\(\int \frac {g+h x}{(a+b x)^5 (c+d x)^3 \sqrt {e+f x}} \, dx\) [149]

Optimal result

Integrand size = 29, antiderivative size = 2077 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Output:

1/192*d*(a^4*d^3*f^3*(-1123*c*f*h+1027*d*e*h+96*d*f*g)+b^4*(1440*d^4*e^4*g 
-2*c^2*d^2*e^2*f*(-256*e*h+167*f*g)-c^3*d*e*f^2*(-232*e*h+185*f*g)-15*c^4* 
f^3*(-8*e*h+7*f*g)-240*c*d^3*e^3*(4*e*h+3*f*g))-a*b^3*(15*c^4*f^4*h-c^2*d^ 
2*e*f^2*(-1898*e*h+1223*f*g)-11*c^3*d*f^3*(-69*e*h+55*f*g)+240*d^4*e^3*(2* 
e*h+21*f*g)-4*c*d^3*e^2*f*(884*e*h+707*f*g))-a^3*b*d^2*f^2*(409*c^2*f^2*h+ 
d^2*e*(2154*e*h+2747*f*g)-c*d*f*(2947*e*h+2363*f*g))+a^2*b^2*d*f*(107*c^3* 
f^3*h-c^2*d*f^2*(-2425*e*h+1519*f*g)+2*d^3*e^2*(856*e*h+3073*f*g)-c*d^2*e* 
f*(4820*e*h+4051*f*g)))*(f*x+e)^(1/2)/(-a*d+b*c)^5/(-a*f+b*e)^4/(-c*f+d*e) 
/(d*x+c)^2-1/4*(-a*h+b*g)*(f*x+e)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^4/(d 
*x+c)^2+1/24*(11*a^2*d*f*h+b^2*(-8*c*e*h+7*c*f*g+12*d*e*g)-a*b*(-c*f*h+4*d 
*e*h+19*d*f*g))*(f*x+e)^(1/2)/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)^3/(d*x+c)^ 
2+1/96*(99*a^3*d^2*f^2*h-a^2*b*d*f*(-26*c*f*h+104*d*e*h+219*d*f*g)-b^3*(12 
0*d^2*e^2*g+5*c^2*f*(-8*e*h+7*f*g)+16*c*d*e*(-5*e*h+4*f*g))-a*b^2*(5*c^2*f 
^2*h-2*c*d*f*(-88*e*h+67*f*g)-8*d^2*e*(5*e*h+38*f*g)))*(f*x+e)^(1/2)/(-a*d 
+b*c)^3/(-a*f+b*e)^3/(b*x+a)^2/(d*x+c)^2+1/192*(693*a^4*d^3*f^3*h-a^3*b*d^ 
2*f^2*(-349*c*f*h+1396*d*e*h+1725*d*f*g)+b^4*(960*d^3*e^3*g+4*c^2*d*e*f*(- 
68*e*h+55*f*g)+40*c*d^2*e^2*(-16*e*h+11*f*g)+15*c^3*f^2*(-8*e*h+7*f*g))+a* 
b^3*(15*c^3*f^3*h-c^2*d*f^2*(-684*e*h+535*f*g)-88*c*d^2*e*f*(-23*e*h+15*f* 
g)-40*d^3*e^2*(8*e*h+83*f*g))-a^2*b^2*d*f*(97*c^2*f^2*h-c*d*f*(-2048*e*h+1 
195*f*g)-4*d^2*e*(282*e*h+995*f*g)))*(f*x+e)^(1/2)/(-a*d+b*c)^4/(-a*f+b...
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(14739\) vs. \(2(2077)=4154\).

Time = 16.58 (sec) , antiderivative size = 14739, normalized size of antiderivative = 7.10 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Result too large to show} \] Input:

Integrate[(g + h*x)/((a + b*x)^5*(c + d*x)^3*Sqrt[e + f*x]),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 5.54 (sec) , antiderivative size = 2212, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {168, 27, 168, 27, 168, 27, 168, 27, 168, 27, 168, 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.

