\(\int \frac {g+h x}{(a+b x) (c+d x)^3 (e+f x)^{3/2}} \, dx\) [190]

Optimal result

Integrand size = 29, antiderivative size = 516 \[ \int \frac {g+h x}{(a+b x) (c+d x)^3 (e+f x)^{3/2}} \, dx=-\frac {f \left (3 a^2 d f (5 d f g-4 d e h-c f h)-b^2 \left (4 d^2 e^2 g-11 c d e f g-c^2 f (8 f g-15 e h)\right )+a b \left (7 c^2 f^2 h-c d f (27 f g-19 e h)-d^2 e (3 f g-4 e h)\right )\right )}{4 (b c-a d)^2 (b e-a f) (d e-c f)^3 \sqrt {e+f x}}+\frac {d g-c h}{2 (b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {a d (5 d f g-4 d e h-c f h)+b \left (4 d^2 e g-9 c d f g+5 c^2 f h\right )}{4 (b c-a d)^2 (d e-c f)^2 (c+d x) \sqrt {e+f x}}-\frac {2 b^{5/2} (b g-a h) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d)^3 (b e-a f)^{3/2}}+\frac {\sqrt {d} \left (3 a^2 d^2 f (5 d f g-4 d e h-c f h)+b^2 \left (8 d^3 e^2 g-28 c d^2 e f g+35 c^2 d f^2 g-15 c^3 f^2 h\right )+2 a b d \left (5 c^2 f^2 h-7 c d f (3 f g-2 e h)+d^2 \left (6 e f g-4 e^2 h\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 (b c-a d)^3 (d e-c f)^{7/2}} \] Output:

-1/4*f*(3*a^2*d*f*(-c*f*h-4*d*e*h+5*d*f*g)-b^2*(4*d^2*e^2*g-11*c*d*e*f*g-c 
^2*f*(-15*e*h+8*f*g))+a*b*(7*c^2*f^2*h-c*d*f*(-19*e*h+27*f*g)-d^2*e*(-4*e* 
h+3*f*g)))/(-a*d+b*c)^2/(-a*f+b*e)/(-c*f+d*e)^3/(f*x+e)^(1/2)+1/2*(-c*h+d* 
g)/(-a*d+b*c)/(-c*f+d*e)/(d*x+c)^2/(f*x+e)^(1/2)+1/4*(a*d*(-c*f*h-4*d*e*h+ 
5*d*f*g)+b*(5*c^2*f*h-9*c*d*f*g+4*d^2*e*g))/(-a*d+b*c)^2/(-c*f+d*e)^2/(d*x 
+c)/(f*x+e)^(1/2)-2*b^(5/2)*(-a*h+b*g)*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f 
+b*e)^(1/2))/(-a*d+b*c)^3/(-a*f+b*e)^(3/2)+1/4*d^(1/2)*(3*a^2*d^2*f*(-c*f* 
h-4*d*e*h+5*d*f*g)+b^2*(-15*c^3*f^2*h+35*c^2*d*f^2*g-28*c*d^2*e*f*g+8*d^3* 
e^2*g)+2*a*b*d*(5*c^2*f^2*h-7*c*d*f*(-2*e*h+3*f*g)+d^2*(-4*e^2*h+6*e*f*g)) 
)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/(-a*d+b*c)^3/(-c*f+d*e)^ 
(7/2)
 

Mathematica [A] (verified)

Time = 5.70 (sec) , antiderivative size = 649, normalized size of antiderivative = 1.26 \[ \int \frac {g+h x}{(a+b x) (c+d x)^3 (e+f x)^{3/2}} \, dx=\frac {\frac {(b c-a d) \left (b^2 \left (8 c^4 f^2 (-f g+e h)+4 d^4 e^2 g x (e+f x)+c^3 d f \left (9 e^2 h-16 f^2 g x+25 e f h x\right )+c d^3 e g \left (6 e^2-5 e f x-11 f^2 x^2\right )-c^2 d^2 \left (2 e^3 h+8 f^3 g x^2+e f^2 x (13 g-15 h x)+e^2 f (13 g-5 h x)\right )\right )+a^2 d^2 f \left (c^2 f (-8 f g+13 e h+5 f h x)+d^2 \left (-15 f^2 g x^2+2 e^2 (g+2 h x)+e f x (-5 g+12 h x)\right )+c d \left (2 e^2 h+f^2 x (-25 g+3 h x)+e f (-9 g+21 h x)\right )\right )-a b d \left (c^3 f^2 (-16 f g+25 e h+9 f h x)+d^3 e (e+f x) (-3 f g x+2 e (g+2 h x))+c d^2 \left (2 e^3 h-27 f^3 g x^2+e^2 f (-3 g+5 h x)+e f^2 x (-14 g+19 h x)\right )+c^2 d f \left (3 e^2 h+f^2 x (-45 g+7 h x)+e f (-13 g+42 h x)\right )\right )\right )}{(b e-a f) (d e-c f)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {8 b^{5/2} (b g-a h) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{(-b e+a f)^{3/2}}+\frac {\sqrt {d} \left (-3 a^2 d^2 f (-5 d f g+4 d e h+c f h)+b^2 \left (8 d^3 e^2 g-28 c d^2 e f g+35 c^2 d f^2 g-15 c^3 f^2 h\right )+2 a b d \left (5 c^2 f^2 h-7 c d f (3 f g-2 e h)+d^2 \left (6 e f g-4 e^2 h\right )\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{(-d e+c f)^{7/2}}}{4 (b c-a d)^3} \] Input:

