\(\int \frac {g+h x}{(a+b x)^4 (c+d x)^2 \sqrt {e+f x}} \, dx\) [139]

Optimal result

Integrand size = 29, antiderivative size = 964 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 \sqrt {e+f x}} \, dx=\frac {d \left (a^3 d^2 f^2 (8 d f g+19 d e h-27 c f h)-b^3 \left (32 d^3 e^3 g-c^2 d e f (7 f g-10 e h)-c^3 f^2 (5 f g-6 e h)-12 c d^2 e^2 (f g+2 e h)\right )+a b^2 \left (c^3 f^3 h-c^2 d f^2 (22 f g-31 e h)+4 d^3 e^2 (21 f g+2 e h)-2 c d^2 e f (19 f g+32 e h)\right )-a^2 b d f \left (6 c^2 f^2 h+d^2 e (65 f g+22 e h)-c d f (41 f g+52 e h)\right )\right ) \sqrt {e+f x}}{8 (b c-a d)^4 (b e-a f)^3 (d e-c f) (c+d x)}-\frac {(b g-a h) \sqrt {e+f x}}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}+\frac {\left (7 a^2 d f h+b^2 (8 d e g+5 c f g-6 c e h)-a b (13 d f g+2 d e h-c f h)\right ) \sqrt {e+f x}}{12 (b c-a d)^2 (b e-a f)^2 (a+b x)^2 (c+d x)}+\frac {\left (35 a^3 d^2 f^2 h-a^2 b d f (89 d f g+32 d e h-16 c f h)-b^3 \left (48 d^2 e^2 g+2 c d e (13 f g-18 e h)+3 c^2 f (5 f g-6 e h)\right )-a b^2 \left (3 c^2 f^2 h-2 c d f (28 f g-41 e h)-2 d^2 e (61 f g+6 e h)\right )\right ) \sqrt {e+f x}}{24 (b c-a d)^3 (b e-a f)^3 (a+b x) (c+d x)}+\frac {\sqrt {b} \left (35 a^4 d^3 f^3 h-35 a^3 b d^2 f^2 (3 d f g+2 d e h-c f h)+b^4 \left (64 d^3 e^3 g+c^3 f^2 (5 f g-6 e h)+4 c^2 d e f (3 f g-4 e h)+24 c d^2 e^2 (f g-2 e h)\right )-7 a^2 b^2 d f \left (c^2 f^2 h-c d f (9 f g-22 e h)-4 d^2 e (9 f g+2 e h)\right )+a b^3 \left (c^3 f^3 h-c^2 d f^2 (27 f g-38 e h)-8 c d^2 e f (9 f g-19 e h)-8 d^3 e^2 (27 f g+2 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{8 (b c-a d)^5 (b e-a f)^{7/2}}-\frac {d^{5/2} \left (a d (d f g-2 d e h+c f h)+b \left (8 d^2 e g+7 c^2 f h-3 c d (3 f g+2 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d)^5 (d e-c f)^{3/2}} \] Output:

1/8*d*(a^3*d^2*f^2*(-27*c*f*h+19*d*e*h+8*d*f*g)-b^3*(32*d^3*e^3*g-c^2*d*e* 
f*(-10*e*h+7*f*g)-c^3*f^2*(-6*e*h+5*f*g)-12*c*d^2*e^2*(2*e*h+f*g))+a*b^2*( 
c^3*f^3*h-c^2*d*f^2*(-31*e*h+22*f*g)+4*d^3*e^2*(2*e*h+21*f*g)-2*c*d^2*e*f* 
(32*e*h+19*f*g))-a^2*b*d*f*(6*c^2*f^2*h+d^2*e*(22*e*h+65*f*g)-c*d*f*(52*e* 
h+41*f*g)))*(f*x+e)^(1/2)/(-a*d+b*c)^4/(-a*f+b*e)^3/(-c*f+d*e)/(d*x+c)-1/3 
*(-a*h+b*g)*(f*x+e)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^3/(d*x+c)+1/12*(7* 
a^2*d*f*h+b^2*(-6*c*e*h+5*c*f*g+8*d*e*g)-a*b*(-c*f*h+2*d*e*h+13*d*f*g))*(f 
*x+e)^(1/2)/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)^2/(d*x+c)+1/24*(35*a^3*d^2*f 
^2*h-a^2*b*d*f*(-16*c*f*h+32*d*e*h+89*d*f*g)-b^3*(48*d^2*e^2*g+2*c*d*e*(-1 
8*e*h+13*f*g)+3*c^2*f*(-6*e*h+5*f*g))-a*b^2*(3*c^2*f^2*h-2*c*d*f*(-41*e*h+ 
28*f*g)-2*d^2*e*(6*e*h+61*f*g)))*(f*x+e)^(1/2)/(-a*d+b*c)^3/(-a*f+b*e)^3/( 
b*x+a)/(d*x+c)+1/8*b^(1/2)*(35*a^4*d^3*f^3*h-35*a^3*b*d^2*f^2*(-c*f*h+2*d* 
e*h+3*d*f*g)+b^4*(64*d^3*e^3*g+c^3*f^2*(-6*e*h+5*f*g)+4*c^2*d*e*f*(-4*e*h+ 
3*f*g)+24*c*d^2*e^2*(-2*e*h+f*g))-7*a^2*b^2*d*f*(c^2*f^2*h-c*d*f*(-22*e*h+ 
9*f*g)-4*d^2*e*(2*e*h+9*f*g))+a*b^3*(c^3*f^3*h-c^2*d*f^2*(-38*e*h+27*f*g)- 
8*c*d^2*e*f*(-19*e*h+9*f*g)-8*d^3*e^2*(2*e*h+27*f*g)))*arctanh(b^(1/2)*(f* 
x+e)^(1/2)/(-a*f+b*e)^(1/2))/(-a*d+b*c)^5/(-a*f+b*e)^(7/2)-d^(5/2)*(a*d*(c 
*f*h-2*d*e*h+d*f*g)+b*(8*d^2*e*g+7*c^2*f*h-3*c*d*(2*e*h+3*f*g)))*arctanh(d 
^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/(-a*d+b*c)^5/(-c*f+d*e)^(3/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(5396\) vs. \(2(964)=1928\).