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

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\int \frac {a c f h-4 a d (2 f g+e h)+b (12 d e g+7 c f g-8 c e h)+11 d f (b g-a h) x}{2 (a+b x)^4 (c+d x)^3 \sqrt {e+f x}}dx}{4 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{4 (a+b x)^4 (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {a c f h-4 a d (2 f g+e h)+b (12 d e g+7 c f g-8 c e h)+11 d f (b g-a h) x}{(a+b x)^4 (c+d x)^3 \sqrt {e+f x}}dx}{8 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{4 (a+b x)^4 (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\int \frac {d f (48 d f g+68 d e h-17 c f h) a^2+b \left (-4 e (49 f g+10 e h) d^2-c f (71 f g-104 e h) d+5 c^2 f^2 h\right ) a+b^2 \left (5 f (7 f g-8 e h) c^2+16 d e (4 f g-5 e h) c+120 d^2 e^2 g\right )+9 d f \left (11 d f h a^2-b (19 d f g+4 d e h-c f h) a+b^2 (12 d e g+7 c f g-8 c e h)\right ) x}{2 (a+b x)^3 (c+d x)^3 \sqrt {e+f x}}dx}{3 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (11 a^2 d f h-a b (-c f h+4 d e h+19 d f g)+b^2 (-8 c e h+7 c f g+12 d e g)\right )}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{4 (a+b x)^4 (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {d f (48 d f g+68 d e h-17 c f h) a^2+b \left (-4 e (49 f g+10 e h) d^2-c f (71 f g-104 e h) d+5 c^2 f^2 h\right ) a+b^2 \left (5 f (7 f g-8 e h) c^2+16 d e (4 f g-5 e h) c+120 d^2 e^2 g\right )+9 d f \left (11 d f h a^2-b (19 d f g+4 d e h-c f h) a+b^2 (12 d e g+7 c f g-8 c e h)\right ) x}{(a+b x)^3 (c+d x)^3 \sqrt {e+f x}}dx}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (11 a^2 d f h-a b (-c f h+4 d e h+19 d f g)+b^2 (-8 c e h+7 c f g+12 d e g)\right )}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{4 (a+b x)^4 (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\frac {\sqrt {e+f x} \left (99 a^3 d^2 f^2 h-a^2 b d f (-26 c f h+104 d e h+219 d f g)-a b^2 \left (5 c^2 f^2 h-2 c d f (67 f g-88 e h)-8 d^2 e (5 e h+38 f g)\right )-b^3 \left (5 c^2 f (7 f g-8 e h)+16 c d e (4 f g-5 e h)+120 d^2 e^2 g\right )\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}-\frac {\int \frac {d^2 f^2 (167 c f h-4 d (48 f g+167 e h)) a^3-b d f \left (-4 e (463 f g+212 e h) d^2-c f (257 f g-816 e h) d+62 c^2 f^2 h\right ) a^2+b^2 \left (-80 e^2 (31 f g+4 e h) d^3-8 c e f (109 f g-183 e h) d^2-2 c^2 f^2 (145 f g-202 e h) d+15 c^3 f^3 h\right ) a+b^3 \left (15 f^2 (7 f g-8 e h) c^3+4 d e f (55 f g-68 e h) c^2+40 d^2 e^2 (11 f g-16 e h) c+960 d^3 e^3 g\right )-7 d f \left (99 d^2 f^2 h a^3-b d f (219 d f g+104 d e h-26 c f h) a^2-b^2 \left (-8 e (38 f g+5 e h) d^2-2 c f (67 f g-88 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+16 d e (4 f g-5 e h) c+120 d^2 e^2 g\right )\right ) x}{2 (a+b x)^2 (c+d x)^3 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (11 a^2 d f h-a b (-c f h+4 d e h+19 d f g)+b^2 (-8 c e h+7 c f g+12 d e g)\right )}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{4 (a+b x)^4 (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\sqrt {e+f x} \left (99 a^3 d^2 f^2 h-a^2 b d f (-26 c f h+104 d e h+219 d f g)-a b^2 \left (5 c^2 f^2 h-2 c d f (67 f g-88 e h)-8 d^2 e (5 e h+38 f g)\right )-b^3 \left (5 c^2 f (7 f g-8 e h)+16 c d e (4 f g-5 e h)+120 d^2 e^2 g\right )\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}-\frac {\int \frac {d^2 f^2 (167 c f h-4 d (48 f g+167 e h)) a^3-b d f \left (-4 e (463 f g+212 e h) d^2-c f (257 f g-816 e h) d+62 c^2 f^2 h\right ) a^2+b^2 \left (-80 e^2 (31 f g+4 e h) d^3-8 c e f (109 f g-183 e h) d^2-2 c^2 f^2 (145 f g-202 e h) d+15 c^3 f^3 h\right ) a+b^3 \left (15 f^2 (7 f g-8 e h) c^3+4 d e f (55 f g-68 e h) c^2+40 d^2 e^2 (11 f g-16 e h) c+960 d^3 e^3 g\right )-7 d f \left (99 d^2 f^2 h a^3-b d f (219 d f g+104 d e h-26 c f h) a^2-b^2 \left (-8 e (38 f g+5 e h) d^2-2 c f (67 f g-88 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+16 d e (4 f g-5 e h) c+120 d^2 e^2 g\right )\right ) x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}}dx}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (11 a^2 d f h-a b (-c f h+4 d e h+19 d f g)+b^2 (-8 c e h+7 c f g+12 d e g)\right )}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{4 (a+b x)^4 (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (11 d f h a^2-b (19 d f g+4 d e h-c f h) a+b^2 (12 d e g+7 c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}-\frac {\frac {\left (99 d^2 f^2 h a^3-b d f (219 d f g+104 d e h-26 c f h) a^2-b^2 \left (-8 e (38 f g+5 e h) d^2-2 c f (67 f g-88 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+16 d e (4 f g-5 e h) c+120 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+1396 d e h-349 c f h) a^3-b^2 d f \left (-4 e (995 f g+282 e h) d^2-c f (1195 f g-2048 e h) d+97 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (83 f g+8 e h) d^3-88 c e f (15 f g-23 e h) d^2-c^2 f^2 (535 f g-684 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (15 f^2 (7 f g-8 e h) c^3+4 d e f (55 f g-68 e h) c^2+40 d^2 e^2 (11 f g-16 e h) c+960 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}-\frac {\int \frac {d^3 f^3 (384 d f g+4108 d e h-1027 c f h) a^4+b d^2 f^2 \left (-4 e (2747 f g+2154 e h) d^2+c f (827 f g+4808 e h) d+109 c^2 f^2 h\right ) a^3-b^2 d f \left (-8 e^2 (3073 f g+856 e h) d^3-88 c e f (42 f g-155 e h) d^2+c^2 f^2 (101 f g+540 e h) d+57 c^3 f^3 h\right ) a^2+b^3 \left (-960 e^3 (21 f g+2 e h) d^4-8 c e^2 f (661 f g-1568 e h) d^3-4 c^2 e f^2 (427 f g-632 e h) d^2-3 c^3 f^3 (85 f g-128 e h) d+15 c^4 f^4 h\right ) a+3 b^4 \left (5 f^3 (7 f g-8 e h) c^4+24 d e f^2 (5 f g-6 e h) c^3+96 d^2 e^2 f (3 f g-4 e h) c^2+640 d^3 e^3 (f g-2 e h) c+1920 d^4 e^4 g\right )+5 d f \left (693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+1396 d e h-349 c f h) a^3-b^2 d f \left (-4 e (995 f g+282 e h) d^2-c f (1195 f g-2048 e h) d+97 