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

Output:

(((b*c - a*d)*(b^2*(8*c^4*f^2*(-(f*g) + e*h) + 4*d^4*e^2*g*x*(e + f*x) + c 
^3*d*f*(9*e^2*h - 16*f^2*g*x + 25*e*f*h*x) + c*d^3*e*g*(6*e^2 - 5*e*f*x - 
11*f^2*x^2) - c^2*d^2*(2*e^3*h + 8*f^3*g*x^2 + e*f^2*x*(13*g - 15*h*x) + e 
^2*f*(13*g - 5*h*x))) + a^2*d^2*f*(c^2*f*(-8*f*g + 13*e*h + 5*f*h*x) + d^2 
*(-15*f^2*g*x^2 + 2*e^2*(g + 2*h*x) + e*f*x*(-5*g + 12*h*x)) + c*d*(2*e^2* 
h + f^2*x*(-25*g + 3*h*x) + e*f*(-9*g + 21*h*x))) - a*b*d*(c^3*f^2*(-16*f* 
g + 25*e*h + 9*f*h*x) + d^3*e*(e + f*x)*(-3*f*g*x + 2*e*(g + 2*h*x)) + c*d 
^2*(2*e^3*h - 27*f^3*g*x^2 + e^2*f*(-3*g + 5*h*x) + e*f^2*x*(-14*g + 19*h* 
x)) + c^2*d*f*(3*e^2*h + f^2*x*(-45*g + 7*h*x) + e*f*(-13*g + 42*h*x)))))/ 
((b*e - a*f)*(d*e - c*f)^3*(c + d*x)^2*Sqrt[e + f*x]) - (8*b^(5/2)*(b*g - 
a*h)*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/(-(b*e) + a*f)^(3 
/2) + (Sqrt[d]*(-3*a^2*d^2*f*(-5*d*f*g + 4*d*e*h + c*f*h) + b^2*(8*d^3*e^2 
*g - 28*c*d^2*e*f*g + 35*c^2*d*f^2*g - 15*c^3*f^2*h) + 2*a*b*d*(5*c^2*f^2* 
h - 7*c*d*f*(3*f*g - 2*e*h) + d^2*(6*e*f*g - 4*e^2*h)))*ArcTan[(Sqrt[d]*Sq 
rt[e + f*x])/Sqrt[-(d*e) + c*f]])/(-(d*e) + c*f)^(7/2))/(4*(b*c - a*d)^3)
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {168, 27, 168, 27, 169, 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.