Time = 16.21 (sec) , antiderivative size = 5396, normalized size of antiderivative = 5.60 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 \sqrt {e+f x}} \, dx=\text {Result too large to show} \] Input:

Integrate[(g + h*x)/((a + b*x)^4*(c + d*x)^2*Sqrt[e + f*x]),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.45 (sec) , antiderivative size = 1063, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {168, 27, 168, 27, 168, 27, 168, 25, 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.

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\int \frac {d f (24 d f g+22 d e h-11 c f h) a^2+b \left (-2 e (41 f g+6 e h) d^2-c f (31 f g-52 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (3 f (5 f g-6 e h) c^2+2 d e (13 f g-18 e h) c+48 d^2 e^2 g\right )+5 d f \left (7 d f h a^2-b (13 d f g+2 d e h-c f h) a+b^2 (8 d e g+5 c f g-6 c e h)\right ) x}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (7 a^2 d f h-a b (-c f h+2 d e h+13 d f g)+b^2 (-6 c e h+5 c f g+8 d e g)\right )}{2 (a+b x)^2 (c+d x) (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{3 (a+b x)^3 (c+d x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {d f (24 d f g+22 d e h-11 c f h) a^2+b \left (-2 e (41 f g+6 e h) d^2-c f (31 f g-52 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (3 f (5 f g-6 e h) c^2+2 d e (13 f g-18 e h) c+48 d^2 e^2 g\right )+5 d f \left (7 d f h a^2-b (13 d f g+2 d e h-c f h) a+b^2 (8 d e g+5 c f g-6 c e h)\right ) x}{(a+b x)^2 (c+d x)^2 \sqrt {e+f x}}dx}{4 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (7 a^2 d f h-a b (-c f h+2 d e h+13 d f g)+b^2 (-6 c e h+5 c f g+8 d e g)\right )}{2 (a+b x)^2 (c+d x) (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{3 (a+b x)^3 (c+d x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\frac {\sqrt {e+f x} \left (35 a^3 d^2 f^2 h-a^2 b d f (-16 c f h+32 d e h+89 d f g)-a b^2 \left (3 c^2 f^2 h-2 c d f (28 f g-41 e h)-2 d^2 e (6 e h+61 f g)\right )-b^3 \left (3 c^2 f (5 f g-6 e h)+2 c d e (13 f g-18 e h)+48 d^2 e^2 g\right )\right )}{(a+b x) (c+d x) (b c-a d) (b e-a f)}-\frac {\int \frac {3 \left (d^2 f^2 (19 c f h-2 d (8 f g+19 e h)) a^3-b d f \left (-2 e (65 f g+22 e h) d^2-c f (7 f g-72 e h) d+4 c^2 f^2 h\right ) a^2+b^2 \left (-8 e^2 (21 f g+2 e h) d^3-2 c e f (23 f g-58 e h) d^2-4 c^2 f^2 (3 f g-5 e h) d+c^3 f^3 h\right ) a+b^3 \left (f^2 (5 f g-6 e h) c^3+4 d e f (3 f g-4 e h) c^2+24 d^2 e^2 (f g-2 e h) c+64 d^3 e^3 g\right )-d f \left (35 d^2 f^2 h a^3-b d f (89 d f g+32 d e h-16 c f h) a^2-b^2 \left (-2 e (61 f g+6 e h) d^2-2 c f (28 f g-41 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (3 f (5 f g-6 e h) c^2+2 d e (13 f g-18 e h) c+48 d^2 e^2 g\right )\right ) x\right )}{2 (a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (7 a^2 d f h-a b (-c f h+2 d e h+13 d f g)+b^2 (-6 c e h+5 c f g+8 d e g)\right )}{2 (a+b x)^2 (c+d x) (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{3 (a+b x)^3 (c+d x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\sqrt {e+f x} \left (35 a^3 d^2 f^2 h-a^2 b d f (-16 c f h+32 d e h+89 d f g)-a b^2 \left (3 c^2 f^2 h-2 