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (83 f g+8 e h) d^3-88 c e f (15 f g-23 e h) d^2-c^2 f^2 (535 f g-684 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (15 f^2 (7 f g-8 e h) c^3+4 d e f (55 f g-68 e h) c^2+40 d^2 e^2 (11 f g-16 e h) c+960 d^3 e^3 g\right )\right ) x}{2 (a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{(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)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (11 d f h a^2-b (19 d f g+4 d e h-c f h) a+b^2 (12 d e g+7 c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}-\frac {\frac {\left (99 d^2 f^2 h a^3-b d f (219 d f g+104 d e h-26 c f h) a^2-b^2 \left (-8 e (38 f g+5 e h) d^2-2 c f (67 f g-88 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+16 d e (4 f g-5 e h) c+120 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+1396 d e h-349 c f h) a^3-b^2 d f \left (-4 e (995 f g+282 e h) d^2-c f (1195 f g-2048 e h) d+97 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (83 f g+8 e h) d^3-88 c e f (15 f g-23 e h) d^2-c^2 f^2 (535 f g-684 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (15 f^2 (7 f g-8 e h) c^3+4 d e f (55 f g-68 e h) c^2+40 d^2 e^2 (11 f g-16 e h) c+960 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}-\frac {\int \frac {d^3 f^3 (384 d f g+4108 d e h-1027 c f h) a^4+b d^2 f^2 \left (-4 e (2747 f g+2154 e h) d^2+c f (827 f g+4808 e h) d+109 c^2 f^2 h\right ) a^3-b^2 d f \left (-8 e^2 (3073 f g+856 e h) d^3-88 c e f (42 f g-155 e h) d^2+c^2 f^2 (101 f g+540 e h) d+57 c^3 f^3 h\right ) a^2+b^3 \left (-960 e^3 (21 f g+2 e h) d^4-8 c e^2 f (661 f g-1568 e h) d^3-4 c^2 e f^2 (427 f g-632 e h) d^2-3 c^3 f^3 (85 f g-128 e h) d+15 c^4 f^4 h\right ) a+3 b^4 \left (5 f^3 (7 f g-8 e h) c^4+24 d e f^2 (5 f g-6 e h) c^3+96 d^2 e^2 f (3 f g-4 e h) c^2+640 d^3 e^3 (f g-2 e h) c+1920 d^4 e^4 g\right )+5 d f \left (693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+1396 d e h-349 c f h) a^3-b^2 d f \left (-4 e (995 f g+282 e h) d^2-c f (1195 f g-2048 e h) d+97 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (83 f g+8 e h) d^3-88 c e f (15 f g-23 e h) d^2-c^2 f^2 (535 f g-684 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (15 f^2 (7 f g-8 e h) c^3+4 d e f (55 f g-68 e h) c^2+40 d^2 e^2 (11 f g-16 e h) c+960 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{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)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (11 d f h a^2-b (19 d f g+4 d e h-c f h) a+b^2 (12 d e g+7 c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}-\frac {\frac {\left (99 d^2 f^2 h a^3-b d f (219 d f g+104 d e h-26 c f h) a^2-b^2 \left (-8 e (38 f g+5 e h) d^2-2 c f (67 f g-88 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+16 d e (4 f g-5 e h) c+120 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+1396 d e h-349 c f h) a^3-b^2 d f \left (-4 e (995 f g+282 e h) d^2-c f (1195 f g-2048 e h) d+97 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (83 f g+8 e h) d^3-88 c e f (15 f g-23 e h) d^2-c^2 f^2 (535 f g-684 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (15 f^2 (7 f g-8 e h) c^3+4 d e f (55 f g-68 e h) c^2+40 d^2 e^2 (11 f g-16 e h) c+960 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}-\frac {\frac {2 d \sqrt {e+f x} \left (d^3 f^3 (96 d f g+1027 d e h-1123 c f h) a^4-b d^2 f^2 \left (e (2747 f g+2154 e h) d^2-c f (2363 f g+2947 e h) d+409 c^2 f^2 h\right ) a^3+b^2 d f \left (2 e^2 (3073 f g+856 e h) d^3-c e f (4051 f g+4820 e h) d^2-c^2 f^2 (1519 f g-2425 e h) d+107 c^3 f^3 h\right ) a^2-b^3 \left (240 e^3 (21 f g+2 e h) d^4-4 c e^2 f (707 f g+884 e h) d^3-c^2 e f^2 (1223 f g-1898 e h) d^2-11 c^3 f^3 (55 f g-69 e h) d+15 c^4 f^4 h\right ) a+b^4 \left (-15 f^3 (7 f g-8 e h) c^4-d e f^2 (185 f g-232 e h) c^3-2 d^2 e^2 f (167 f g-256 e h) c^2-240 d^3 e^3 (3 f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2}+\frac {\int \frac {6 \left (32 d^4 f^4 (3 d f g-4 d e h+c f h) a^5+b d^3 f^3 \left (2 e (128 f g+771 e h) d^2-c f (640 f g+1673 e h) d+515 c^2 f^2 h\right ) a^4-b^2 d^2 f^2 \left (10 e^2 (415 f g+304 e h) d^3-c e f (4529 f g+4948 e h) d^2-c^2 f^2 (197 f g-2393 e h) d+91 c^3 f^3 h\right ) a^3+b^3 d f \left (16 e^3 (543 f g+146 e h) d^4-4 c e^2 f (1867 f g+1800 e h) d^3-c^2 e f^2 (1351 f g-4702 e h) d^2-11 c^3 f^3 (23 f g-47 e h) d+29 c^4 f^4 h\right ) a^2-b^4 \left (160 e^4 (43 f g+4 e h) d^5-176 c e^3 f (29 f g+28 e h) d^4-2 c^2 e^2 f^2 (613 f g-1720 e h) d^3-c^3 e f^3 (491 f g-736 e h) d^2-c^4 f^4 (155 f g-203 e h) d+5 c^5 f^5 h\right ) a+b^5 (d e-c f) \left (5 f^3 (7 f g-8 e h) c^4+24 d e f^2 (5 f g-6 e h) c^3+96 d^2 e^2 f (3 f g-4 e h) c^2+640 d^3 e^3 (f g-2 e h) c+1920 d^4 e^4 g\right )+b d f \left (d^3 f^3 (96 d f g+1027 d e h-1123 c f h) a^4-b d^2 f^2 \left (e (2747 f g+2154 e h) d^2-c f (2363 f g+2947 e h) d+409 c^2 f^2 h\right ) a^3+b^2 d f \left (2 e^2 (3073 f g+856 e h) d^3-c e f (4051 f g+4820 e h) d^2-c^2 f^2 (1519 f g-2425 e h) d+107 c^3 f^3 h\right ) a^2-b^3 \left (240 e^3 (21 f g+2 e h) d^4-4 c e^2 f (707 f g+884 e h) d^3-c^2 e f^2 (1223 f g-1898 e h) d^2-11 c^3 f^3 (55 f g-69 e h) d+15 c^4 f^4 h\right ) a+b^4 \left (-15 f^3 (7 f g-8 e h) c^4-d e f^2 (185 f g-232 e h) c^3-2 d^2 e^2 f (167 f g-256 e h) c^2-240 d^3 e^3 (3 f g+4 e h) c+1440 d^4 e^4 g\right )\right ) x\right )}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (d e-c f)}}{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)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (11 d f h a^2-b (19 d f g+4 d e h-c f h) a+b^2 (12 d e g+7 c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}-\frac {\frac {\left (99 d^2 f^2 h a^3-b d f (219 d f g+104 d e h-26 c f h) a^2-b^2 \left (-8 e (38 f g+5 e h) d^2-2 c f (67 f g-88 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+16 d e (4 f g-5 e h) c+120 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+1396 