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\int \frac {3 d f (5 d f g-4 d e h-c f h) a^2+b \left (4 e (3 f g-2 e h) d^2-c f (27 f g-16 e h) d+7 c^2 f^2 h\right ) a+8 b^2 (d e-c f)^2 g+3 b f \left (a d (5 d f g-4 d e h-c f h)+b \left (5 f h c^2-9 d f g c+4 d^2 e g\right )\right ) x}{2 (a+b x) (c+d x) (e+f x)^{3/2}}dx}{(b c-a d) (d e-c f)}+\frac {a d (-c f h-4 d e h+5 d f g)+b \left (5 c^2 f h-9 c d f g+4 d^2 e g\right )}{(c+d x) \sqrt {e+f x} (b c-a d) (d e-c f)}}{4 (b c-a d) (d e-c f)}+\frac {d g-c h}{2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {3 d f (5 d f g-4 d e h-c f h) a^2+b \left (4 e (3 f g-2 e h) d^2-c f (27 f g-16 e h) d+7 c^2 f^2 h\right ) a+8 b^2 (d e-c f)^2 g+3 b f \left (a d (5 d f g-4 d e h-c f h)+b \left (5 f h c^2-9 d f g c+4 d^2 e g\right )\right ) x}{(a+b x) (c+d x) (e+f x)^{3/2}}dx}{2 (b c-a d) (d e-c f)}+\frac {a d (-c f h-4 d e h+5 d f g)+b \left (5 c^2 f h-9 c d f g+4 d^2 e g\right )}{(c+d x) \sqrt {e+f x} (b c-a d) (d e-c f)}}{4 (b c-a d) (d e-c f)}+\frac {d g-c h}{2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {-\frac {2 \int -\frac {-3 d^2 f^2 (5 d f g-4 d e h-c f h) a^3-b d f \left (-e (3 f g-4 e h) d^2-c f (27 f g-19 e h) d+7 c^2 f^2 h\right ) a^2+b^2 \left (4 e^2 (f g-2 e h) d^3-c e f (11 f g-24 e h) d^2-c^2 f^2 (8 f g+9 e h) d+8 c^3 f^3 h\right ) a+8 b^3 (d e-c f)^3 g-b d f \left (3 d f (5 d f g-4 d e h-c f h) a^2+b \left (-e (3 f g-4 e h) d^2-c f (27 f g-19 e h) d+7 c^2 f^2 h\right ) a-b^2 \left (-f (8 f g-15 e h) c^2-11 d e f g c+4 d^2 e^2 g\right )\right ) x}{2 (a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}-\frac {2 f \left (3 a^2 d f (-c f h-4 d e h+5 d f g)+a b \left (7 c^2 f^2 h-c d f (27 f g-19 e h)+d^2 (-e) (3 f g-4 e h)\right )-b^2 \left (c^2 (-f) (8 f g-15 e h)-11 c d e f g+4 d^2 e^2 g\right )\right )}{\sqrt {e+f x} (b e-a f) (d e-c f)}}{2 (b c-a d) (d e-c f)}+\frac {a d (-c f h-4 d e h+5 d f g)+b \left (5 c^2 f h-9 c d f g+4 d^2 e g\right )}{(c+d x) \sqrt {e+f x} (b c-a d) (d e-c f)}}{4 (b c-a d) (d e-c f)}+\frac {d g-c h}{2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {-3 d^2 f^2 (5 d f g-4 d e h-c f h) a^3-b d f \left (-e (3 f g-4 e h) d^2-c f (27 f g-19 e h) d+7 c^2 f^2 h\right ) a^2+b^2 \left (4 e^2 (f g-2 e h) d^3-c e f (11 f g-24 e h) d^2-c^2 f^2 (8 f g+9 e h) d+8 c^3 f^3 h\right ) a+8 b^3 (d e-c f)^3 g-b d f \left (3 d f (5 d f g-4 d e h-c f h) a^2+b \left (-e (3 f g-4 e h) d^2-c f (27 f g-19 e h) d+7 c^2 f^2 h\right ) a-b^2 \left (-f (8 f g-15 e h) c^2-11 d e f g c+4 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}-\frac {2 f \left (3 a^2 d f (-c f h-4 d e h+5 d f g)+a b \left (7 c^2 f^2 h-c d f (27 f g-19 e h)+d^2 (-e) (3 f g-4 e h)\right )-b^2 \left (c^2 (-f) (8 f g-15 e h)-11 c d e f g+4 d^2 e^2 g\right )\right )}{\sqrt {e+f x} (b e-a f) (d e-c f)}}{2 (b c-a d) (d e-c f)}+\frac {a d (-c f h-4 d e h+5 d f g)+b \left (5 c^2 f h-9 c d f g+4 d^2 