}+\frac {\sqrt {d} \left (a^2 d^2 f (3 d f g-4 d e h+c f h)-2 a b d \left (5 c^2 f^2 h+c d f (7 f g-16 e h)+4 d^2 e (-f g+2 e h)\right )+b^2 \left (24 d^3 e^2 g-15 c^3 f^2 h-8 c d^2 e (7 f g+e h)+5 c^2 d f (7 f g+4 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{(-d e+c f)^{5/2}}}{4 (b c-a d)^4} \] Input:

Integrate[(g + h*x)/((a + b*x)^2*(c + d*x)^3*Sqrt[e + f*x]),x]
 

Output:

(((b*c - a*d)*Sqrt[e + f*x]*(a^3*d^2*f*(-(c^2*f*h) + c*d*(5*f*g - 2*e*h + 
f*h*x) + d^2*(3*f*g*x - 2*e*(g + 2*h*x))) + a^2*b*d*(9*c^3*f^2*h + c^2*d*f 
*(-13*f*g - 5*e*h + 6*f*h*x) + c*d^2*(2*e^2*h + e*f*(5*g - 7*h*x) + f^2*x* 
(-6*g + h*x)) + d^3*(3*f^2*g*x^2 + e*f*x*(3*g - 4*h*x) + 2*e^2*(g + 2*h*x) 
)) - b^3*(4*c^4*f^2*g + 12*d^4*e^2*g*x^2 + c*d^3*e*x*(18*e*g - 19*f*g*x - 
4*e*h*x) + c^3*d*f*(-8*e*g + 8*f*g*x + 9*e*h*x) + c^2*d^2*(4*f^2*g*x^2 + e 
^2*(4*g - 6*h*x) + e*f*x*(-29*g + 7*h*x))) + a*b^2*(4*c^4*f^2*h + 17*c^3*d 
*f*h*(-e + f*x) + d^4*e*x*(-6*e*g + 5*f*g*x + 8*e*h*x) + c*d^3*(-11*f^2*g* 
x^2 + e*f*x*(16*g - 13*h*x) - 2*e^2*(5*g - 7*h*x)) + c^2*d^2*(10*e^2*h + e 
*f*(13*g - 28*h*x) + f^2*x*(-13*g + 11*h*x)))))/((b*e - a*f)*(d*e - c*f)^2 
*(a + b*x)*(c + d*x)^2) + (4*b^(3/2)*(5*a^2*d*f*h + b^2*(6*d*e*g + c*f*g - 
 2*c*e*h) + a*b*(-7*d*f*g - 4*d*e*h + c*f*h))*ArcTan[(Sqrt[b]*Sqrt[e + f*x 
])/Sqrt[-(b*e) + a*f]])/(-(b*e) + a*f)^(3/2) + (Sqrt[d]*(a^2*d^2*f*(3*d*f* 
g - 4*d*e*h + c*f*h) - 2*a*b*d*(5*c^2*f^2*h + c*d*f*(7*f*g - 16*e*h) + 4*d 
^2*e*(-(f*g) + 2*e*h)) + b^2*(24*d^3*e^2*g - 15*c^3*f^2*h - 8*c*d^2*e*(7*f 
*g + e*h) + 5*c^2*d*f*(7*f*g + 4*e*h)))*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqr 
t[-(d*e) + c*f]])/(-(d*e) + c*f)^(5/2))/(4*(b*c - a*d)^4)
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 649, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {168, 27, 168, 168, 27, 174, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {g+h x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}} \, dx\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\int \frac {a c f h-2 a d (f g+2 e h)+b (6 d e g+c f g-2 c e h)+5 d f (b g-a h) x}{2 (a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {a c f h-2 a d (f g+2 e h)+b (6 d e g+c f g-2 c e h)+5 d f (b g-a h) x}{(a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\frac {\int \frac {-d f (3 d f g-4 d e h+c f h) a^2-b \left (e (5 f g+8 e h) d^2-c f (8 f g+7 e h) d+2 c^2 f^2 h\right ) a+2 b^2 (d e-c f) (6 d e g+c f g-2 c e h)+3 b d f (3 b d e g-b c (2 f g+e h)-a (d f g+2 d e h-3 c f h)) x}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (d e-c f)}+\frac {d \sqrt {e+f x} (b (-c e h-2 c f g+3 d e g)-a (-3 c f h+2 d e h+d f g))}{(c+d x)^2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\frac {\frac {\int \frac {-d^2 f^2 (3 d f g-4 d e h+c f h) a^3+b d f \left (-e (5 f g-12 e h) d^2+c f (11 f g-27 e h) d+9 c^2 f^2 h\right ) a^2+b^2 \left (-16 e^2 (f g+e h) d^3+c e f (37 f g+32 e h) d^2-c^2 f^2 (24 f g+17 e h) d+4 c^3 f^3 h\right ) a+4 b^3 (d e-c f)^2 (6 d e g+c f g-2 c e h)-b d f \left (d f (3 d f g-4 d e h+c f h) a^2+b \left (e (5 f g+8 e h) d^2-c f (11 f g+13 e h) d+11 c^2 f^2 h\right ) a-b^2 \left (f (4 f g+7 e h) c^2-d e (19 f g+4 e h) c+12 d^2 e^2 g\right )\right ) x}{2 (a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}-\frac {d \sqrt {e+f x} \left (a^2 d f (c f h-4 d e h+3 d f g)+a b \left (11 c^2 f^2 h-c d f (13 e h+11 f g)+d^2 e (8 e h+5 f g)\right )-b^2 \left (c^2 f (7 e h+4 f g)-c d e (4 e h+19 f g)+12 d^2 e^2 g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}}{2 (b c-a d) (d e-c f)}+\frac {d \sqrt {e+f x} (b (-c e h-2 c f g+3 d e g)-a (-3 c f h+2 d e h+d f g))}{(c+d x)^2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {\int \frac {-d^2 f^2 (3 d f g-4 d e