d e h-349 c f h) a^3-b^2 d f \left (-4 e (995 f g+282 e h) d^2-c f (1195 f g-2048 e h) d+97 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (83 f g+8 e h) d^3-88 c e f (15 f g-23 e h) d^2-c^2 f^2 (535 f g-684 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (15 f^2 (7 f g-8 e h) c^3+4 d e f (55 f g-68 e h) c^2+40 d^2 e^2 (11 f g-16 e h) c+960 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}-\frac {\frac {2 d \sqrt {e+f x} \left (d^3 f^3 (96 d f g+1027 d e h-1123 c f h) a^4-b d^2 f^2 \left (e (2747 f g+2154 e h) d^2-c f (2363 f g+2947 e h) d+409 c^2 f^2 h\right ) a^3+b^2 d f \left (2 e^2 (3073 f g+856 e h) d^3-c e f (4051 f g+4820 e h) d^2-c^2 f^2 (1519 f g-2425 e h) d+107 c^3 f^3 h\right ) a^2-b^3 \left (240 e^3 (21 f g+2 e h) d^4-4 c e^2 f (707 f g+884 e h) d^3-c^2 e f^2 (1223 f g-1898 e h) d^2-11 c^3 f^3 (55 f g-69 e h) d+15 c^4 f^4 h\right ) a+b^4 \left (-15 f^3 (7 f g-8 e h) c^4-d e f^2 (185 f g-232 e h) c^3-2 d^2 e^2 f (167 f g-256 e h) c^2-240 d^3 e^3 (3 f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2}+\frac {3 \int \frac {32 d^4 f^4 (3 d f g-4 d e h+c f h) a^5+b d^3 f^3 \left (2 e (128 f g+771 e h) d^2-c f (640 f g+1673 e h) d+515 c^2 f^2 h\right ) a^4-b^2 d^2 f^2 \left (10 e^2 (415 f g+304 e h) d^3-c e f (4529 f g+4948 e h) d^2-c^2 f^2 (197 f g-2393 e h) d+91 c^3 f^3 h\right ) a^3+b^3 d f \left (16 e^3 (543 f g+146 e h) d^4-4 c e^2 f (1867 f g+1800 e h) d^3-c^2 e f^2 (1351 f g-4702 e h) d^2-11 c^3 f^3 (23 f g-47 e h) d+29 c^4 f^4 h\right ) a^2-b^4 \left (160 e^4 (43 f g+4 e h) d^5-176 c e^3 f (29 f g+28 e h) d^4-2 c^2 e^2 f^2 (613 f g-1720 e h) d^3-c^3 e f^3 (491 f g-736 e h) d^2-c^4 f^4 (155 f g-203 e h) d+5 c^5 f^5 h\right ) a+b^5 (d e-c f) \left (5 f^3 (7 f g-8 e h) c^4+24 d e f^2 (5 f g-6 e h) c^3+96 d^2 e^2 f (3 f g-4 e h) c^2+640 d^3 e^3 (f g-2 e h) c+1920 d^4 e^4 g\right )+b d f \left (d^3 f^3 (96 d f g+1027 d e h-1123 c f h) a^4-b d^2 f^2 \left (e (2747 f g+2154 e h) d^2-c f (2363 f g+2947 e h) d+409 c^2 f^2 h\right ) a^3+b^2 d f \left (2 e^2 (3073 f g+856 e h) d^3-c e f (4051 f g+4820 e h) d^2-c^2 f^2 (1519 f g-2425 e h) d+107 c^3 f^3 h\right ) a^2-b^3 \left (240 e^3 (21 f g+2 e h) d^4-4 c e^2 f (707 f g+884 e h) d^3-c^2 e f^2 (1223 f g-1898 e h) d^2-11 c^3 f^3 (55 f g-69 e h) d+15 c^4 f^4 h\right ) a+b^4 \left (-15 f^3 (7 f g-8 e h) c^4-d e f^2 (185 f g-232 e h) c^3-2 d^2 e^2 f (167 f g-256 e h) c^2-240 d^3 e^3 (3 f g+4 e h) c+1440 d^4 e^4 g\right )\right ) x}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}}{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)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (11 d f h a^2-b (19 d f g+4 d e h-c f h) a+b^2 (12 d e g+7 c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}-\frac {\frac {\left (99 d^2 f^2 h a^3-b d f (219 d f g+104 d e h-26 c f h) a^2-b^2 \left (-8 e (38 f g+5 e h) d^2-2 c f (67 f g-88 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+16 d e (4 f g-5 e h) c+120 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+1396 d e h-349 c f h) a^3-b^2 d f \left (-4 e (995 f g+282 e h) d^2-c f (1195 f g-2048 e h) d+97 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (83 f g+8 e h) d^3-88 c e f (15 f g-23 e h) d^2-c^2 f^2 (535 f g-684 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (15 f^2 (7 f g-8 e h) c^3+4 d e f (55 f g-68 e h) c^2+40 d^2 e^2 (11 f g-16 e h) c+960 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}-\frac {\frac {2 d \sqrt {e+f x} \left (d^3 f^3 (96 d f g+1027 d e h-1123 c f h) a^4-b d^2 f^2 \left (e (2747 f g+2154 e h) d^2-c f (2363 f g+2947 e h) d+409 c^2 f^2 h\right ) a^3+b^2 d f \left (2 e^2 (3073 f g+856 e h) d^3-c e f (4051 f g+4820 e h) d^2-c^2 f^2 (1519 f g-2425 e h) d+107 c^3 f^3 h\right ) a^2-b^3 \left (240 e^3 (21 f g+2 e h) d^4-4 c e^2 f (707 f g+884 e h) d^3-c^2 e f^2 (1223 f g-1898 e h) d^2-11 c^3 f^3 (55 f g-69 e h) d+15 c^4 f^4 h\right ) a+b^4 \left (-15 f^3 (7 f g-8 e h) c^4-d e f^2 (185 f g-232 e h) c^3-2 d^2 e^2 f (167 f g-256 e h) c^2-240 d^3 e^3 (3 f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2}+\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (16 d^4 f^4 (3 d f g-4 d e h+c f h) a^5+b d^3 f^3 \left (e (128 f g+771 e h) d^2-2 c f (184 f g+675 e h) d+819 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-5 e^2 (415 f g+304 e h) d^3+c e f (3638 f g+3551 e h) d^2-3 c^2 f^2 (361 f g+890 e h) d+159 c^3 f^3 h\right ) a^3-b^3 d f \left (-8 e^3 (543 f g+146 e h) d^4+c e^2 f (6807 f g+4456 e h) d^3-9 c^2 e f^2 (150 f g+529 e h) d^2-3 c^3 f^3 (211 f g-318 e h) d+39 c^4 f^4 h\right ) a^2+b^4 \left (-80 e^4 (43 f g+4 e h) d^5+16 c e^3 f (317 f g+169 e h) d^4-c^2 e^2 f^2 (801 f g+3488 e h) d^3-c^3 e f^3 (366 f g-581 e h) d^2-c^4 f^4 (225 f g-278 e h) d+5 c^5 f^5 h\right ) a+b^5 \left (5 f^4 (7 f g-8 e h) c^5+2 d e f^3 (25 f g-32 e h) c^4+d^2 e^2 f^2 (83 f g-136 e h) c^3+8 d^3 e^3 f (23 f g+116 e h) c^2-80 d^4 e^4 (17 f g+8 e h) c+960 d^5 e^5 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\int \frac {16 d^5 f^5 (3 d f g-4 d e h+c f h) a^6-16 b d^4 f^4 \left (-8 e (f g-3 e h) d^2+c f (23 f g-60 e h) d+21 c^2 f^2 h\right ) a^5-b^2 d^3 f^3 \left (-32 e^2 (29 f g+68 e h) d^3+c e f (2368 f g+4765 e h) d^2-6 c^2 f^2 (320 f g+479 e h) d+765 c^3 f^3 h\right ) a^4+b^3 d^2 f^2 \left (-64 e^3 (93 f g+56 e h) d^4+c e^2 f (12069 f g+10064 e h) d^3-3 c^2 e f^2 (1838 f g+3275 e h) d^2-3 c^3 f^3 (361 f g-1222 e h) d+159 c^4 f^4 h\right ) a^3-b^4 d f \left (-16 e^4 (643 f g+156 e h) d^5+8 c e^3 f (2369 f g+1356 e h) d^4-3 c^2 e^2 f^2 (2307 f g+4616 e h) d^3-9 c^3 e f^3 (150 f g-527 e h) d^2-3 c^4 f^4 (211 f g-318 e h) d+39 c^5 f^5 h\right ) a^2+b^5 \left (-320 e^5 (23 f g+2 e h) d^6+16 c e^4 f (799 f g+368 e h) d^5-16 c^2 e^3 f^2 (255 f g+557 e h) d^4-89 c^3 e^2 f^3 (9 f g-32 e h) d^3-c^4 e f^4 (366 f g-581 e h) d^2-c^5 