e g\right )}{(c+d x) \sqrt {e+f x} (b c-a d) (d e-c f)}}{4 (b c-a d) (d e-c f)}+\frac {d g-c h}{2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {\frac {8 b^3 (b g-a h) (d e-c f)^3 \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}-\frac {d (b e-a f) \left (3 a^2 d^2 f (-c f h-4 d e h+5 d f g)+2 a b d \left (5 c^2 f^2 h-7 c d f (3 f g-2 e h)+d^2 \left (6 e f g-4 e^2 h\right )\right )+b^2 \left (-15 c^3 f^2 h+35 c^2 d f^2 g-28 c d^2 e f g+8 d^3 e^2 g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}}{(b e-a f) (d e-c f)}-\frac {2 f \left (3 a^2 d f (-c f h-4 d e h+5 d f g)+a b \left (7 c^2 f^2 h-c d f (27 f g-19 e h)+d^2 (-e) (3 f g-4 e h)\right )-b^2 \left (c^2 (-f) (8 f g-15 e h)-11 c d e f g+4 d^2 e^2 g\right )\right )}{\sqrt {e+f x} (b e-a f) (d e-c f)}}{2 (b c-a d) (d e-c f)}+\frac {a d (-c f h-4 d e h+5 d f g)+b \left (5 c^2 f h-9 c d f g+4 d^2 e g\right )}{(c+d x) \sqrt {e+f x} (b c-a d) (d e-c f)}}{4 (b c-a d) (d e-c f)}+\frac {d g-c h}{2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {\frac {16 b^3 (b g-a h) (d e-c f)^3 \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{f (b c-a d)}-\frac {2 d (b e-a f) \left (3 a^2 d^2 f (-c f h-4 d e h+5 d f g)+2 a b d \left (5 c^2 f^2 h-7 c d f (3 f g-2 e h)+d^2 \left (6 e f g-4 e^2 h\right )\right )+b^2 \left (-15 c^3 f^2 h+35 c^2 d f^2 g-28 c d^2 e f g+8 d^3 e^2 g\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{f (b c-a d)}}{(b e-a f) (d e-c f)}-\frac {2 f \left (3 a^2 d f (-c f h-4 d e h+5 d f g)+a b \left (7 c^2 f^2 h-c d f (27 f g-19 e h)+d^2 (-e) (3 f g-4 e h)\right )-b^2 \left (c^2 (-f) (8 f g-15 e h)-11 c d e f g+4 d^2 e^2 g\right )\right )}{\sqrt {e+f x} (b e-a f) (d e-c f)}}{2 (b c-a d) (d e-c f)}+\frac {a d (-c f h-4 d e h+5 d f g)+b \left (5 c^2 f h-9 c d f g+4 d^2 e g\right )}{(c+d x) \sqrt {e+f x} (b c-a d) (d e-c f)}}{4 (b c-a d) (d e-c f)}+\frac {d g-c h}{2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {\frac {2 \sqrt {d} (b e-a f) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) \left (3 a^2 d^2 f (-c f h-4 d e h+5 d f g)+2 a b d \left (5 c^2 f^2 h-7 c d f (3 f g-2 e h)+d^2 \left (6 e f g-4 e^2 h\right )\right )+b^2 \left (-15 c^3 f^2 h+35 c^2 d f^2 g-28 c d^2 e f g+8 d^3 e^2 g\right )\right )}{(b c-a d) \sqrt {d e-c f}}-\frac {16 b^{5/2} (b g-a h) (d e-c f)^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d) \sqrt {b e-a f}}}{(b e-a f) (d e-c f)}-\frac {2 f \left (3 a^2 d f (-c f h-4 d e h+5 d f g)+a b \left (7 c^2 f^2 h-c d f (27 f g-19 e h)+d^2 (-e) (3 f g-4 e h)\right )-b^2 \left (c^2 (-f) (8 f g-15 e h)-11 c d e f g+4 d^2 e^2 g\right )\right )}{\sqrt {e+f x} (b e-a f) (d e-c f)}}{2 (b c-a d) (d e-c f)}+\frac {a d (-c f h-4 d e h+5 d f g)+b \left (5 c^2 f h-9 c d f g+4 d^2 e g\right )}{(c+d x) \sqrt {e+f x} (b c-a d) (d e-c f)}}{4 (b c-a d) (d e-c f)}+\frac {d g-c h}{2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (d e-c f)}\)