c d f (28 f g-41 e h)-2 d^2 e (6 e h+61 f g)\right )-b^3 \left (3 c^2 f (5 f g-6 e h)+2 c d e (13 f g-18 e h)+48 d^2 e^2 g\right )\right )}{(a+b x) (c+d x) (b c-a d) (b e-a f)}-\frac {3 \int \frac {d^2 f^2 (19 c f h-2 d (8 f g+19 e h)) a^3-b d f \left (-2 e (65 f g+22 e h) d^2-c f (7 f g-72 e h) d+4 c^2 f^2 h\right ) a^2+b^2 \left (-8 e^2 (21 f g+2 e h) d^3-2 c e f (23 f g-58 e h) d^2-4 c^2 f^2 (3 f g-5 e h) d+c^3 f^3 h\right ) a+b^3 \left (f^2 (5 f g-6 e h) c^3+4 d e f (3 f g-4 e h) c^2+24 d^2 e^2 (f g-2 e h) c+64 d^3 e^3 g\right )-d f \left (35 d^2 f^2 h a^3-b d f (89 d f g+32 d e h-16 c f h) a^2-b^2 \left (-2 e (61 f g+6 e h) d^2-2 c f (28 f g-41 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (3 f (5 f g-6 e h) c^2+2 d e (13 f g-18 e h) c+48 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (7 a^2 d f h-a b (-c f h+2 d e h+13 d f g)+b^2 (-6 c e h+5 c f g+8 d e g)\right )}{2 (a+b x)^2 (c+d x) (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{3 (a+b x)^3 (c+d x) (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (7 d f h a^2-b (13 d f g+2 d e h-c f h) a+b^2 (8 d e g+5 c f g-6 c e h)\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)}-\frac {\frac {\left (35 d^2 f^2 h a^3-b d f (89 d f g+32 d e h-16 c f h) a^2-b^2 \left (-2 e (61 f g+6 e h) d^2-2 c f (28 f g-41 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (3 f (5 f g-6 e h) c^2+2 d e (13 f g-18 e h) c+48 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x) (c+d x)}-\frac {3 \left (\frac {\int -\frac {8 d^3 f^3 (d f g-2 d e h+c f h) a^4+b d^2 f^2 \left (8 e (5 f g+6 e h) d^2-c f (64 f g+53 e h) d+29 c^2 f^2 h\right ) a^3-b^2 d f \left (24 e^2 (7 f g+2 e h) d^3-c e f (151 f g+146 e h) d^2-c^2 f^2 (41 f g-116 e h) d+6 c^3 f^3 h\right ) a^2+b^3 \left (8 e^3 (23 f g+2 e h) d^4-12 c e^2 f (11 f g+12 e h) d^3-2 c^2 e f^2 (19 f g-52 e h) d^2-c^3 f^3 (22 f g-31 e h) d+c^4 f^4 h\right ) a-b^4 (d e-c f) \left (f^2 (5 f g-6 e h) c^3+4 d e f (3 f g-4 e h) c^2+24 d^2 e^2 (f g-2 e h) c+64 d^3 e^3 g\right )+b d f \left (d^2 f^2 (8 d f g+19 d e h-27 c f h) a^3-b d f \left (e (65 f g+22 e h) d^2-c f (41 f g+52 e h) d+6 c^2 f^2 h\right ) a^2+b^2 \left (4 e^2 (21 f g+2 e h) d^3-2 c e f (19 f g+32 e h) d^2-c^2 f^2 (22 f g-31 e h) d+c^3 f^3 h\right ) a-b^3 \left (-f^2 (5 f g-6 e h) c^3-d e f (7 f g-10 e h) c^2-12 d^2 e^2 (f g+2 e h) c+32 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}-\frac {2 d \left (d^2 f^2 (8 d f g+19 d e h-27 c f h) a^3-b d f \left (e (65 f g+22 e h) d^2-c f (41 f g+52 e h) d+6 c^2 f^2 h\right ) a^2+b^2 \left (4 e^2 (21 f g+2 e h) d^3-2 c e f (19 f g+32 e h) d^2-c^2 f^2 (22 f g-31 e h) d+c^3 f^3 h\right ) a-b^3 \left (-f^2 (5 f g-6 e h) c^3-d e f (7 f g-10 e h) c^2-12 d^2 e^2 (f g+2 e h) c+32 d^3 e^3 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (d e-c f) (c+d x)}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (7 d f h a^2-b (13 d f g+2 d e h-c f h) a+b^2 (8 d e g+5 c f g-6 c e h)\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)}-\frac {\frac {\left (35 d^2 f^2 h a^3-b d f (89 d f g+32 d e h-16 c f h) a^2-b^2 \left (-2 e (61 