f^5 (225 f g-278 e h) d+5 c^6 f^6 h\right ) a+b^6 (d e-c f)^2 \left (5 f^3 (7 f g-8 e h) c^4+24 d e f^2 (5 f g-6 e h) c^3+96 d^2 e^2 f (3 f g-4 e h) c^2+640 d^3 e^3 (f g-2 e h) c+1920 d^4 e^4 g\right )+b d f \left (16 d^4 f^4 (3 d f g-4 d e h+c f h) a^5+b d^3 f^3 \left (e (128 f g+771 e h) d^2-2 c f (184 f g+675 e h) d+819 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-5 e^2 (415 f g+304 e h) d^3+c e f (3638 f g+3551 e h) d^2-3 c^2 f^2 (361 f g+890 e h) d+159 c^3 f^3 h\right ) a^3-b^3 d f \left (-8 e^3 (543 f g+146 e h) d^4+c e^2 f (6807 f g+4456 e h) d^3-9 c^2 e f^2 (150 f g+529 e h) d^2-3 c^3 f^3 (211 f g-318 e h) d+39 c^4 f^4 h\right ) a^2+b^4 \left (-80 e^4 (43 f g+4 e h) d^5+16 c e^3 f (317 f g+169 e h) d^4-c^2 e^2 f^2 (801 f g+3488 e h) d^3-c^3 e f^3 (366 f g-581 e h) d^2-c^4 f^4 (225 f g-278 e h) d+5 c^5 f^5 h\right ) a+b^5 \left (5 f^4 (7 f g-8 e h) c^5+2 d e f^3 (25 f g-32 e h) c^4+d^2 e^2 f^2 (83 f g-136 e h) c^3+8 d^3 e^3 f (23 f g+116 e h) c^2-80 d^4 e^4 (17 f g+8 e h) c+960 d^5 e^5 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}\right )}{(b c-a d) (d e-c f)}}{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)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (11 d f h a^2-b (19 d f g+4 d e h-c f h) a+b^2 (12 d e g+7 c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}-\frac {\frac {\left (99 d^2 f^2 h a^3-b d f (219 d f g+104 d e h-26 c f h) a^2-b^2 \left (-8 e (38 f g+5 e h) d^2-2 c f (67 f g-88 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+16 d e (4 f g-5 e h) c+120 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+1396 d e h-349 c f h) a^3-b^2 d f \left (-4 e (995 f g+282 e h) d^2-c f (1195 f g-2048 e h) d+97 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (83 f g+8 e h) d^3-88 c e f (15 f g-23 e h) d^2-c^2 f^2 (535 f g-684 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (15 f^2 (7 f g-8 e h) c^3+4 d e f (55 f g-68 e h) c^2+40 d^2 e^2 (11 f g-16 e h) c+960 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}-\frac {\frac {2 d \sqrt {e+f x} \left (d^3 f^3 (96 d f g+1027 d e h-1123 c f h) a^4-b d^2 f^2 \left (e (2747 f g+2154 e h) d^2-c f (2363 f g+2947 e h) d+409 c^2 f^2 h\right ) a^3+b^2 d f \left (2 e^2 (3073 f g+856 e h) d^3-c e f (4051 f g+4820 e h) d^2-c^2 f^2 (1519 f g-2425 e h) d+107 c^3 f^3 h\right ) a^2-b^3 \left (240 e^3 (21 f g+2 e h) d^4-4 c e^2 f (707 f g+884 e h) d^3-c^2 e f^2 (1223 f g-1898 e h) d^2-11 c^3 f^3 (55 f g-69 e h) d+15 c^4 f^4 h\right ) a+b^4 \left (-15 f^3 (7 f g-8 e h) c^4-d e f^2 (185 f g-232 e h) c^3-2 d^2 e^2 f (167 f g-256 e h) c^2-240 d^3 e^3 (3 f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2}+\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (16 d^4 f^4 (3 d f g-4 d e h+c f h) a^5+b d^3 f^3 \left (e (128 f g+771 e h) d^2-2 c f (184 f g+675 e h) d+819 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-5 e^2 (415 f g+304 e h) d^3+c e f (3638 f g+3551 e h) d^2-3 c^2 f^2 (361 f g+890 e h) d+159 c^3 f^3 h\right ) a^3-b^3 d f \left (-8 e^3 (543 f g+146 e h) d^4+c e^2 f (6807 f g+4456 e h) d^3-9 c^2 e f^2 (150 f g+529 e h) d^2-3 c^3 f^3 (211 f g-318 e h) d+39 c^4 f^4 h\right ) a^2+b^4 \left (-80 e^4 (43 f g+4 e h) d^5+16 c e^3 f (317 f g+169 e h) d^4-c^2 e^2 f^2 (801 f g+3488 e h) d^3-c^3 e f^3 (366 f g-581 e h) d^2-c^4 f^4 (225 f g-278 e h) d+5 c^5 f^5 h\right ) a+b^5 \left (5 f^4 (7 f g-8 e h) c^5+2 d e f^3 (25 f g-32 e h) c^4+d^2 e^2 f^2 (83 f g-136 e h) c^3+8 d^3 e^3 f (23 f g+116 e h) c^2-80 d^4 e^4 (17 f g+8 e h) c+960 d^5 e^5 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {-\frac {16 d^4 \left (\left (-99 f^2 h c^3+11 d f (13 f g+16 e h) c^2-20 d^2 e (13 f g+4 e h) c+120 d^3 e^2 g\right ) b^2-2 a d \left (-10 e (f g-2 e h) d^2+c f (13 f g-34 e h) d+11 c^2 f^2 h\right ) b+a^2 d^2 f (3 d f g-4 d e h+c f h)\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx (b e-a f)^4}{b c-a d}-\frac {b^2 (d e-c f)^2 \left (1155 d^4 f^4 h a^5-231 b d^3 f^3 (13 d f g+16 d e h-4 c f h) a^4-66 b^2 d^2 f^2 \left (-12 e (13 f g+6 e h) d^2-2 c f (13 f g-38 e h) d+3 c^2 f^2 h\right ) a^3+22 b^3 d f \left (-16 e^2 (39 f g+8 e h) d^3-12 c e f (13 f g-30 e h) d^2-3 c^2 f^2 (13 f g-20 e h) d+2 c^3 f^3 h\right ) a^2-b^4 \left (-640 e^3 (13 f g+e h) d^4-64 c e^2 f (39 f g-82 e h) d^3-72 c^2 e f^2 (13 f g-18 e h) d^2-4 c^3 f^3 (65 f g-82 e h) d+5 c^4 f^4 h\right ) a-b^5 \left (5 f^3 (7 f g-8 e h) c^4+24 d e f^2 (5 f g-6 e h) c^3+96 d^2 e^2 f (3 f g-4 e h) c^2+640 d^3 e^3 (f g-2 e h) c+1920 d^4 e^4 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 )}{(b c-a d) (d e-c f)}}{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)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (11 d f h a^2-b (19 d f g+4 d e h-c f h) a+b^2 (12 d e g+7 c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}-\frac {\frac {\left (99 d^2 f^2 h a^3-b d f (219 d f g+104 d e h-26 c f h) a^2-b^2 \left (-8 e (38 f g+5 e h) d^2-2 c f (67 f g-88 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+16 d e (4 f g-5 e h) c+120 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+1396 d e h-349 c f h) a^3-b^2 d f \left (-4 e (995 f g+282 e h) d^2-c f (1195 f g-2048 e h) d+97 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (83 f g+8 e h) d^3-88 c e f (15 f g-23 e h) d^2-c^2 f^2 (535 f g-684 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (15 f^2 (7 f g-8 e h) c^3+4 d e f (55 f g-68 e h) c^2+40 d^2 e^2 (11 f g-16 e h) c+960 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}-\frac {\frac {2 d \sqrt {e+f x} \left (d^3 f^3 (96 d f g+1027 d e h-1123 c f h) a^4-b d^2 f^2 \left (e (2747 f g+2154 e h) d^2-c f (2363 f g+2947 e h) d+409 c^2 f^2 h\right ) a^3+b^2 d f \left (2 e^2 (3073 f g+856 e h) d^3-c e f (4051 f g+4820 e h) d^2-c^2 f^2 (1519 f g-2425 e h) d+107 c^3 f^3 h\right ) a^2-b^3 \left (240 e^3 (21 f g+2 