Input:

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

Output:

(d*g - c*h)/(2*(b*c - a*d)*(d*e - c*f)*(c + d*x)^2*Sqrt[e + f*x]) + ((a*d* 
(5*d*f*g - 4*d*e*h - c*f*h) + b*(4*d^2*e*g - 9*c*d*f*g + 5*c^2*f*h))/((b*c 
 - a*d)*(d*e - c*f)*(c + d*x)*Sqrt[e + f*x]) + ((-2*f*(3*a^2*d*f*(5*d*f*g 
- 4*d*e*h - c*f*h) - b^2*(4*d^2*e^2*g - 11*c*d*e*f*g - c^2*f*(8*f*g - 15*e 
*h)) + a*b*(7*c^2*f^2*h - c*d*f*(27*f*g - 19*e*h) - d^2*e*(3*f*g - 4*e*h)) 
))/((b*e - a*f)*(d*e - c*f)*Sqrt[e + f*x]) + ((-16*b^(5/2)*(d*e - c*f)^3*( 
b*g - a*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)*(3*a^2*d^2*f*(5*d*f*g - 4*d*e*h 
- c*f*h) + b^2*(8*d^3*e^2*g - 28*c*d^2*e*f*g + 35*c^2*d*f^2*g - 15*c^3*f^2 
*h) + 2*a*b*d*(5*c^2*f^2*h - 7*c*d*f*(3*f*g - 2*e*h) + d^2*(6*e*f*g - 4*e^ 
2*h)))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/((b*c - a*d)*Sqrt 
[d*e - c*f]))/((b*e - a*f)*(d*e - c*f)))/(2*(b*c - a*d)*(d*e - c*f)))/(4*( 
b*c - a*d)*(d*e - c*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 169