f g+6 e h) d^2-2 c f (28 f g-41 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (3 f (5 f g-6 e h) c^2+2 d e (13 f g-18 e h) c+48 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x) (c+d x)}-\frac {3 \left (-\frac {2 d \sqrt {e+f x} \left (d^2 f^2 (8 d f g+19 d e h-27 c f h) a^3-b d f \left (e (65 f g+22 e h) d^2-c f (41 f g+52 e h) d+6 c^2 f^2 h\right ) a^2+b^2 \left (4 e^2 (21 f g+2 e h) d^3-2 c e f (19 f g+32 e h) d^2-c^2 f^2 (22 f g-31 e h) d+c^3 f^3 h\right ) a-b^3 \left (-f^2 (5 f g-6 e h) c^3-d e f (7 f g-10 e h) c^2-12 d^2 e^2 (f g+2 e h) c+32 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}-\frac {\int \frac {8 d^3 f^3 (d f g-2 d e h+c f h) a^4+b d^2 f^2 \left (8 e (5 f g+6 e h) d^2-c f (64 f g+53 e h) d+29 c^2 f^2 h\right ) a^3-b^2 d f \left (24 e^2 (7 f g+2 e h) d^3-c e f (151 f g+146 e h) d^2-c^2 f^2 (41 f g-116 e h) d+6 c^3 f^3 h\right ) a^2+b^3 \left (8 e^3 (23 f g+2 e h) d^4-12 c e^2 f (11 f g+12 e h) d^3-2 c^2 e f^2 (19 f g-52 e h) d^2-c^3 f^3 (22 f g-31 e h) d+c^4 f^4 h\right ) a-b^4 (d e-c f) \left (f^2 (5 f g-6 e h) c^3+4 d e f (3 f g-4 e h) c^2+24 d^2 e^2 (f g-2 e h) c+64 d^3 e^3 g\right )+b d f \left (d^2 f^2 (8 d f g+19 d e h-27 c f h) a^3-b d f \left (e (65 f g+22 e h) d^2-c f (41 f g+52 e h) d+6 c^2 f^2 h\right ) a^2+b^2 \left (4 e^2 (21 f g+2 e h) d^3-2 c e f (19 f g+32 e h) d^2-c^2 f^2 (22 f g-31 e h) d+c^3 f^3 h\right ) a-b^3 \left (-f^2 (5 f g-6 e h) c^3-d e f (7 f g-10 e h) c^2-12 d^2 e^2 (f g+2 e h) c+32 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (7 d f h a^2-b (13 d f g+2 d e h-c f h) a+b^2 (8 d e g+5 c f g-6 c e h)\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)}-\frac {\frac {\left (35 d^2 f^2 h a^3-b d f (89 d f g+32 d e h-16 c f h) a^2-b^2 \left (-2 e (61 f g+6 e h) d^2-2 c f (28 f g-41 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (3 f (5 f g-6 e h) c^2+2 d e (13 f g-18 e h) c+48 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x) (c+d x)}-\frac {3 \left (-\frac {2 d \sqrt {e+f x} \left (d^2 f^2 (8 d f g+19 d e h-27 c f h) a^3-b d f \left (e (65 f g+22 e h) d^2-c f (41 f g+52 e h) d+6 c^2 f^2 h\right ) a^2+b^2 \left (4 e^2 (21 f g+2 e h) d^3-2 c e f (19 f g+32 e h) d^2-c^2 f^2 (22 f g-31 e h) d+c^3 f^3 h\right ) a-b^3 \left (-f^2 (5 f g-6 e h) c^3-d e f (7 f g-10 e h) c^2-12 d^2 e^2 (f g+2 e h) c+32 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}-\frac {\frac {8 d^3 (b e-a f)^3 \left (a d (d f g-2 d e h+c f h)+b \left (7 f h c^2-3 d (3 f g+2 e h) c+8 d^2 e g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}-\frac {b (d e-c f) \left (35 d^3 f^3 h a^4-35 b d^2 f^2 (3 d f g+2 d e h-c f h) a^3-7 b^2 d f \left (-4 e (9 f g+2 e h) d^2-c f (9 f g-22 e h) d+c^2 f^2 h\right ) a^2+b^3 \left (-8 e^2 (27 f g+2 e h) d^3-8 c e f (9 f g-19 e h) d^2-c^2 f^2 (27 f g-38 e h) d+c^3 f^3 h\right ) a+b^4 \left (f^2 (5 f g-6 e h) c^3+4 d e f (3 f g-4 e h) c^2+24 d^2 e^2 (f g-2 e h) c+64 d^3 e^3 g\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}}{(b c-a d) (d e-c f)}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (7 d f h a^2-b (13 d f g+2 d e h-c f h) a+b^2 (8 d e g+5 c f g-6 c e h)\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)}-\frac {\frac {\left (35 d^2 f^2 h a^3-b d f (89 d f g+32 d e h-16 c f h) a^2-b^2 \left (-2 e (61 f g+6 e h) d^2-2 c f (28 f g-41 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (3 f (5 f g-6 e h) c^2+2 d e (13 f g-18 e h) c+48 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x) (c+d x)}-\frac {3 \left (-\frac {2 d \sqrt {e+f x} \left (d^2 f^2 (8 d f g+19 d e h-27 c f h) a^3-b d f \left (e (65 f g+22 e h) d^2-c f (41 f g+52 e h) d+6 c^2 f^2 h\right ) a^2+b^2 \left (4 e^2 (21 f g+2 e h) d^3-2 c e f (19 f g+32 e h) d^2-c^2 f^2 (22 f g-31 e h) d+c^3 f^3 h\right ) a-b^3 \left (-f^2 (5 f g-6 e h) c^3-d e f (7 f g-10 e h) c^2-12 d^2 e^2 (f g+2 e h) c+32 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}-\frac {\frac {16 d^3 (b e-a f)^3 \left (a d (d f g-2 d e h+c f h)+b \left (7 f h c^2-3 d (3 f g+2 e h) c+8 d^2 e g\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{(b c-a d) f}-\frac {2 b (d e-c f) \left (35 d^3 f^3 h a^4-35 b d^2 f^2 (3 d f g+2 d e h-c f h) a^3-7 b^2 d f \left (-4 e (9 f g+2 e h) d^2-c f (9 f g-22 e h) d+c^2 f^2 h\right ) a^2+b^3 \left (-8 e^2 (27 f g+2 e h) d^3-8 c e f (9 f g-19 e h) d^2-c^2 f^2 (27 f g-38 e h) d+c^3 f^3 h\right ) a+b^4 \left (f^2 (5 f g-6 e h) c^3+4 d e f (3 f g-4 e h) c^2+24 d^2 e^2 (f g-2 e h) c+64 d^3 e^3 g\right )\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{(b c-a d) f}}{(b c-a d) (d e-c f)}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (7 d f h a^2-b (13 d f g+2 d e h-c f h) a+b^2 (8 d e g+5 c f g-6 c e h)\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)}-\frac {\frac {\left (35 d^2 f^2 h a^3-b d f (89 d f g+32 d e h-16 c f h) a^2-b^2 \left (-2 e (61 f g+6 e h) d^2-2 c f (28 f g-41 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (3 f (5 f g-6 e h) c^2+2 d e (13 f g-18 e h) c+48 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x) (c+d x)}-\frac {3 \left (-\frac {2 d \sqrt {e+f x} \left (d^2 f^2 (8 d f g+19 d e h-27 c f h) a^3-b d f \left (e (65 f g+22 e h) d^2-c f (41 f g+52 e h) d+6 c^2 f^2 h\right ) a^2+b^2 \left (4 e^2 (21 f g+2 e h) d^3-2 c e f (19 f g+32 e h) d^2-c^2 f^2 (22 f g-31 e h) d+c^3 f^3 h\right ) a-b^3 \left (-f^2 (5 f g-6 e h) c^3-d e f (7 f g-10 e h) c^2-12 d^2 e^2 (f g+2 e h) c+32 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}-\frac {\frac {2 \sqrt {b} (d e-c f) \left (35 d^3 f^3 h a^4-35 b d^2 f^2 (3 d f g+2 d e h-c f h) a^3-7 b^2 d f \left (-4 e (9 f g+2 e h) d^2-c f (9 f g-22 e h) d+c^2 f^2 h\right ) a^2+b^3 \left (-8 e^2 (27 f g+2 e h) d^3-8 c e f (9 f g-19 e h) d^2-c^2 f^2 (27 f g-38 e h) d+c^3 f^3 h\right ) a+b^4 \left (f^2 (5 f g-6 e h) c^3+4 d e f (3 f g-4 e h) c^2+24 d^2 e^2 (f g-2 e h) c+64 d^3 e^3 g\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d) \sqrt {b e-a f}}-\frac {16 d^{5/2} (b e-a f)^3 \left (a d (d f g-2 d e h+c f h)+b \left (7 f h c^2-3 d (3 f g+2 e h) c+8 d^2 e g\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d) \sqrt {d e-c f}}}{(b c-a d) (d e-c f)}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}\)