e h) d^4-4 c e^2 f (707 f g+884 e h) d^3-c^2 e f^2 (1223 f g-1898 e h) d^2-11 c^3 f^3 (55 f g-69 e h) d+15 c^4 f^4 h\right ) a+b^4 \left (-15 f^3 (7 f g-8 e h) c^4-d e f^2 (185 f g-232 e h) c^3-2 d^2 e^2 f (167 f g-256 e h) c^2-240 d^3 e^3 (3 f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2}+\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (16 d^4 f^4 (3 d f g-4 d e h+c f h) a^5+b d^3 f^3 \left (e (128 f g+771 e h) d^2-2 c f (184 f g+675 e h) d+819 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-5 e^2 (415 f g+304 e h) d^3+c e f (3638 f g+3551 e h) d^2-3 c^2 f^2 (361 f g+890 e h) d+159 c^3 f^3 h\right ) a^3-b^3 d f \left (-8 e^3 (543 f g+146 e h) d^4+c e^2 f (6807 f g+4456 e h) d^3-9 c^2 e f^2 (150 f g+529 e h) d^2-3 c^3 f^3 (211 f g-318 e h) d+39 c^4 f^4 h\right ) a^2+b^4 \left (-80 e^4 (43 f g+4 e h) d^5+16 c e^3 f (317 f g+169 e h) d^4-c^2 e^2 f^2 (801 f g+3488 e h) d^3-c^3 e f^3 (366 f g-581 e h) d^2-c^4 f^4 (225 f g-278 e h) d+5 c^5 f^5 h\right ) a+b^5 \left (5 f^4 (7 f g-8 e h) c^5+2 d e f^3 (25 f g-32 e h) c^4+d^2 e^2 f^2 (83 f g-136 e h) c^3+8 d^3 e^3 f (23 f g+116 e h) c^2-80 d^4 e^4 (17 f g+8 e h) c+960 d^5 e^5 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {-\frac {32 d^4 \left (\left (-99 f^2 h c^3+11 d f (13 f g+16 e h) c^2-20 d^2 e (13 f g+4 e h) c+120 d^3 e^2 g\right ) b^2-2 a d \left (-10 e (f g-2 e h) d^2+c f (13 f g-34 e h) d+11 c^2 f^2 h\right ) b+a^2 d^2 f (3 d f g-4 d e h+c f h)\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x} (b e-a f)^4}{(b c-a d) f}-\frac {2 b^2 (d e-c f)^2 \left (1155 d^4 f^4 h a^5-231 b d^3 f^3 (13 d f g+16 d e h-4 c f h) a^4-66 b^2 d^2 f^2 \left (-12 e (13 f g+6 e h) d^2-2 c f (13 f g-38 e h) d+3 c^2 f^2 h\right ) a^3+22 b^3 d f \left (-16 e^2 (39 f g+8 e h) d^3-12 c e f (13 f g-30 e h) d^2-3 c^2 f^2 (13 f g-20 e h) d+2 c^3 f^3 h\right ) a^2-b^4 \left (-640 e^3 (13 f g+e h) d^4-64 c e^2 f (39 f g-82 e h) d^3-72 c^2 e f^2 (13 f g-18 e h) d^2-4 c^3 f^3 (65 f g-82 e h) d+5 c^4 f^4 h\right ) a-b^5 \left (5 f^3 (7 f g-8 e h) c^4+24 d e f^2 (5 f g-6 e h) c^3+96 d^2 e^2 f (3 f g-4 e h) c^2+640 d^3 e^3 (f g-2 e h) c+1920 d^4 e^4 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 )}{(b c-a d) (d e-c f)}}{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)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (11 d f h a^2-b (19 d f g+4 d e h-c f h) a+b^2 (12 d e g+7 c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}-\frac {\frac {\left (99 d^2 f^2 h a^3-b d f (219 d f g+104 d e h-26 c f h) a^2-b^2 \left (-8 e (38 f g+5 e h) d^2-2 c f (67 f g-88 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+16 d e (4 f g-5 e h) c+120 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+1396 d e h-349 c f h) a^3-b^2 d f \left (-4 e (995 f g+282 e h) d^2-c f (1195 f g-2048 e h) d+97 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (83 f g+8 e h) d^3-88 c e f (15 f g-23 e h) d^2-c^2 f^2 (535 f g-684 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (15 f^2 (7 f g-8 e h) c^3+4 d e f (55 f g-68 e h) c^2+40 d^2 e^2 (11 f g-16 e h) c+960 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}-\frac {\frac {2 d \sqrt {e+f x} \left (d^3 f^3 (96 d f g+1027 d e h-1123 c f h) a^4-b d^2 f^2 \left (e (2747 f g+2154 e h) d^2-c f (2363 f g+2947 e h) d+409 c^2 f^2 h\right ) a^3+b^2 d f \left (2 e^2 (3073 f g+856 e h) d^3-c e f (4051 f g+4820 e h) d^2-c^2 f^2 (1519 f g-2425 e h) d+107 c^3 f^3 h\right ) a^2-b^3 \left (240 e^3 (21 f g+2 e h) d^4-4 c e^2 f (707 f g+884 e h) d^3-c^2 e f^2 (1223 f g-1898 e h) d^2-11 c^3 f^3 (55 f g-69 e h) d+15 c^4 f^4 h\right ) a+b^4 \left (-15 f^3 (7 f g-8 e h) c^4-d e f^2 (185 f g-232 e h) c^3-2 d^2 e^2 f (167 f g-256 e h) c^2-240 d^3 e^3 (3 f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2}+\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (16 d^4 f^4 (3 d f g-4 d e h+c f h) a^5+b d^3 f^3 \left (e (128 f g+771 e h) d^2-2 c f (184 f g+675 e h) d+819 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-5 e^2 (415 f g+304 e h) d^3+c e f (3638 f g+3551 e h) d^2-3 c^2 f^2 (361 f g+890 e h) d+159 c^3 f^3 h\right ) a^3-b^3 d f \left (-8 e^3 (543 f g+146 e h) d^4+c e^2 f (6807 f g+4456 e h) d^3-9 c^2 e f^2 (150 f g+529 e h) d^2-3 c^3 f^3 (211 f g-318 e h) d+39 c^4 f^4 h\right ) a^2+b^4 \left (-80 e^4 (43 f g+4 e h) d^5+16 c e^3 f (317 f g+169 e h) d^4-c^2 e^2 f^2 (801 f g+3488 e h) d^3-c^3 e f^3 (366 f g-581 e h) d^2-c^4 f^4 (225 f g-278 e h) d+5 c^5 f^5 h\right ) a+b^5 \left (5 f^4 (7 f g-8 e h) c^5+2 d e f^3 (25 f g-32 e h) c^4+d^2 e^2 f^2 (83 f g-136 e h) c^3+8 d^3 e^3 f (23 f g+116 e h) c^2-80 d^4 e^4 (17 f g+8 e h) c+960 d^5 e^5 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\frac {32 d^{7/2} \left (\left (-99 f^2 h c^3+11 d f (13 f g+16 e h) c^2-20 d^2 e (13 f g+4 e h) c+120 d^3 e^2 g\right ) b^2-2 a d \left (-10 e (f g-2 e h) d^2+c f (13 f g-34 e h) d+11 c^2 f^2 h\right ) b+a^2 d^2 f (3 d f g-4 d e h+c f h)\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) (b e-a f)^4}{(b c-a d) \sqrt {d e-c f}}+\frac {2 b^{3/2} (d e-c f)^2 \left (1155 d^4 f^4 h a^5-231 b d^3 f^3 (13 d f g+16 d e h-4 c f h) a^4-66 b^2 d^2 f^2 \left (-12 e (13 f g+6 e h) d^2-2 c f (13 f g-38 e h) d+3 c^2 f^2 h\right ) a^3+22 b^3 d f \left (-16 e^2 (39 f g+8 e h) d^3-12 c e f (13 f g-30 e h) d^2-3 c^2 f^2 (13 f g-20 e h) d+2 c^3 f^3 h\right ) a^2-b^4 \left (-640 e^3 (13 f g+e h) d^4-64 c e^2 f (39 f g-82 e h) d^3-72 c^2 e f^2 (13 f g-18 e h) d^2-4 c^3 f^3 (65 f g-82 e h) d+5 c^4 f^4 h\right ) a-b^5 \left (5 f^3 (7 f g-8 e h) c^4+24 d e f^2 (5 f g-6 e h) c^3+96 d^2 e^2 f (3 f g-4 e h) c^2+640 d^3 e^3 (f g-2 e h) c+1920 d^4 e^4 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}}}{(b c-a d) (d e-c f)}\right )}{(b c-a d) (d e-c f)}}{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)}}{8 (b c-a d) (b e-a f)}\)