h+c f h) a^3+b d f \left (-e (5 f g-12 e h) d^2+c f (11 f g-27 e h) d+9 c^2 f^2 h\right ) a^2+b^2 \left (-16 e^2 (f g+e h) d^3+c e f (37 f g+32 e h) d^2-c^2 f^2 (24 f g+17 e h) d+4 c^3 f^3 h\right ) a+4 b^3 (d e-c f)^2 (6 d e g+c f g-2 c e h)-b d f \left (d f (3 d f g-4 d e h+c f h) a^2+b \left (e (5 f g+8 e h) d^2-c f (11 f g+13 e h) d+11 c^2 f^2 h\right ) a-b^2 \left (f (4 f g+7 e h) c^2-d e (19 f g+4 e h) c+12 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{2 (b c-a d) (d e-c f)}-\frac {d \sqrt {e+f x} \left (a^2 d f (c f h-4 d e h+3 d f g)+a b \left (11 c^2 f^2 h-c d f (13 e h+11 f g)+d^2 e (8 e h+5 f g)\right )-b^2 \left (c^2 f (7 e h+4 f g)-c d e (4 e h+19 f g)+12 d^2 e^2 g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}}{2 (b c-a d) (d e-c f)}+\frac {d \sqrt {e+f x} (b (-c e h-2 c f g+3 d e g)-a (-3 c f h+2 d e h+d f g))}{(c+d x)^2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {\frac {\frac {\frac {4 b^2 (d e-c f)^2 \left (5 a^2 d f h-a b (-c f h+4 d e h+7 d f g)+b^2 (-2 c e h+c f g+6 d e g)\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}-\frac {d (b e-a f) \left (a^2 d^2 f (c f h-4 d e h+3 d f g)-2 a b d \left (5 c^2 f^2 h+c d f (7 f g-16 e h)-4 d^2 e (f g-2 e h)\right )+b^2 \left (-15 c^3 f^2 h+5 c^2 d f (4 e h+7 f g)-8 c d^2 e (e h+7 f g)+24 d^3 e^2 g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}}{2 (b c-a d) (d e-c f)}-\frac {d \sqrt {e+f x} \left (a^2 d f (c f h-4 d e h+3 d f g)+a b \left (11 c^2 f^2 h-c d f (13 e h+11 f g)+d^2 e (8 e h+5 f g)\right )-b^2 \left (c^2 f (7 e h+4 f g)-c d e (4 e h+19 f g)+12 d^2 e^2 g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}}{2 (b c-a d) (d e-c f)}+\frac {d \sqrt {e+f x} (b (-c e h-2 c f g+3 d e g)-a (-3 c f h+2 d e h+d f g))}{(c+d x)^2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\frac {\frac {\frac {8 b^2 (d e-c f)^2 \left (5 a^2 d f h-a b (-c f h+4 d e h+7 d f g)+b^2 (-2 c e h+c f g+6 d e g)\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{f (b c-a d)}-\frac {2 d (b e-a f) \left (a^2 d^2 f (c f h-4 d e h+3 d f g)-2 a b d \left (5 c^2 f^2 h+c d f (7 f g-16 e h)-4 d^2 e (f g-2 e h)\right )+b^2 \left (-15 c^3 f^2 h+5 c^2 d f (4 e h+7 f g)-8 c d^2 e (e h+7 f g)+24 d^3 e^2 g\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{f (b c-a d)}}{2 (b c-a d) (d e-c f)}-\frac {d \sqrt {e+f x} \left (a^2 d f (c f h-4 d e h+3 d f g)+a b \left (11 c^2 f^2 h-c d f (13 e h+11 f g)+d^2 e (8 e h+5 f g)\right )-b^2 \left (c^2 f (7 e h+4 f g)-c d e (4 e h+19 f g)+12 d^2 e^2 g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}}{2 (b c-a d) (d e-c f)}+\frac {d \sqrt {e+f x} (b (-c e h-2 c f g+3 d e g)-a (-3 c f h+2 d e h+d f g))}{(c+d x)^2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {\frac {\frac {2 \sqrt {d} (b e-a f) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) \left (a^2 d^2 f (c f h-4 d e h+3 d f g)-2 a b d \left (5 c^2 f^2 h+c d f (7 f g-16 e h)-4 d^2 e (f g-2 e h)\right )+b^2 \left (-15 c^3 f^2 h+5 c^2 d f (4 e h+7 f g)-8 c d^2 e (e h+7 f g)+24 d^3 e^2 g\right )\right )}{(b c-a d) \sqrt {d e-c f}}-\frac {8 b^{3/2} (d e-c f)^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right ) \left (5 a^2 d f h-a b (-c f h+4 d e h+7 d f g)+b^2 (-2 c e h+c f g+6 d e g)\right )}{(b c-a d) \sqrt {b e-a f}}}{2 (b c-a d) (d e-c f)}-\frac {d \sqrt {e+f x} \left (a^2 d f (c f h-4 d e h+3 d f g)+a b \left (11 c^2 f^2 h-c d f (13 e h+11 f g)+d^2 e (8 e h+5 f g)\right )-b^2 \left (c^2 f (7 e h+4 f g)-c d e (4 e h+19 f g)+12 d^2 e^2 g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}}{2 (b c-a d) (d e-c f)}+\frac {d \sqrt {e+f x} (b (-c e h-2 c f g+3 d e g)-a (-3 c f h+2 d e h+d f g))}{(c+d x)^2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}\)