Input:

Int[(g + h*x)/((a + b*x)^4*(c + d*x)^2*Sqrt[e + f*x]),x]
 

Output:

-1/3*((b*g - a*h)*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^3*(c + 
 d*x)) - (-1/2*((7*a^2*d*f*h + b^2*(8*d*e*g + 5*c*f*g - 6*c*e*h) - a*b*(13 
*d*f*g + 2*d*e*h - c*f*h))*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b* 
x)^2*(c + d*x)) - (((35*a^3*d^2*f^2*h - a^2*b*d*f*(89*d*f*g + 32*d*e*h - 1 
6*c*f*h) - b^3*(48*d^2*e^2*g + 2*c*d*e*(13*f*g - 18*e*h) + 3*c^2*f*(5*f*g 
- 6*e*h)) - a*b^2*(3*c^2*f^2*h - 2*c*d*f*(28*f*g - 41*e*h) - 2*d^2*e*(61*f 
*g + 6*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x)) 
 - (3*((-2*d*(a^3*d^2*f^2*(8*d*f*g + 19*d*e*h - 27*c*f*h) - b^3*(32*d^3*e^ 
3*g - c^2*d*e*f*(7*f*g - 10*e*h) - c^3*f^2*(5*f*g - 6*e*h) - 12*c*d^2*e^2* 
(f*g + 2*e*h)) + a*b^2*(c^3*f^3*h - c^2*d*f^2*(22*f*g - 31*e*h) + 4*d^3*e^ 
2*(21*f*g + 2*e*h) - 2*c*d^2*e*f*(19*f*g + 32*e*h)) - a^2*b*d*f*(6*c^2*f^2 
*h + d^2*e*(65*f*g + 22*e*h) - c*d*f*(41*f*g + 52*e*h)))*Sqrt[e + f*x])/(( 
b*c - a*d)*(d*e - c*f)*(c + d*x)) - ((2*Sqrt[b]*(d*e - c*f)*(35*a^4*d^3*f^ 
3*h - 35*a^3*b*d^2*f^2*(3*d*f*g + 2*d*e*h - c*f*h) + b^4*(64*d^3*e^3*g + c 
^3*f^2*(5*f*g - 6*e*h) + 4*c^2*d*e*f*(3*f*g - 4*e*h) + 24*c*d^2*e^2*(f*g - 
 2*e*h)) - 7*a^2*b^2*d*f*(c^2*f^2*h - c*d*f*(9*f*g - 22*e*h) - 4*d^2*e*(9* 
f*g + 2*e*h)) + a*b^3*(c^3*f^3*h - c^2*d*f^2*(27*f*g - 38*e*h) - 8*c*d^2*e 
*f*(9*f*g - 19*e*h) - 8*d^3*e^2*(27*f*g + 2*e*h)))*ArcTanh[(Sqrt[b]*Sqrt[e 
 + f*x])/Sqrt[b*e - a*f]])/((b*c - a*d)*Sqrt[b*e - a*f]) - (16*d^(5/2)*(b* 
e - a*f)^3*(a*d*(d*f*g - 2*d*e*h + c*f*h) + b*(8*d^2*e*g + 7*c^2*f*h - ...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 = 50.12 (sec) , antiderivative size = 1477, normalized size of antiderivative = 1.53

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(1477\)
derivativedivides	\(\text {Expression too large to display}\)	\(1531\)
default	\(\text {Expression too large to display}\)	\(1531\)

Input:

int((h*x+g)/(b*x+a)^4/(d*x+c)^2/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/((a*f-b*e)*b)^(1/2)*(35/8*(d*x+c)*((c*f-d*e)*d)^(1/2)*(c*f-d*e)*(b*x+a) 
^3*b*((64/35*d^3*e^3*g-48/35*(e*h-1/2*f*g)*c*e^2*d^2-16/35*(e*h-3/4*f*g)*c 
^2*f*e*d-6/35*(e*h-5/6*f*g)*c^3*f^2)*b^4+1/35*a*((-16*e^3*h-216*e^2*f*g)*d 
^3+(152*c*e^2*f*h-72*c*e*f^2*g)*d^2+(38*c^2*e*f^2*h-27*c^2*f^3*g)*d+c^3*f^ 
3*h)*b^3-1/5*a^2*d*f*((-8*e^2*h-36*e*f*g)*d^2+(22*c*e*f*h-9*c*f^2*g)*d+c^2 
*f^2*h)*b^2+a^3*((-2*e*h-3*f*g)*d+c*f*h)*d^2*f^2*b+a^4*d^3*f^3*h)*arctan(b 
*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))+(-d^3*(d*x+c)*((8*d^2*e*g+(-6*c*e*h-9* 
c*f*g)*d+7*c^2*f*h)*b+a*((-2*e*h+f*g)*d+c*f*h)*d)*(b*x+a)^3*(a*f-b*e)^3*ar 
ctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))+(a*d-b*c)*(f*x+e)^(1/2)*((c*f-d* 
e)*d)^(1/2)*((4*d^4*e^3*g*x^3+2*x^2*c*(-3/4*f*g*x+e*(-3/2*h*x+g))*e^2*d^3- 
2/3*x*c^2*e*(21/16*x^2*g*f^2+11/8*x*e*(-15/11*h*x+g)*f+e^2*(9/4*h*x+g))*d^ 
2+1/3*(-15/8*f^3*g*x^3-11/8*x^2*(-18/11*h*x+g)*e*f^2+3/4*e^2*x*(3*h*x+g)*f 
+e^3*(3/2*h*x+g))*c^3*d-1/3*c^4*f*(15/8*x^2*g*f^2-5/4*x*(9/5*h*x+g)*e*f+(3 
/2*h*x+g)*e^2))*b^6-1/6*a*((63*e^2*f*g*x^3-60*x^2*(-1/10*h*x+g)*e^3)*d^4-3 
2*x*c*(57/64*x^2*g*f^2-13/8*(-12/13*h*x+g)*x*e*f+e^2*(-3/2*h*x+g))*e*d^3+1 
0*c^2*(-33/20*f^3*g*x^3-7/20*x^2*(-93/14*h*x+g)*e*f^2+7/20*x*(-83/7*h*x+g) 
*e^2*f+e^3*(23/10*h*x+g))*d^2-c^3*((-3/4*h*x^3+13/2*g*x^2)*f^3-37/2*x*(43/ 
74*h*x+g)*e*f^2+7/2*e^2*(3*h*x+g)*f+e^3*h)*d+c^4*((3/4*h*x^2+10*g*x)*f^2-1 
3/2*(25/13*h*x+g)*e*f+e^2*h)*f)*b^5+2/3*a^2*(11*x*e*(195/176*x^2*g*f^2-317 
/88*x*(-33/317*h*x+g)*e*f+e^2*(-15/44*h*x+g))*d^4+13/2*c*(-123/104*f^3*...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)^4/(d*x+c)^2/(f*x+e)^(1/2),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)**4/(d*x+c)**2/(f*x+e)**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((h*x+g)/(b*x+a)^4/(d*x+c)^2/(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(c*f-d*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. 2757 vs. \(2 (926) = 1852\).

