\(\int \frac {g+h x}{(a+b x)^4 (c+d x)^3 (e+f x)^{3/2}} \, dx\) [193]

Optimal result

Integrand size = 29, antiderivative size = 1935 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^3 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Output:

-1/8*f*(6*a^5*d^4*f^4*(-c*f*h-4*d*e*h+5*d*f*g)+2*a^4*b*d^3*f^3*(19*c^2*f^2 
*h-c*d*f*(-40*e*h+63*f*g)-4*d^2*e*(-4*e*h+3*f*g))+b^5*(80*d^5*e^5*g+3*c^3* 
d^2*e^2*f^2*(-10*e*h+7*f*g)+3*c^4*d*e*f^3*(-6*e*h+5*f*g)-5*c^5*f^4*(-6*e*h 
+7*f*g)-4*c*d^4*e^4*(12*e*h+37*f*g)+c^2*d^3*e^3*f*(96*e*h+37*f*g))+a^3*b^2 
*d^2*f^2*(71*c^3*f^3*h+5*c^2*d*f^2*(-105*e*h+32*f*g)-d^3*e^2*(119*e*h+44*f 
*g)+c*d^2*e*f*(273*e*h+184*f*g))-a^2*b^3*d*f*(28*c^4*f^4*h+c^3*d*f^3*(-168 
*e*h+269*f*g)-d^4*e^3*(108*e*h+245*f*g)-3*c^2*d^2*e*f^2*(124*e*h+109*f*g)+ 
c*d^3*e^2*f*(320*e*h+603*f*g))+a*b^4*(5*c^5*f^5*h+c^4*d*f^4*(-129*e*h+160* 
f*g)-3*c^3*d^2*e*f^3*(-47*e*h+34*f*g)-4*d^5*e^4*(8*e*h+63*f*g)+14*c*d^4*e^ 
3*f*(14*e*h+37*f*g)-c^2*d^3*e^2*f^2*(331*e*h+174*f*g)))/(-a*d+b*c)^5/(-a*f 
+b*e)^4/(-c*f+d*e)^3/(f*x+e)^(1/2)+1/24*d*(3*a^3*d^2*f^2*(-47*c*f*h+43*d*e 
*h+4*d*f*g)-b^3*(120*d^3*e^3*g-c^2*d*e*f*(-30*e*h+41*f*g)-5*c^3*f^2*(-6*e* 
h+7*f*g)-8*c*d^2*e^2*(9*e*h+4*f*g))-a*b^2*(5*c^3*f^3*h+c^2*d*f^2*(-109*e*h 
+146*f*g)-8*d^3*e^2*(6*e*h+41*f*g)+2*c*d^2*e*f*(94*e*h+73*f*g))+a^2*b*d*f* 
(26*c^2*f^2*h-d^2*e*(142*e*h+255*f*g)+c*d*f*(152*e*h+219*f*g)))/(-a*d+b*c) 
^4/(-a*f+b*e)^3/(-c*f+d*e)/(d*x+c)^2/(f*x+e)^(1/2)-1/3*(-a*h+b*g)/(-a*d+b* 
c)/(-a*f+b*e)/(b*x+a)^3/(d*x+c)^2/(f*x+e)^(1/2)+1/12*(11*a^2*d*f*h+b^2*(-6 
*c*e*h+7*c*f*g+10*d*e*g)-a*b*(c*f*h+4*d*e*h+17*d*f*g))/(-a*d+b*c)^2/(-a*f+ 
b*e)^2/(b*x+a)^2/(d*x+c)^2/(f*x+e)^(1/2)+1/24*(99*a^3*d^2*f^2*h-3*a^2*b*d* 
f*(8*c*f*h+32*d*e*h+59*d*f*g)-b^3*(80*d^2*e^2*g+2*c*d*e*(-24*e*h+31*f*g...
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 12.76 (sec) , antiderivative size = 1338, normalized size of antiderivative = 0.69 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^3 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

Integrate[(g + h*x)/((a + b*x)^4*(c + d*x)^3*(e + f*x)^(3/2)),x]
 

Output:

((-8*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))/(a + b*x)^3 + (22*a^2*d*f*h + 2* 
b^2*(10*d*e*g + 7*c*f*g - 6*c*e*h) - 2*a*b*(17*d*f*g + 4*d*e*h + c*f*h))/( 
a + b*x)^2 + (99*a^3*d^2*f^2*h - 3*a^2*b*d*f*(59*d*f*g + 32*d*e*h + 8*c*f* 
h) + a*b^2*(5*c^2*f^2*h + 2*c*d*f*(66*f*g - 47*e*h) + 2*d^2*e*(111*f*g + 1 
6*e*h)) + b^3*(-80*d^2*e^2*g + 5*c^2*f*(-7*f*g + 6*e*h) + 2*c*d*e*(-31*f*g 
 + 24*e*h)))/((b*c - a*d)*(b*e - a*f)*(a + b*x)) + (-(d*(b*c - a*d)^2*(b*e 
 - a*f)*(d*e - c*f)^2*(-3*a^3*d^2*f^2*(4*d*f*g + 43*d*e*h - 47*c*f*h) + b^ 
3*(120*d^3*e^3*g + 5*c^3*f^2*(-7*f*g + 6*e*h) - 8*c*d^2*e^2*(4*f*g + 9*e*h 
) + c^2*d*e*f*(-41*f*g + 30*e*h)) + a*b^2*(5*c^3*f^3*h + c^2*d*f^2*(146*f* 
g - 109*e*h) - 8*d^3*e^2*(41*f*g + 6*e*h) + 2*c*d^2*e*f*(73*f*g + 94*e*h)) 
 + a^2*b*d*f*(-26*c^2*f^2*h + d^2*e*(255*f*g + 142*e*h) - c*d*f*(219*f*g + 
 152*e*h)))) + (c + d*x)*(-(d*(b*c - a*d)*(b*e - a*f)*(-(d*e) + c*f)*(-6*a 
^4*d^3*f^3*(-5*d*f*g + 4*d*e*h + c*f*h) + b^4*(-240*d^4*e^4*g + c^2*d^2*e^ 
2*f*(f*g - 216*e*h) + 4*c^3*d*e*f^2*(-5*f*g + 3*e*h) + 36*c*d^3*e^3*(9*f*g 
 + 4*e*h) + 5*c^4*f^3*(-7*f*g + 6*e*h)) + a^3*b*d^2*f^2*(269*c^2*f^2*h - 2 
*c*d*f*(63*f*g + 194*e*h) + d^2*e*(6*f*g + 239*e*h)) - a^2*b^2*d*f*(28*c^3 
*f^3*h + d^3*e^2*(467*f*g + 276*e*h) - 2*c*d^2*e*f*(458*f*g + 289*e*h) + c 
^2*d*f^2*(269*f*g + 454*e*h)) + a*b^3*(5*c^4*f^4*h + 4*c^3*d*f^3*(40*f*g - 
 31*e*h) + 12*d^4*e^3*(53*f*g + 8*e*h) - 2*c*d^3*e^2*f*(487*f*g + 234*e*h) 
 + c^2*d^2*e*f^2*(58*f*g + 611*e*h)))) - 3*(c + d*x)*(b^2*(d*e - c*f)^3...
 

Rubi [A] (verified)

Time = 5.80 (sec) , antiderivative size = 2056, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {168, 27, 168, 27, 168, 27, 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.