Input:

Int[(g + h*x)/((a + b*x)^2*(c + d*x)^3*Sqrt[e + f*x]),x]
 

Output:

-(((b*g - a*h)*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x) 
^2)) - ((d*(b*(3*d*e*g - 2*c*f*g - c*e*h) - a*(d*f*g + 2*d*e*h - 3*c*f*h)) 
*Sqrt[e + f*x])/((b*c - a*d)*(d*e - c*f)*(c + d*x)^2) + (-((d*(a^2*d*f*(3* 
d*f*g - 4*d*e*h + c*f*h) - b^2*(12*d^2*e^2*g - c*d*e*(19*f*g + 4*e*h) + c^ 
2*f*(4*f*g + 7*e*h)) + a*b*(11*c^2*f^2*h + d^2*e*(5*f*g + 8*e*h) - c*d*f*( 
11*f*g + 13*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(d*e - c*f)*(c + d*x))) + ( 
(-8*b^(3/2)*(d*e - c*f)^2*(5*a^2*d*f*h + b^2*(6*d*e*g + c*f*g - 2*c*e*h) - 
 a*b*(7*d*f*g + 4*d*e*h - c*f*h))*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e 
 - a*f]])/((b*c - a*d)*Sqrt[b*e - a*f]) + (2*Sqrt[d]*(b*e - a*f)*(a^2*d^2* 
f*(3*d*f*g - 4*d*e*h + c*f*h) - 2*a*b*d*(5*c^2*f^2*h + c*d*f*(7*f*g - 16*e 
*h) - 4*d^2*e*(f*g - 2*e*h)) + b^2*(24*d^3*e^2*g - 15*c^3*f^2*h - 8*c*d^2* 
e*(7*f*g + e*h) + 5*c^2*d*f*(7*f*g + 4*e*h)))*ArcTanh[(Sqrt[d]*Sqrt[e + f* 
x])/Sqrt[d*e - c*f]])/((b*c - a*d)*Sqrt[d*e - c*f]))/(2*(b*c - a*d)*(d*e - 
 c*f)))/(2*(b*c - a*d)*(d*e - c*f)))/(2*(b*c - a*d)*(b*e - a*f))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