Time = 0.28 (sec) , antiderivative size = 2757, normalized size of antiderivative = 2.86 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)/(b*x+a)^4/(d*x+c)^2/(f*x+e)^(1/2),x, algorithm="giac")
 

Output:

-1/8*(64*b^5*d^3*e^3*g + 24*b^5*c*d^2*e^2*f*g - 216*a*b^4*d^3*e^2*f*g + 12 
*b^5*c^2*d*e*f^2*g - 72*a*b^4*c*d^2*e*f^2*g + 252*a^2*b^3*d^3*e*f^2*g + 5* 
b^5*c^3*f^3*g - 27*a*b^4*c^2*d*f^3*g + 63*a^2*b^3*c*d^2*f^3*g - 105*a^3*b^ 
2*d^3*f^3*g - 48*b^5*c*d^2*e^3*h - 16*a*b^4*d^3*e^3*h - 16*b^5*c^2*d*e^2*f 
*h + 152*a*b^4*c*d^2*e^2*f*h + 56*a^2*b^3*d^3*e^2*f*h - 6*b^5*c^3*e*f^2*h 
+ 38*a*b^4*c^2*d*e*f^2*h - 154*a^2*b^3*c*d^2*e*f^2*h - 70*a^3*b^2*d^3*e*f^ 
2*h + a*b^4*c^3*f^3*h - 7*a^2*b^3*c^2*d*f^3*h + 35*a^3*b^2*c*d^2*f^3*h + 3 
5*a^4*b*d^3*f^3*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^8*c^5* 
e^3 - 5*a*b^7*c^4*d*e^3 + 10*a^2*b^6*c^3*d^2*e^3 - 10*a^3*b^5*c^2*d^3*e^3 
+ 5*a^4*b^4*c*d^4*e^3 - a^5*b^3*d^5*e^3 - 3*a*b^7*c^5*e^2*f + 15*a^2*b^6*c 
^4*d*e^2*f - 30*a^3*b^5*c^3*d^2*e^2*f + 30*a^4*b^4*c^2*d^3*e^2*f - 15*a^5* 
b^3*c*d^4*e^2*f + 3*a^6*b^2*d^5*e^2*f + 3*a^2*b^6*c^5*e*f^2 - 15*a^3*b^5*c 
^4*d*e*f^2 + 30*a^4*b^4*c^3*d^2*e*f^2 - 30*a^5*b^3*c^2*d^3*e*f^2 + 15*a^6* 
b^2*c*d^4*e*f^2 - 3*a^7*b*d^5*e*f^2 - a^3*b^5*c^5*f^3 + 5*a^4*b^4*c^4*d*f^ 
3 - 10*a^5*b^3*c^3*d^2*f^3 + 10*a^6*b^2*c^2*d^3*f^3 - 5*a^7*b*c*d^4*f^3 + 
a^8*d^5*f^3)*sqrt(-b^2*e + a*b*f)) + (8*b*d^5*e*g - 9*b*c*d^4*f*g + a*d^5* 
f*g - 6*b*c*d^4*e*h - 2*a*d^5*e*h + 7*b*c^2*d^3*f*h + a*c*d^4*f*h)*arctan( 
sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b^5*c^5*d*e - 5*a*b^4*c^4*d^2*e + 
10*a^2*b^3*c^3*d^3*e - 10*a^3*b^2*c^2*d^4*e + 5*a^4*b*c*d^5*e - a^5*d^6*e 
- b^5*c^6*f + 5*a*b^4*c^5*d*f - 10*a^2*b^3*c^4*d^2*f + 10*a^3*b^2*c^3*d...
 