Input:

Int[(g + h*x)/((a + b*x)^5*(c + d*x)^3*Sqrt[e + f*x]),x]
 

Output:

-1/4*((b*g - a*h)*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^4*(c + 
 d*x)^2) - (-1/3*((11*a^2*d*f*h + b^2*(12*d*e*g + 7*c*f*g - 8*c*e*h) - a*b 
*(19*d*f*g + 4*d*e*h - c*f*h))*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a 
+ b*x)^3*(c + d*x)^2) - (((99*a^3*d^2*f^2*h - a^2*b*d*f*(219*d*f*g + 104*d 
*e*h - 26*c*f*h) - b^3*(120*d^2*e^2*g + 5*c^2*f*(7*f*g - 8*e*h) + 16*c*d*e 
*(4*f*g - 5*e*h)) - a*b^2*(5*c^2*f^2*h - 2*c*d*f*(67*f*g - 88*e*h) - 8*d^2 
*e*(38*f*g + 5*e*h)))*Sqrt[e + f*x])/(2*(b*c - a*d)*(b*e - a*f)*(a + b*x)^ 
2*(c + d*x)^2) - (-(((693*a^4*d^3*f^3*h - a^3*b*d^2*f^2*(1725*d*f*g + 1396 
*d*e*h - 349*c*f*h) + b^4*(960*d^3*e^3*g + 4*c^2*d*e*f*(55*f*g - 68*e*h) + 
 40*c*d^2*e^2*(11*f*g - 16*e*h) + 15*c^3*f^2*(7*f*g - 8*e*h)) + a*b^3*(15* 
c^3*f^3*h - c^2*d*f^2*(535*f*g - 684*e*h) - 88*c*d^2*e*f*(15*f*g - 23*e*h) 
 - 40*d^3*e^2*(83*f*g + 8*e*h)) - a^2*b^2*d*f*(97*c^2*f^2*h - c*d*f*(1195* 
f*g - 2048*e*h) - 4*d^2*e*(995*f*g + 282*e*h)))*Sqrt[e + f*x])/((b*c - a*d 
)*(b*e - a*f)*(a + b*x)*(c + d*x)^2)) - ((2*d*(a^4*d^3*f^3*(96*d*f*g + 102 
7*d*e*h - 1123*c*f*h) + b^4*(1440*d^4*e^4*g - 2*c^2*d^2*e^2*f*(167*f*g - 2 
56*e*h) - c^3*d*e*f^2*(185*f*g - 232*e*h) - 15*c^4*f^3*(7*f*g - 8*e*h) - 2 
40*c*d^3*e^3*(3*f*g + 4*e*h)) - a*b^3*(15*c^4*f^4*h - c^2*d^2*e*f^2*(1223* 
f*g - 1898*e*h) - 11*c^3*d*f^3*(55*f*g - 69*e*h) + 240*d^4*e^3*(21*f*g + 2 
*e*h) - 4*c*d^3*e^2*f*(707*f*g + 884*e*h)) - a^3*b*d^2*f^2*(409*c^2*f^2*h 
+ d^2*e*(2747*f*g + 2154*e*h) - c*d*f*(2363*f*g + 2947*e*h)) + a^2*b^2*...
 

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 = 0.09 (sec) , antiderivative size = 3250, normalized size of antiderivative = 1.56

\[\text {output too large to display}\]

Input:

int((h*x+g)/(b*x+a)^5/(d*x+c)^3/(f*x+e)^(1/2),x)
 

Output:

2*f^6*(-d^4/f^6/(a*d-b*c)^7*((-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+18*a*b*c^2*d*f*h-12*a*b*c*d^2*e*h-26*a*b*c*d^2*f*g+20*a*b*d^3*e*g- 
19*b^2*c^3*f*h+16*b^2*c^2*d*e*h+23*b^2*c^2*d*f*g-20*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-22*a*b*c^2*d*f*h+12*a*b*c*d^2*e*h+30*a*b*c*d^2*f*g-20*a*b*d^3*e*g+ 
21*b^2*c^3*f*h-16*b^2*c^2*d*e*h-25*b^2*c^2*d*f*g+20*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-22*a*b*c^2*d*f^2*h+68*a*b*c*d^2*e*f*h-26*a*b*c*d^2*f^ 
2*g-40*a*b*d^3*e^2*h+20*a*b*d^3*e*f*g-99*b^2*c^3*f^2*h+176*b^2*c^2*d*e*f*h 
+143*b^2*c^2*d*f^2*g-80*b^2*c*d^2*e^2*h-260*b^2*c*d^2*e*f*g+120*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^6/(a*d-b*c)^7*((1/128*b^3*f*(515*a^5*d^4*f^3 
*h-356*a^4*b*c*d^3*f^3*h-1136*a^4*b*d^4*e*f^2*h-1083*a^4*b*d^4*f^3*g-198*a 
^3*b^2*c^2*d^2*f^3*h+104*a^3*b^2*c*d^3*e*f^2*h+1716*a^3*b^2*c*d^3*f^3*g+91 
2*a^3*b^2*d^4*e^2*f*h+2616*a^3*b^2*d^4*e*f^2*g+44*a^2*b^3*c^3*d*f^3*h+1320 
*a^2*b^3*c^2*d^2*e*f^2*h-858*a^2*b^3*c^2*d^2*f^3*g+240*a^2*b^3*c*d^3*e^2*f 
*h-3432*a^2*b^3*c*d^3*e*f^2*g-256*a^2*b^3*d^4*e^3*h-2208*a^2*b^3*d^4*e^2*f 
*g-5*a*b^4*c^4*f^3*h-328*a*b^4*c^3*d*e*f^2*h+260*a*b^4*c^3*d*f^3*g-1296*a* 
b^4*c^2*d^2*e^2*f*h+936*a*b^4*c^2*d^2*e*f^2*g-128*a*b^4*c*d^3*e^3*h+2496*a 
*b^4*c*d^3*e^2*f*g+640*a*b^4*d^4*e^3*g+40*b^5*c^4*e*f^2*h-35*b^5*c^4*f^...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)^5/(d*x+c)^3/(f*x+e)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)**5/(d*x+c)**3/(f*x+e)**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((h*x+g)/(b*x+a)^5/(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(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. 6287 vs. \(2 (2029) = 4058\).

Time = 0.53 (sec) , antiderivative size = 6287, normalized size of antiderivative = 3.03 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)/(b*x+a)^5/(d*x+c)^3/(f*x+e)^(1/2),x, algorithm="giac")
 

Output:

1/64*(1920*b^7*d^4*e^4*g + 640*b^7*c*d^3*e^3*f*g - 8320*a*b^6*d^4*e^3*f*g 
+ 288*b^7*c^2*d^2*e^2*f^2*g - 2496*a*b^6*c*d^3*e^2*f^2*g + 13728*a^2*b^5*d 
^4*e^2*f^2*g + 120*b^7*c^3*d*e*f^3*g - 936*a*b^6*c^2*d^2*e*f^3*g + 3432*a^ 
2*b^5*c*d^3*e*f^3*g - 10296*a^3*b^4*d^4*e*f^3*g + 35*b^7*c^4*f^4*g - 260*a 
*b^6*c^3*d*f^4*g + 858*a^2*b^5*c^2*d^2*f^4*g - 1716*a^3*b^4*c*d^3*f^4*g + 
3003*a^4*b^3*d^4*f^4*g - 1280*b^7*c*d^3*e^4*h - 640*a*b^6*d^4*e^4*h - 384* 
b^7*c^2*d^2*e^3*f*h + 5248*a*b^6*c*d^3*e^3*f*h + 2816*a^2*b^5*d^4*e^3*f*h 
- 144*b^7*c^3*d*e^2*f^2*h + 1296*a*b^6*c^2*d^2*e^2*f^2*h - 7920*a^2*b^5*c* 
d^3*e^2*f^2*h - 4752*a^3*b^4*d^4*e^2*f^2*h - 40*b^7*c^4*e*f^3*h + 328*a*b^ 
6*c^3*d*e*f^3*h - 1320*a^2*b^5*c^2*d^2*e*f^3*h + 5016*a^3*b^4*c*d^3*e*f^3* 
h + 3696*a^4*b^3*d^4*e*f^3*h + 5*a*b^6*c^4*f^4*h - 44*a^2*b^5*c^3*d*f^4*h 
+ 198*a^3*b^4*c^2*d^2*f^4*h - 924*a^4*b^3*c*d^3*f^4*h - 1155*a^5*b^2*d^4*f 
^4*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^11*c^7*e^4 - 7*a*b^ 
10*c^6*d*e^4 + 21*a^2*b^9*c^5*d^2*e^4 - 35*a^3*b^8*c^4*d^3*e^4 + 35*a^4*b^ 
7*c^3*d^4*e^4 - 21*a^5*b^6*c^2*d^5*e^4 + 7*a^6*b^5*c*d^6*e^4 - a^7*b^4*d^7 
*e^4 - 4*a*b^10*c^7*e^3*f + 28*a^2*b^9*c^6*d*e^3*f - 84*a^3*b^8*c^5*d^2*e^ 
3*f + 140*a^4*b^7*c^4*d^3*e^3*f - 140*a^5*b^6*c^3*d^4*e^3*f + 84*a^6*b^5*c 
^2*d^5*e^3*f - 28*a^7*b^4*c*d^6*e^3*f + 4*a^8*b^3*d^7*e^3*f + 6*a^2*b^9*c^ 
7*e^2*f^2 - 42*a^3*b^8*c^6*d*e^2*f^2 + 126*a^4*b^7*c^5*d^2*e^2*f^2 - 210*a 
^5*b^6*c^4*d^3*e^2*f^2 + 210*a^6*b^5*c^3*d^4*e^2*f^2 - 126*a^7*b^4*c^2*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Hanged} \] Input:

int((g + h*x)/((e + f*x)^(1/2)*(a + b*x)^5*(c + d*x)^3),x)
 

Output:

\text{Hanged}
 

Reduce [B] (verification not implemented)

Time = 9.73 (sec) , antiderivative size = 92781, normalized size of antiderivative = 44.67 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

int((h*x+g)/(b*x+a)^5/(d*x+c)^3/(f*x+e)^(1/2),x)
 

Output:

(3465*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b 
*e)))*a**9*b*c**5*d**4*f**7*h - 10395*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e 
 + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b*c**4*d**5*e*f**6*h + 6930*sqr 
t(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a** 
9*b*c**4*d**5*f**7*h*x + 10395*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x) 
*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b*c**3*d**6*e**2*f**5*h - 20790*sqrt(b 
)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b 
*c**3*d**6*e*f**6*h*x + 3465*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b 
)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b*c**3*d**6*f**7*h*x**2 - 3465*sqrt(b)*s 
qrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b*c* 
*2*d**7*e**3*f**4*h + 20790*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b) 
/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b*c**2*d**7*e**2*f**5*h*x - 10395*sqrt(b) 
*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b* 
c**2*d**7*e*f**6*h*x**2 - 6930*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x) 
*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b*c*d**8*e**3*f**4*h*x + 10395*sqrt(b) 
*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b* 
c*d**8*e**2*f**5*h*x**2 - 3465*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x) 
*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b*d**9*e**3*f**4*h*x**2 + 2772*sqrt(b) 
*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**8*b* 
*2*c**6*d**3*f**7*h - 19404*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)...