Maple [A] (verified)

Time = 3.36 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(2 f^{3} \left (-\frac {d \left (\frac {-\frac {d f \left (a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h +3 a^{2} d^{3} f g +6 a b \,c^{2} d f h -14 a b c \,d^{2} f g +8 a b \,d^{3} e g -7 b^{2} c^{3} f h +4 b^{2} c^{2} d e h +11 b^{2} c^{2} d f g -8 b^{2} c \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}+\frac {\left (a^{2} c \,d^{2} f h +4 a^{2} d^{3} e h -5 a^{2} d^{3} f g -10 a b \,c^{2} d f h +18 a b c \,d^{2} f g -8 a b \,d^{3} e g +9 b^{2} c^{3} f h -4 b^{2} c^{2} d e h -13 b^{2} c^{2} d f g +8 b^{2} c \,d^{2} e g \right ) f \sqrt {f x +e}}{8 c f -8 d e}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}-\frac {\left (a^{2} c \,d^{2} f^{2} h -4 a^{2} d^{3} e f h +3 a^{2} d^{3} f^{2} g -10 a b \,c^{2} d \,f^{2} h +32 a b c \,d^{2} e f h -14 a b c \,d^{2} f^{2} g -16 e^{2} h b a \,d^{3}+8 a b \,d^{3} e f g -15 b^{2} c^{3} f^{2} h +20 b^{2} c^{2} d e f h +35 b^{2} c^{2} d \,f^{2} g -8 b^{2} c \,d^{2} e^{2} h -56 b^{2} c \,d^{2} e f g +24 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {\left (c f -d e \right ) d}}\right )}{f^{3} \left (a d -b c \right )^{4}}+\frac {b^{2} \left (\frac {f \left (a^{2} d h -a b c h -a b d g +b^{2} c g \right ) \sqrt {f x +e}}{2 \left (a f -b e \right ) \left (\left (f x +e \right ) b +a f -b e \right )}+\frac {\left (5 a^{2} d f h +a b c f h -4 a b d e h -7 a b d f g -2 b^{2} c e h +b^{2} c f g +6 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}\right )}{f^{3} \left (a d -b c \right )^{4}}\right )\)	\(706\)
default	\(2 f^{3} \left (-\frac {d \left (\frac {-\frac {d f \left (a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h +3 a^{2} d^{3} f g +6 a b \,c^{2} d f h -14 a b c \,d^{2} f g +8 a b \,d^{3} e g -7 b^{2} c^{3} f h +4 b^{2} c^{2} d e h +11 b^{2} c^{2} d f g -8 b^{2} c \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}+\frac {\left (a^{2} c \,d^{2} f h +4 a^{2} d^{3} e h -5 a^{2} d^{3} f g -10 a b \,c^{2} d f h +18 a b c \,d^{2} f g -8 a b \,d^{3} e g +9 b^{2} c^{3} f h -4 b^{2} c^{2} d e h -13 b^{2} c^{2} d f g +8 b^{2} c \,d^{2} e g \right ) f \sqrt {f x +e}}{8 c f -8 d e}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}-\frac {\left (a^{2} c \,d^{2} f^{2} h -4 a^{2} d^{3} e f h +3 a^{2} d^{3} f^{2} g -10 a b \,c^{2} d \,f^{2} h +32 a b c \,d^{2} e f h -14 a b c \,d^{2} f^{2} g -16 e^{2} h b a \,d^{3}+8 a b \,d^{3} e f g -15 b^{2} c^{3} f^{2} h +20 b^{2} c^{2} d e f h +35 b^{2} c^{2} d \,f^{2} g -8 b^{2} c \,d^{2} e^{2} h -56 b^{2} c \,d^{2} e f g +24 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {\left (c f -d e \right ) d}}\right )}{f^{3} \left (a d -b c \right )^{4}}+\frac {b^{2} \left (\frac {f \left (a^{2} d h -a b c h -a b d g +b^{2} c g \right ) \sqrt {f x +e}}{2 \left (a f -b e \right ) \left (\left (f x +e \right ) b +a f -b e \right )}+\frac {\left (5 a^{2} d f h +a b c f h -4 a b d e h -7 a b d f g -2 b^{2} c e h +b^{2} c f g +6 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}\right )}{f^{3} \left (a d -b c \right )^{4}}\right )\)	\(706\)