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] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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 = 1.42 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.32

method	result	size
derivativedivides	\(2 f^{2} \left (\frac {d \left (\frac {\left (\frac {3}{8} a^{2} c \,d^{3} f^{2} h +\frac {1}{2} a^{2} d^{4} e f h -\frac {7}{8} a^{2} d^{4} f^{2} g -\frac {5}{4} a b \,c^{2} d^{2} f^{2} h -\frac {1}{2} a b c \,d^{3} e f h +\frac {9}{4} a b c \,d^{3} f^{2} g -\frac {1}{2} a b \,d^{4} e f g +\frac {7}{8} b^{2} c^{3} d \,f^{2} h -\frac {11}{8} b^{2} c^{2} d^{2} f^{2} g +\frac {1}{2} b^{2} c \,d^{3} e f g \right ) \left (f x +e \right )^{\frac {3}{2}}+\frac {f \left (5 a^{2} c^{2} d^{2} f^{2} h -a^{2} c \,d^{3} e f h -9 a^{2} c \,d^{3} f^{2} g -4 a^{2} d^{4} e^{2} h +9 a^{2} d^{4} e f g -14 a b \,c^{3} d \,f^{2} h +10 a b \,c^{2} d^{2} e f h +22 a b \,c^{2} d^{2} f^{2} g +4 a b c \,d^{3} e^{2} h -26 a b c \,d^{3} e f g +4 a b \,d^{4} e^{2} g +9 b^{2} c^{4} f^{2} h -9 b^{2} c^{3} d e f h -13 b^{2} c^{3} d \,f^{2} g +17 b^{2} c^{2} d^{2} e f g -4 b^{2} c \,d^{3} e^{2} g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (3 a^{2} c \,d^{2} f^{2} h +12 a^{2} d^{3} e f h -15 a^{2} d^{3} f^{2} g -10 a b \,c^{2} d \,f^{2} h -28 a b c \,d^{2} e f h +42 a b c \,d^{2} f^{2} g +8 e^{2} h b a \,d^{3}-12 a b \,d^{3} e f g +15 b^{2} c^{3} f^{2} h -35 b^{2} c^{2} d \,f^{2} g +28 b^{2} c \,d^{2} e f g -8 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right )^{3} \left (c f -d e \right )^{3} f^{2}}-\frac {-e h +f g}{\left (c f -d e \right )^{3} \left (a f -b e \right ) \sqrt {f x +e}}-\frac {\left (a h -b g \right ) b^{3} \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f^{2} \left (a d -b c \right )^{3} \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}\right )\)	\(680\)
default	\(2 f^{2} \left (\frac {d \left (\frac {\left (\frac {3}{8} a^{2} c \,d^{3} f^{2} h +\frac {1}{2} a^{2} d^{4} e f h -\frac {7}{8} a^{2} d^{4} f^{2} g -\frac {5}{4} a b \,c^{2} d^{2} f^{2} h -\frac {1}{2} a b c \,d^{3} e f h +\frac {9}{4} a b c \,d^{3} f^{2} g -\frac {1}{2} a b \,d^{4} e f g +\frac {7}{8} b^{2} c^{3} d \,f^{2} h -\frac {11}{8} b^{2} c^{2} d^{2} f^{2} g +\frac {1}{2} b^{2} c \,d^{3} e f g \right ) \left (f x +e \right )^{\frac {3}{2}}+\frac {f \left (5 a^{2} c^{2} d^{2} f^{2} h -a^{2} c \,d^{3} e f h -9 a^{2} c \,d^{3} f^{2} g -4 a^{2} d^{4} e^{2} h +9 a^{2} d^{4} e f g -14 a b \,c^{3} d \,f^{2} h +10 a b \,c^{2} d^{2} e f h +22 a b \,c^{2} d^{2} f^{2} g +4 a b c \,d^{3} e^{2} h -26 a b c \,d^{3} e f g +4 a b \,d^{4} e^{2} g +9 b^{2} c^{4} f^{2} h -9 b^{2} c^{3} d e f h -13 b^{2} c^{3} d \,f^{2} g +17 b^{2} c^{2} d^{2} e f g -4 b^{2} c \,d^{3} e^{2} g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (3 a^{2} c \,d^{2} f^{2} h +12 a^{2} d^{3} e f h -15 a^{2} d^{3} f^{2} g -10 a b \,c^{2} d \,f^{2} h -28 a b c \,d^{2} e f h +42 a b c \,d^{2} f^{2} g +8 e^{2} h b a \,d^{3}-12 a b \,d^{3} e f g +15 b^{2} c^{3} f^{2} h -35 b^{2} c^{2} d \,f^{2} g +28 b^{2} c \,d^{2} e f g -8 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right )^{3} \left (c f -d e \right )^{3} f^{2}}-\frac {-e h +f g}{\left (c f -d e \right )^{3} \left (a f -b e \right ) \sqrt {f x +e}}-\frac {\left (a h -b g \right ) b^{3} \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f^{2} \left (a d -b c \right )^{3} \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}\right )\)	\(680\)
pseudoelliptic	\(\frac {\frac {3 \left (\left (5 \left (c^{3} h -\frac {7}{3} c^{2} d g \right ) b^{2}-\frac {10 a c d \left (c h -\frac {21 d g}{5}\right ) b}{3}+a^{2} d^{2} \left (c h -5 d g \right )\right ) f^{2}+4 \left (a h -b g \right ) d^{2} e \left (a d -\frac {7 b c}{3}\right ) f +\frac {8 b \,d^{3} e^{2} \left (a h -b g \right )}{3}\right ) \left (x d +c \right )^{2} \sqrt {\left (a f -b e \right ) b}\, \left (a f -b e \right ) d \sqrt {f x +e}\, \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{4}+\frac {13 \sqrt {\left (c f -d e \right ) d}\, \left (-\frac {8 \sqrt {f x +e}\, b^{3} \left (x d +c \right )^{2} \left (c f -d e \right )^{3} \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{13}+\sqrt {\left (a f -b e \right ) b}\, \left (a d -b c \right ) \left (\frac {8 \left (-c^{2} g \left (x d +c \right )^{2} b^{2}+2 a c \left (\frac {27 d^{2} g \,x^{2}}{16}+\frac {45 \left (-\frac {7 h x}{45}+g \right ) x c d}{16}+c^{2} \left (-\frac {9 h x}{16}+g \right )\right ) d b -a^{2} d^{2} \left (\frac {15 d^{2} g \,x^{2}}{8}+\frac {25 x c \left (-\frac {3 h x}{25}+g \right ) d}{8}+c^{2} \left (-\frac {5 h x}{8}+g \right )\right )\right ) f^{3}}{13}+\left (\frac {8 c \left (-\frac {11 d^{3} g \,x^{2}}{8}-\frac {13 x c \left (-\frac {15 h x}{13}+g \right ) d^{2}}{8}+\frac {25 c^{2} d h x}{8}+c^{3} h \right ) b^{2}}{13}-\frac {25 a d \left (-\frac {3 d^{3} g \,x^{2}}{25}-\frac {14 \left (-\frac {19 h x}{14}+g \right ) x c \,d^{2}}{25}-\frac {13 c^{2} \left (-\frac {42 h x}{13}+g \right ) d}{25}+c^{3} h \right ) b}{13}+a^{2} \left (-\frac {5 x \left (-\frac {12 h x}{5}+g \right ) d^{2}}{13}-\frac {9 c \left (-\frac {7 h x}{3}+g \right ) d}{13}+h \,c^{2}\right ) d^{2}\right ) e \,f^{2}+\frac {2 d \left (\left (2 d^{3} g \,x^{2}-\frac {5 c g x \,d^{2}}{2}-\frac {13 c^{2} \left (-\frac {5 h x}{13}+g \right ) d}{2}+\frac {9 c^{3} h}{2}\right ) b^{2}-\frac {3 \left (-\frac {x \left (-4 h x +g \right ) d^{2}}{3}-c \left (-\frac {5 h x}{3}+g \right ) d +h \,c^{2}\right ) a d b}{2}+a^{2} d^{2} \left (\left (2 h x +g \right ) d +c h \right )\right ) e^{2} f}{13}-\frac {2 \left (\left (-2 d^{2} g x +h \,c^{2}-3 c d g \right ) b +a \left (\left (2 h x +g \right ) d +c h \right ) d \right ) d^{2} b \,e^{3}}{13}\right )\right )}{4}}{\left (x d +c \right )^{2} \left (a d -b c \right )^{3} \left (c f -d e \right )^{3} \sqrt {\left (c f -d e \right ) d}\, \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}\, \sqrt {f x +e}}\)	\(690\)