Mupad [B] (verification not implemented)

Time = 44.96 (sec) , antiderivative size = 1088917, normalized size of antiderivative = 1129.58 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

int((g + h*x)/((e + f*x)^(1/2)*(a + b*x)^4*(c + d*x)^2),x)
 

Output:

- atan(((((256*a^17*b^2*d^17*e*f^10*g - 256*a^17*b^2*c*d^16*f^11*g - 160*a 
^3*b^16*c^15*d^2*f^11*g + 2304*a^4*b^15*c^14*d^3*f^11*g - 15552*a^5*b^14*c 
^13*d^4*f^11*g + 66048*a^6*b^13*c^12*d^5*f^11*g - 197856*a^7*b^12*c^11*d^6 
*f^11*g + 440320*a^8*b^11*c^10*d^7*f^11*g - 743808*a^9*b^10*c^9*d^8*f^11*g 
 + 958464*a^10*b^9*c^8*d^9*f^11*g - 937056*a^11*b^8*c^7*d^10*f^11*g + 6853 
12*a^12*b^7*c^6*d^11*f^11*g - 365760*a^13*b^6*c^5*d^12*f^11*g + 136704*a^1 
4*b^5*c^4*d^13*f^11*g - 33312*a^15*b^4*c^3*d^14*f^11*g + 4608*a^16*b^3*c^2 
*d^15*f^11*g - 32*a^4*b^15*c^15*d^2*f^11*h + 512*a^5*b^14*c^14*d^3*f^11*h 
- 4288*a^6*b^13*c^13*d^4*f^11*h + 21504*a^7*b^12*c^12*d^5*f^11*h - 68960*a 
^8*b^11*c^11*d^6*f^11*h + 148224*a^9*b^10*c^10*d^7*f^11*h - 219264*a^10*b^ 
9*c^9*d^8*f^11*h + 224256*a^11*b^8*c^8*d^9*f^11*h - 154848*a^12*b^7*c^7*d^ 
10*f^11*h + 66560*a^13*b^6*c^6*d^11*f^11*h - 12992*a^14*b^5*c^5*d^12*f^11* 
h - 2048*a^15*b^4*c^4*d^13*f^11*h + 1632*a^16*b^3*c^3*d^14*f^11*h - 256*a^ 
17*b^2*c^2*d^15*f^11*h + 1024*a^10*b^9*d^17*e^8*f^3*g - 6272*a^11*b^8*d^17 
*e^7*f^4*g + 15968*a^12*b^7*d^17*e^6*f^5*g - 21536*a^13*b^6*d^17*e^5*f^6*g 
 + 15904*a^14*b^5*d^17*e^4*f^7*g - 5600*a^15*b^4*d^17*e^3*f^8*g + 256*a^16 
*b^3*d^17*e^2*f^9*g - 256*a^11*b^8*d^17*e^8*f^3*h + 1600*a^12*b^7*d^17*e^7 
*f^4*h - 4192*a^13*b^6*d^17*e^6*f^5*h + 6048*a^14*b^5*d^17*e^5*f^6*h - 515 
2*a^15*b^4*d^17*e^4*f^7*h + 2464*a^16*b^3*d^17*e^3*f^8*h - 512*a^17*b^2*d^ 
17*e^2*f^9*h + 1024*b^19*c^10*d^7*e^8*f^3*g - 1920*b^19*c^11*d^6*e^7*f^...
 

Reduce [B] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 34058, normalized size of antiderivative = 35.33 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

int((h*x+g)/(b*x+a)^4/(d*x+c)^2/(f*x+e)^(1/2),x)
 

Output:

( - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - 
 b*e)))*a**7*c**3*d**3*f**5*h + 210*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + 
 f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*c**2*d**4*e*f**4*h - 105*sqrt(b)* 
sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*c** 
2*d**4*f**5*h*x - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt 
(b)*sqrt(a*f - b*e)))*a**7*c*d**5*e**2*f**3*h + 210*sqrt(b)*sqrt(a*f - b*e 
)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*c*d**5*e*f**4*h*x 
 - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - 
b*e)))*a**7*d**6*e**2*f**3*h*x - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e 
+ f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**4*d**2*f**5*h + 420*sqrt(b) 
*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b* 
c**3*d**3*e*f**4*h + 315*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(s 
qrt(b)*sqrt(a*f - b*e)))*a**6*b*c**3*d**3*f**5*g - 420*sqrt(b)*sqrt(a*f - 
b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**3*d**3*f* 
*5*h*x - 525*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt( 
a*f - b*e)))*a**6*b*c**2*d**4*e**2*f**3*h - 630*sqrt(b)*sqrt(a*f - b*e)*at 
an((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**2*d**4*e*f**4*g 
+ 1050*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - 
b*e)))*a**6*b*c**2*d**4*e*f**4*h*x + 315*sqrt(b)*sqrt(a*f - b*e)*atan((sqr 
t(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**2*d**4*f**5*g*x - 31...