pseudoelliptic	\(-\frac {-\sqrt {\left (a f -b e \right ) b}\, \left (\left (24 b^{2} e^{2} g +8 a \left (-2 e^{2} h +e f g \right ) b +\left (-4 e f h +3 f^{2} g \right ) a^{2}\right ) d^{3}+c \left (8 \left (-e^{2} h -7 e f g \right ) b^{2}+2 \left (16 e f h -7 f^{2} g \right ) a b +a^{2} f^{2} h \right ) d^{2}-10 c^{2} \left (\left (-2 e h -\frac {7 f g}{2}\right ) b +a f h \right ) b f d -15 b^{2} c^{3} f^{2} h \right ) d \left (b x +a \right ) \left (x d +c \right )^{2} \left (a f -b e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\left (-20 \left (\left (\frac {6 b^{2} e g}{5}-\frac {4 a \left (e h +\frac {7 f g}{4}\right ) b}{5}+a^{2} f h \right ) d +\frac {c b \left (\left (-2 e h +f g \right ) b +a f h \right )}{5}\right ) \left (b x +a \right ) \left (x d +c \right )^{2} \left (c f -d e \right )^{2} b^{2} \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+\sqrt {\left (a f -b e \right ) b}\, \left (a d -b c \right ) \left (2 \left (6 b^{3} e^{2} g \,x^{2}+3 a x \left (-\frac {5 f g x}{6}+e \left (-\frac {4 h x}{3}+g \right )\right ) e \,b^{2}-a^{2} \left (\frac {3 x^{2} g \,f^{2}}{2}+\frac {3 x \left (-\frac {4 h x}{3}+g \right ) e f}{2}+e^{2} \left (2 h x +g \right )\right ) b +\left (-\frac {3 f g x}{2}+e \left (2 h x +g \right )\right ) a^{3} f \right ) d^{4}+2 c \left (\left (-\frac {19 x^{2} g f e}{2}+9 x \left (-\frac {2 h x}{9}+g \right ) e^{2}\right ) b^{3}+5 a \left (\frac {11 x^{2} g \,f^{2}}{10}-\frac {8 x \left (-\frac {13 h x}{16}+g \right ) e f}{5}+e^{2} \left (-\frac {7 h x}{5}+g \right )\right ) b^{2}-a^{2} \left (\left (\frac {1}{2} h \,x^{2}-3 g x \right ) f^{2}+\frac {5 \left (-\frac {7 h x}{5}+g \right ) e f}{2}+e^{2} h \right ) b +a^{3} \left (\frac {\left (-h x -5 g \right ) f}{2}+e h \right ) f \right ) d^{3}+c^{2} \left (\left (4 x^{2} g \,f^{2}-29 x e \left (-\frac {7 h x}{29}+g \right ) f +4 e^{2} \left (-\frac {3 h x}{2}+g \right )\right ) b^{3}-10 a \left (-\frac {13 x \left (-\frac {11 h x}{13}+g \right ) f^{2}}{10}+\frac {13 e \left (-\frac {28 h x}{13}+g \right ) f}{10}+e^{2} h \right ) b^{2}+5 a^{2} \left (\frac {\left (-6 h x +13 g \right ) f}{5}+e h \right ) f b +f^{2} a^{3} h \right ) d^{2}-9 c^{3} \left (\frac {8 \left (-f g x +e \left (-\frac {9 h x}{8}+g \right )\right ) b^{2}}{9}-\frac {17 a h \left (-f x +e \right ) b}{9}+a^{2} f h \right ) b f d -4 b^{2} c^{4} f^{2} \left (a h -b g \right )\right ) \sqrt {f x +e}\right ) \sqrt {\left (c f -d e \right ) d}}{4 \sqrt {\left (a f -b e \right ) b}\, \sqrt {\left (c f -d e \right ) d}\, \left (c f -d e \right )^{2} \left (x d +c \right )^{2} \left (b x +a \right ) \left (a d -b c \right )^{4} \left (a f -b e \right )}\)	\(792\)