Input:

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

Output:

2*f^2*(d/(a*d-b*c)^3/(c*f-d*e)^3/f^2*(((3/8*a^2*c*d^3*f^2*h+1/2*a^2*d^4*e* 
f*h-7/8*a^2*d^4*f^2*g-5/4*a*b*c^2*d^2*f^2*h-1/2*a*b*c*d^3*e*f*h+9/4*a*b*c* 
d^3*f^2*g-1/2*a*b*d^4*e*f*g+7/8*b^2*c^3*d*f^2*h-11/8*b^2*c^2*d^2*f^2*g+1/2 
*b^2*c*d^3*e*f*g)*(f*x+e)^(3/2)+1/8*f*(5*a^2*c^2*d^2*f^2*h-a^2*c*d^3*e*f*h 
-9*a^2*c*d^3*f^2*g-4*a^2*d^4*e^2*h+9*a^2*d^4*e*f*g-14*a*b*c^3*d*f^2*h+10*a 
*b*c^2*d^2*e*f*h+22*a*b*c^2*d^2*f^2*g+4*a*b*c*d^3*e^2*h-26*a*b*c*d^3*e*f*g 
+4*a*b*d^4*e^2*g+9*b^2*c^4*f^2*h-9*b^2*c^3*d*e*f*h-13*b^2*c^3*d*f^2*g+17*b 
^2*c^2*d^2*e*f*g-4*b^2*c*d^3*e^2*g)*(f*x+e)^(1/2))/((f*x+e)*d+c*f-d*e)^2+1 
/8*(3*a^2*c*d^2*f^2*h+12*a^2*d^3*e*f*h-15*a^2*d^3*f^2*g-10*a*b*c^2*d*f^2*h 
-28*a*b*c*d^2*e*f*h+42*a*b*c*d^2*f^2*g+8*a*b*d^3*e^2*h-12*a*b*d^3*e*f*g+15 
*b^2*c^3*f^2*h-35*b^2*c^2*d*f^2*g+28*b^2*c*d^2*e*f*g-8*b^2*d^3*e^2*g)/((c* 
f-d*e)*d)^(1/2)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)))-(-e*h+f*g)/(c 
*f-d*e)^3/(a*f-b*e)/(f*x+e)^(1/2)-(a*h-b*g)*b^3/f^2/(a*d-b*c)^3/(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. \[ \int \frac {g+h x}{(a+b x) (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((h*x+g)/(b*x+a)/(d*x+c)^3/(f*x+e)^(3/2),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. 1130 vs. \(2 (483) = 966\).

Time = 0.23 (sec) , antiderivative size = 1130, normalized size of antiderivative = 2.19 \[ \int \frac {g+h x}{(a+b x) (c+d x)^3 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

2*(b^4*g - a*b^3*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((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)*sqrt(-b^2*e + a*b*f)) - 1/4* 
(8*b^2*d^4*e^2*g - 28*b^2*c*d^3*e*f*g + 12*a*b*d^4*e*f*g + 35*b^2*c^2*d^2* 
f^2*g - 42*a*b*c*d^3*f^2*g + 15*a^2*d^4*f^2*g - 8*a*b*d^4*e^2*h + 28*a*b*c 
*d^3*e*f*h - 12*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 
- 3*a^2*c*d^3*f^2*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b^3*c^ 
3*d^3*e^3 - 3*a*b^2*c^2*d^4*e^3 + 3*a^2*b*c*d^5*e^3 - a^3*d^6*e^3 - 3*b^3* 
c^4*d^2*e^2*f + 9*a*b^2*c^3*d^3*e^2*f - 9*a^2*b*c^2*d^4*e^2*f + 3*a^3*c*d^ 
5*e^2*f + 3*b^3*c^5*d*e*f^2 - 9*a*b^2*c^4*d^2*e*f^2 + 9*a^2*b*c^3*d^3*e*f^ 
2 - 3*a^3*c^2*d^4*e*f^2 - b^3*c^6*f^3 + 3*a*b^2*c^5*d*f^3 - 3*a^2*b*c^4*d^ 
2*f^3 + a^3*c^3*d^3*f^3)*sqrt(-d^2*e + c*d*f)) - 2*(f^3*g - e*f^2*h)/((b*d 
^3*e^4 - 3*b*c*d^2*e^3*f - a*d^3*e^3*f + 3*b*c^2*d*e^2*f^2 + 3*a*c*d^2*e^2 
*f^2 - b*c^3*e*f^3 - 3*a*c^2*d*e*f^3 + a*c^3*f^4)*sqrt(f*x + e)) + 1/4*(4* 
(f*x + e)^(3/2)*b*d^4*e*f*g - 4*sqrt(f*x + e)*b*d^4*e^2*f*g - 11*(f*x + e) 
^(3/2)*b*c*d^3*f^2*g + 7*(f*x + e)^(3/2)*a*d^4*f^2*g + 17*sqrt(f*x + e)*b* 
c*d^3*e*f^2*g - 9*sqrt(f*x + e)*a*d^4*e*f^2*g - 13*sqrt(f*x + e)*b*c^2*d^2 
*f^3*g + 9*sqrt(f*x + e)*a*c*d^3*f^3*g - 4*(f*x + e)^(3/2)*a*d^4*e*f*h + 4 
*sqrt(f*x + e)*a*d^4*e^2*f*h + 7*(f*x + e)^(3/2)*b*c^2*d^2*f^2*h - 3*(f*x 
+ e)^(3/2)*a*c*d^3*f^2*h - 9*sqrt(f*x + e)*b*c^2*d^2*e*f^2*h + sqrt(f*x...
 

Mupad [B] (verification not implemented)

Time = 84.56 (sec) , antiderivative size = 802140, normalized size of antiderivative = 1554.53 \[ \int \frac {g+h x}{(a+b x) (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

- ((2*(f^3*g - e*f^2*h))/(a*c*f^2 + b*d*e^2 - a*d*e*f - b*c*e*f) + ((e + f 
*x)^2*(15*a^2*d^4*f^3*g - 3*a^2*c*d^3*f^3*h - 12*a^2*d^4*e*f^2*h - 4*b^2*d 
^4*e^2*f*g + 8*b^2*c^2*d^2*f^3*g - 15*b^2*c^2*d^2*e*f^2*h - 27*a*b*c*d^3*f 
^3*g - 3*a*b*d^4*e*f^2*g + 4*a*b*d^4*e^2*f*h + 7*a*b*c^2*d^2*f^3*h + 11*b^ 
2*c*d^3*e*f^2*g + 19*a*b*c*d^3*e*f^2*h))/(4*(c*f - d*e)^2*(a^2*d^2 + b^2*c 
^2 - 2*a*b*c*d)*(a*c*f^2 + b*d*e^2 - a*d*e*f - b*c*e*f)) + ((e + f*x)*(25* 
a^2*d^3*f^3*g - 5*a^2*c*d^2*f^3*h + 16*b^2*c^2*d*f^3*g - 20*a^2*d^3*e*f^2* 
h - 4*b^2*d^3*e^2*f*g - 45*a*b*c*d^2*f^3*g + 9*a*b*c^2*d*f^3*h - 5*a*b*d^3 
*e*f^2*g + 4*a*b*d^3*e^2*f*h + 13*b^2*c*d^2*e*f^2*g - 25*b^2*c^2*d*e*f^2*h 
 + 37*a*b*c*d^2*e*f^2*h))/(4*(c*f - d*e)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*( 
a*c*f^2 + b*d*e^2 - a*d*e*f - b*c*e*f)))/(d^2*(e + f*x)^(5/2) - (e + f*x)^ 
(3/2)*(2*d^2*e - 2*c*d*f) + (e + f*x)^(1/2)*(c^2*f^2 + d^2*e^2 - 2*c*d*e*f 
)) - atan((((e + f*x)^(1/2)*(8192*a^3*b^15*c^21*d^3*f^20*g^2 - 49152*a^4*b 
^14*c^20*d^4*f^20*g^2 + 279680*a^5*b^13*c^19*d^5*f^20*g^2 - 1480960*a^6*b^ 
12*c^18*d^6*f^20*g^2 + 5092992*a^7*b^11*c^17*d^7*f^20*g^2 - 11152384*a^8*b 
^10*c^16*d^8*f^20*g^2 + 16285952*a^9*b^9*c^15*d^9*f^20*g^2 - 16381440*a^10 
*b^8*c^14*d^10*f^20*g^2 + 11475200*a^11*b^7*c^13*d^11*f^20*g^2 - 5532672*a 
^12*b^6*c^12*d^12*f^20*g^2 + 1759872*a^13*b^5*c^11*d^13*f^20*g^2 - 334080* 
a^14*b^4*c^10*d^14*f^20*g^2 + 28800*a^15*b^3*c^9*d^15*f^20*g^2 + 36992*a^5 
*b^13*c^21*d^3*f^20*h^2 - 260352*a^6*b^12*c^20*d^4*f^20*h^2 + 809600*a^...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 10467, normalized size of antiderivative = 20.28 \[ \int \frac {g+h x}{(a+b x) (c+d x)^3 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 8*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b 
)*sqrt(a*f - b*e)))*a*b**2*c**6*f**4*h + 32*sqrt(b)*sqrt(e + f*x)*sqrt(a*f 
 - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2*c**5*d*e* 
f**3*h - 16*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/( 
sqrt(b)*sqrt(a*f - b*e)))*a*b**2*c**5*d*f**4*h*x - 48*sqrt(b)*sqrt(e + f*x 
)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2 
*c**4*d**2*e**2*f**2*h + 64*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sq 
rt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2*c**4*d**2*e*f**3*h*x - 8* 
sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt 
(a*f - b*e)))*a*b**2*c**4*d**2*f**4*h*x**2 + 32*sqrt(b)*sqrt(e + f*x)*sqrt 
(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2*c**3* 
d**3*e**3*f*h - 96*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f* 
x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2*c**3*d**3*e**2*f**2*h*x + 32*sqrt( 
b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f 
- b*e)))*a*b**2*c**3*d**3*e*f**3*h*x**2 - 8*sqrt(b)*sqrt(e + f*x)*sqrt(a*f 
 - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2*c**2*d**4 
*e**4*h + 64*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/ 
(sqrt(b)*sqrt(a*f - b*e)))*a*b**2*c**2*d**4*e**3*f*h*x - 48*sqrt(b)*sqrt(e 
 + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))* 
a*b**2*c**2*d**4*e**2*f**2*h*x**2 - 16*sqrt(b)*sqrt(e + f*x)*sqrt(a*f -...