Input:

int((h*x+g)/(b*x+a)^2/(d*x+c)^3/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2*f^3*(-d/f^3/(a*d-b*c)^4*((-1/8*d*f*(a^2*c*d^2*f*h-4*a^2*d^3*e*h+3*a^2*d^ 
3*f*g+6*a*b*c^2*d*f*h-14*a*b*c*d^2*f*g+8*a*b*d^3*e*g-7*b^2*c^3*f*h+4*b^2*c 
^2*d*e*h+11*b^2*c^2*d*f*g-8*b^2*c*d^2*e*g)/(c^2*f^2-2*c*d*e*f+d^2*e^2)*(f* 
x+e)^(3/2)+1/8*(a^2*c*d^2*f*h+4*a^2*d^3*e*h-5*a^2*d^3*f*g-10*a*b*c^2*d*f*h 
+18*a*b*c*d^2*f*g-8*a*b*d^3*e*g+9*b^2*c^3*f*h-4*b^2*c^2*d*e*h-13*b^2*c^2*d 
*f*g+8*b^2*c*d^2*e*g)*f/(c*f-d*e)*(f*x+e)^(1/2))/((f*x+e)*d+c*f-d*e)^2-1/8 
*(a^2*c*d^2*f^2*h-4*a^2*d^3*e*f*h+3*a^2*d^3*f^2*g-10*a*b*c^2*d*f^2*h+32*a* 
b*c*d^2*e*f*h-14*a*b*c*d^2*f^2*g-16*a*b*d^3*e^2*h+8*a*b*d^3*e*f*g-15*b^2*c 
^3*f^2*h+20*b^2*c^2*d*e*f*h+35*b^2*c^2*d*f^2*g-8*b^2*c*d^2*e^2*h-56*b^2*c* 
d^2*e*f*g+24*b^2*d^3*e^2*g)/(c^2*f^2-2*c*d*e*f+d^2*e^2)/((c*f-d*e)*d)^(1/2 
)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)))+b^2/f^3/(a*d-b*c)^4*(1/2*f* 
(a^2*d*h-a*b*c*h-a*b*d*g+b^2*c*g)/(a*f-b*e)*(f*x+e)^(1/2)/((f*x+e)*b+a*f-b 
*e)+1/2*(5*a^2*d*f*h+a*b*c*f*h-4*a*b*d*e*h-7*a*b*d*f*g-2*b^2*c*e*h+b^2*c*f 
*g+6*b^2*d*e*g)/(a*f-b*e)/((a*f-b*e)*b)^(1/2)*arctan(b*(f*x+e)^(1/2)/((a*f 
-b*e)*b)^(1/2))))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)^2/(d*x+c)^3/(f*x+e)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)**2/(d*x+c)**3/(f*x+e)**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {g+h x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((h*x+g)/(b*x+a)^2/(d*x+c)^3/(f*x+e)^(1/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*f-b*e>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1274 vs. \(2 (550) = 1100\).

Time = 0.19 (sec) , antiderivative size = 1274, normalized size of antiderivative = 2.19 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)/(b*x+a)^2/(d*x+c)^3/(f*x+e)^(1/2),x, algorithm="giac")
 

Output:

-(6*b^4*d*e*g + b^4*c*f*g - 7*a*b^3*d*f*g - 2*b^4*c*e*h - 4*a*b^3*d*e*h + 
a*b^3*c*f*h + 5*a^2*b^2*d*f*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f) 
)/((b^5*c^4*e - 4*a*b^4*c^3*d*e + 6*a^2*b^3*c^2*d^2*e - 4*a^3*b^2*c*d^3*e 
+ a^4*b*d^4*e - a*b^4*c^4*f + 4*a^2*b^3*c^3*d*f - 6*a^3*b^2*c^2*d^2*f + 4* 
a^4*b*c*d^3*f - a^5*d^4*f)*sqrt(-b^2*e + a*b*f)) + 1/4*(24*b^2*d^4*e^2*g - 
 56*b^2*c*d^3*e*f*g + 8*a*b*d^4*e*f*g + 35*b^2*c^2*d^2*f^2*g - 14*a*b*c*d^ 
3*f^2*g + 3*a^2*d^4*f^2*g - 8*b^2*c*d^3*e^2*h - 16*a*b*d^4*e^2*h + 20*b^2* 
c^2*d^2*e*f*h + 32*a*b*c*d^3*e*f*h - 4*a^2*d^4*e*f*h - 15*b^2*c^3*d*f^2*h 
- 10*a*b*c^2*d^2*f^2*h + a^2*c*d^3*f^2*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2 
*e + c*d*f))/((b^4*c^4*d^2*e^2 - 4*a*b^3*c^3*d^3*e^2 + 6*a^2*b^2*c^2*d^4*e 
^2 - 4*a^3*b*c*d^5*e^2 + a^4*d^6*e^2 - 2*b^4*c^5*d*e*f + 8*a*b^3*c^4*d^2*e 
*f - 12*a^2*b^2*c^3*d^3*e*f + 8*a^3*b*c^2*d^4*e*f - 2*a^4*c*d^5*e*f + b^4* 
c^6*f^2 - 4*a*b^3*c^5*d*f^2 + 6*a^2*b^2*c^4*d^2*f^2 - 4*a^3*b*c^3*d^3*f^2 
+ a^4*c^2*d^4*f^2)*sqrt(-d^2*e + c*d*f)) - (sqrt(f*x + e)*b^3*f*g - sqrt(f 
*x + e)*a*b^2*f*h)/((b^4*c^3*e - 3*a*b^3*c^2*d*e + 3*a^2*b^2*c*d^2*e - a^3 
*b*d^3*e - a*b^3*c^3*f + 3*a^2*b^2*c^2*d*f - 3*a^3*b*c*d^2*f + a^4*d^3*f)* 
((f*x + e)*b - b*e + a*f)) - 1/4*(8*(f*x + e)^(3/2)*b*d^4*e*f*g - 8*sqrt(f 
*x + e)*b*d^4*e^2*f*g - 11*(f*x + e)^(3/2)*b*c*d^3*f^2*g + 3*(f*x + e)^(3/ 
2)*a*d^4*f^2*g + 21*sqrt(f*x + e)*b*c*d^3*e*f^2*g - 5*sqrt(f*x + e)*a*d^4* 
e*f^2*g - 13*sqrt(f*x + e)*b*c^2*d^2*f^3*g + 5*sqrt(f*x + e)*a*c*d^3*f^...
 

Mupad [B] (verification not implemented)

Time = 33.23 (sec) , antiderivative size = 691757, normalized size of antiderivative = 1188.59 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

int((g + h*x)/((e + f*x)^(1/2)*(a + b*x)^2*(c + d*x)^3),x)
 

Output:

- (((e + f*x)^(1/2)*(4*b^3*c^3*f^4*g - 5*a^3*d^3*f^4*g - 4*a*b^2*c^3*f^4*h 
 + a^3*c*d^2*f^4*h + 4*a^3*d^3*e*f^3*h - 12*b^3*d^3*e^3*f*g + 11*a*b^2*d^3 
*e^2*f^2*g - 8*a^2*b*d^3*e^2*f^2*h + 25*b^3*c*d^2*e^2*f^2*g - 9*b^3*c^2*d* 
e^2*f^2*h + 13*a^2*b*c*d^2*f^4*g - 9*a^2*b*c^2*d*f^4*h + 2*a^2*b*d^3*e*f^3 
*g + 8*a*b^2*d^3*e^3*f*h - 12*b^3*c^2*d*e*f^3*g + 4*b^3*c*d^2*e^3*f*h - 26 
*a*b^2*c*d^2*e*f^3*g + 30*a*b^2*c^2*d*e*f^3*h + 2*a^2*b*c*d^2*e*f^3*h - 19 
*a*b^2*c*d^2*e^2*f^2*h))/(4*(a*d - b*c)*(a*f - b*e)*(c*f - d*e)*(a^2*d^2 + 
 b^2*c^2 - 2*a*b*c*d)) - (f*(e + f*x)^(5/2)*(8*a*b^2*d^4*e^2*h - 12*b^3*d^ 
4*e^2*g + 3*a^2*b*d^4*f^2*g + 4*b^3*c*d^3*e^2*h - 4*b^3*c^2*d^2*f^2*g + 11 
*a*b^2*c^2*d^2*f^2*h + 5*a*b^2*d^4*e*f*g - 4*a^2*b*d^4*e*f*h + 19*b^3*c*d^ 
3*e*f*g - 11*a*b^2*c*d^3*f^2*g + a^2*b*c*d^3*f^2*h - 7*b^3*c^2*d^2*e*f*h - 
 13*a*b^2*c*d^3*e*f*h))/(4*(a*f - b*e)*(c*f - d*e)^2*(a^3*d^3 - b^3*c^3 + 
3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (d*(e + f*x)^(3/2)*(8*b^3*c^3*f^4*g - 3* 
a^3*d^3*f^4*g - 17*a*b^2*c^3*f^4*h - a^3*c*d^2*f^4*h + 4*a^3*d^3*e*f^3*h + 
 9*b^3*c^3*e*f^3*h - 24*b^3*d^3*e^3*f*g + 16*a*b^2*d^3*e^2*f^2*g - 12*a^2* 
b*d^3*e^2*f^2*h + 56*b^3*c*d^2*e^2*f^2*g - 20*b^3*c^2*d*e^2*f^2*h + 13*a*b 
^2*c^2*d*f^4*g + 6*a^2*b*c*d^2*f^4*g - 6*a^2*b*c^2*d*f^4*h + 3*a^2*b*d^3*e 
*f^3*g + 16*a*b^2*d^3*e^3*f*h - 37*b^3*c^2*d*e*f^3*g + 8*b^3*c*d^2*e^3*f*h 
 - 38*a*b^2*c*d^2*e*f^3*g + 50*a*b^2*c^2*d*e*f^3*h + 9*a^2*b*c*d^2*e*f^3*h 
 - 40*a*b^2*c*d^2*e^2*f^2*h))/(4*(a*d - b*c)*(c*f - d*e)*(a^2*d^2 + b^2...
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 18732, normalized size of antiderivative = 32.19 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

int((h*x+g)/(b*x+a)^2/(d*x+c)^3/(f*x+e)^(1/2),x)
 

Output:

(20*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e 
)))*a**3*b*c**5*d*f**4*h - 60*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)* 
b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*c**4*d**2*e*f**3*h + 40*sqrt(b)*sqrt( 
a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*c**4*d 
**2*f**4*h*x + 60*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)* 
sqrt(a*f - b*e)))*a**3*b*c**3*d**3*e**2*f**2*h - 120*sqrt(b)*sqrt(a*f - b* 
e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*c**3*d**3*e*f* 
*3*h*x + 20*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a 
*f - b*e)))*a**3*b*c**3*d**3*f**4*h*x**2 - 20*sqrt(b)*sqrt(a*f - b*e)*atan 
((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*c**2*d**4*e**3*f*h + 
120*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e 
)))*a**3*b*c**2*d**4*e**2*f**2*h*x - 60*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt 
(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*c**2*d**4*e*f**3*h*x**2 - 4 
0*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)) 
)*a**3*b*c*d**5*e**3*f*h*x + 60*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x 
)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*c*d**5*e**2*f**2*h*x**2 - 20*sqrt(b 
)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b 
*d**6*e**3*f*h*x**2 + 4*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sq 
rt(b)*sqrt(a*f - b*e)))*a**2*b**2*c**6*f**4*h - 28*sqrt(b)*sqrt(a*f - b*e) 
*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*c**5*d*e*f...