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

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\int \frac {2 b \left (5 d e g+\frac {7 c f g}{2}-3 c e h\right )-a (6 d f g+4 d e h+c f h)+11 d f (b g-a h) x}{2 (a+b x)^3 (c+d x)^3 (e+f x)^{3/2}}dx}{3 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {b (10 d e g+7 c f g-6 c e h)-a (6 d f g+4 d e h+c f h)+11 d f (b g-a h) x}{(a+b x)^3 (c+d x)^3 (e+f x)^{3/2}}dx}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\int \frac {3 d f (8 d f g+20 d e h+5 c f h) a^2-b \left (4 e (33 f g+8 e h) d^2+c f (69 f g-40 e h) d+5 c^2 f^2 h\right ) a+b^2 \left (5 f (7 f g-6 e h) c^2+2 d e (31 f g-24 e h) c+80 d^2 e^2 g\right )+9 d f \left (11 d f h a^2-b (17 d f g+4 d e h+c f h) a+b^2 (10 d e g+7 c f g-6 c e h)\right ) x}{2 (a+b x)^2 (c+d x)^3 (e+f x)^{3/2}}dx}{2 (b c-a d) (b e-a f)}-\frac {11 a^2 d f h-a b (c f h+4 d e h+17 d f g)+b^2 (-6 c e h+7 c f g+10 d e g)}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {3 d f (8 d f g+20 d e h+5 c f h) a^2-b \left (4 e (33 f g+8 e h) d^2+c f (69 f g-40 e h) d+5 c^2 f^2 h\right ) a+b^2 \left (5 f (7 f g-6 e h) c^2+2 d e (31 f g-24 e h) c+80 d^2 e^2 g\right )+9 d f \left (11 d f h a^2-b (17 d f g+4 d e h+c f h) a+b^2 (10 d e g+7 c f g-6 c e h)\right ) x}{(a+b x)^2 (c+d x)^3 (e+f x)^{3/2}}dx}{4 (b c-a d) (b e-a f)}-\frac {11 a^2 d f h-a b (c f h+4 d e h+17 d f g)+b^2 (-6 c e h+7 c f g+10 d e g)}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\frac {99 a^3 d^2 f^2 h-3 a^2 b d f (8 c f h+32 d e h+59 d f g)+a b^2 \left (5 c^2 f^2 h+2 c d f (66 f g-47 e h)+2 d^2 e (16 e h+111 f g)\right )-b^3 \left (5 c^2 f (7 f g-6 e h)+2 c d e (31 f g-24 e h)+80 d^2 e^2 g\right )}{(a+b x) (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}-\frac {\int \frac {-3 d^2 f^2 (43 c f h+4 d (4 f g+43 e h)) a^3+b d f \left (4 e (255 f g+142 e h) d^2+c f (363 f g+64 e h) d+64 c^2 f^2 h\right ) a^2-b^2 \left (32 e^2 (41 f g+6 e h) d^3+2 c e f (485 f g-264 e h) d^2+2 c^2 f^2 (170 f g-111 e h) d+15 c^3 f^3 h\right ) a+3 b^3 \left (5 f^2 (7 f g-6 e h) c^3+18 d e f (5 f g-4 e h) c^2+48 d^2 e^2 (3 f g-2 e h) c+160 d^3 e^3 g\right )-7 d f \left (99 d^2 f^2 h a^3-3 b d f (59 d f g+32 d e h+8 c f h) a^2+b^2 \left (2 e (111 f g+16 e h) d^2+2 c f (66 f g-47 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (31 f g-24 e h) c+80 d^2 e^2 g\right )\right ) x}{2 (a+b x) (c+d x)^3 (e+f x)^{3/2}}dx}{(b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {11 a^2 d f h-a b (c f h+4 d e h+17 d f g)+b^2 (-6 c e h+7 c f g+10 d e g)}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {99 a^3 d^2 f^2 h-3 a^2 b d f (8 c f h+32 d e h+59 d f g)+a b^2 \left (5 c^2 f^2 h+2 c d f (66 f g-47 e h)+2 d^2 e (16 e h+111 f g)\right )-b^3 \left (5 c^2 f (7 f g-6 e h)+2 c d e (31 f g-24 e h)+80 d^2 e^2 g\right )}{(a+b x) (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}-\frac {\int \frac {-3 d^2 f^2 (43 c f h+4 d (4 f g+43 e h)) a^3+b d f \left (4 e (255 f g+142 e h) d^2+c f (363 f g+64 e h) d+64 c^2 f^2 h\right ) a^2-b^2 \left (32 e^2 (41 f g+6 e h) d^3+2 c e f (485 f g-264 e h) d^2+2 c^2 f^2 (170 f g-111 e h) d+15 c^3 f^3 h\right ) a+3 b^3 \left (5 f^2 (7 f g-6 e h) c^3+18 d e f (5 f g-4 e h) c^2+48 d^2 e^2 (3 f g-2 e h) c+160 d^3 e^3 g\right )-7 d f \left (99 d^2 f^2 h a^3-3 b d f (59 d f g+32 d e h+8 c f h) a^2+b^2 \left (2 e (111 f g+16 e h) d^2+2 c f (66 f g-47 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (31 f g-24 e h) c+80 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x)^3 (e+f x)^{3/2}}dx}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {11 a^2 d f h-a b (c f h+4 d e h+17 d f g)+b^2 (-6 c e h+7 c f g+10 d e g)}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (17 d f g+4 d e h+c f h) a+b^2 (10 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-3 b d f (59 d f g+32 d e h+8 c f h) a^2+b^2 \left (2 e (111 f g+16 e h) d^2+2 c f (66 f g-47 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (31 f g-24 e h) c+80 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {\int -\frac {2 \left (12 d^3 f^3 (5 d f g-4 d e h-c f h) a^4-b d^2 f^2 \left (-2 e (6 f g+239 e h) d^2+c f (192 f g+131 e h) d+167 c^2 f^2 h\right ) a^3+b^2 d f \left (-2 e^2 (467 f g+276 e h) d^3+c e f (557 f g+446 e h) d^2+c^2 f^2 (557 f g-148 e h) d+74 c^3 f^3 h\right ) a^2-b^3 \left (-24 e^3 (53 f g+8 e h) d^4+4 c e^2 f (77 f g+174 e h) d^3+2 c^2 e f^2 (307 f g-141 e h) d^2+c^3 f^3 (410 f g-297 e h) d+15 c^4 f^4 h\right ) a-3 b^4 (d e-c f) \left (5 f^2 (7 f g-6 e h) c^3+18 d e f (5 f g-4 e h) c^2+48 d^2 e^2 (3 f g-2 e h) c+160 d^3 e^3 g\right )+5 b d f \left (3 d^2 f^2 (4 d f g+43 d e h-47 c f h) a^3+b d f \left (-e (255 f g+142 e h) d^2+c f (219 f g+152 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (-8 e^2 (41 f g+6 e h) d^3+2 c e f (73 f g+94 e h) d^2+c^2 f^2 (146 f g-109 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (41 f g-30 e h) c^2-8 d^2 e^2 (4 f g+9 e h) c+120 d^3 e^3 g\right )\right ) x\right )}{(a+b x) (c+d x)^2 (e+f x)^{3/2}}dx}{2 (b c-a d) (d e-c f)}-\frac {2 d \left (3 d^2 f^2 (4 d f g+43 d e h-47 c f h) a^3+b d f \left (-e (255 f g+142 e h) d^2+c f (219 f g+152 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (-8 e^2 (41 f g+6 e h) d^3+2 c e f (73 f g+94 e h) d^2+c^2 f^2 (146 f g-109 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (41 f g-30 e h) c^2-8 d^2 e^2 (4 f g+9 e h) c+120 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}}{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)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (17 d f g+4 d e h+c f h) a+b^2 (10 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-3 b d f (59 d f g+32 d e h+8 c f h) a^2+b^2 \left (2 e (111 f g+16 e h) d^2+2 c f (66 f g-47 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (31 f g-24 e h) c+80 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {2 d \left (3 d^2 f^2 (4 d f g+43 d e h-47 c f h) a^3+b d f \left (-e (255 f g+142 e h) d^2+c f (219 f g+152 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (-8 e^2 (41 f g+6 e h) d^3+2 c e f (73 f g+94 e h) d^2+c^2 f^2 (146 f g-109 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (41 f g-30 e h) c^2-8 d^2 e^2 (4 f g+9 e h) c+120 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}-\frac {\int \frac {12 d^3 f^3 (5 d f g-4 d e h-c f h) a^4-b d^2 f^2 \left (-2 e (6 f g+239 e h) d^2+c f (192 f g+131 e h) d+167 c^2 f^2 h\right ) a^3+b^2 d f \left (-2 e^2 (467 f g+276 e h) d^3+c e f (557 f g+446 e h) d^2+c^2 f^2 (557 f g-148 e h) d+74 c^3 f^3 h\right ) a^2-b^3 \left (-24 e^3 (53 f g+8 e h) d^4+4 c e^2 f (77 f g+174 e h) d^3+2 c^2 e f^2 (307 f g-141 e h) d^2+c^3 f^3 (410 f g-297 e h) d+15 c^4 f^4 h\right ) a-3 b^4 (d e-c f) \left (5 f^2 (7 f g-6 e h) c^3+18 d e f (5 f g-4 e h) c^2+48 d^2 e^2 (3 f g-2 e h) c+160 d^3 e^3 g\right )+5 b d f \left (3 d^2 f^2 (4 d f g+43 d e h-47 c f h) a^3+b d f \left (-e (255 f g+142 e h) d^2+c f (219 f g+152 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (-8 e^2 (41 f g+6 e h) d^3+2 c e f (73 f g+94 e h) d^2+c^2 f^2 (146 f g-109 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (41 f g-30 e h) c^2-8 d^2 e^2 (4 f g+9 e h) c+120 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x)^2 (e+f x)^{3/2}}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)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (17 d f g+4 d e h+c f h) a+b^2 (10 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-3 b d f (59 d f g+32 d e h+8 c f h) a^2+b^2 \left (2 e (111 f g+16 e h) d^2+2 c f (66 f g-47 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (31 f g-24 e h) c+80 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {2 d \left (3 d^2 f^2 (4 d f g+43 d e h-47 c f h) a^3+b d f \left (-e (255 f g+142 e h) d^2+c f (219 f g+152 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (-8 e^2 (41 f g+6 e h) d^3+2 c e f (73 f g+94 e h) d^2+c^2 f^2 (146 f g-109 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (41 f g-30 e h) c^2-8 d^2 e^2 (4 f g+9 e h) c+120 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (6 d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (e (6 f g+239 e h) d^2-2 c f (63 f g+194 e h) d+269 c^2 f^2 h\right ) a^3-b^2 d f \left (e^2 (467 f g+276 e h) d^3-2 c e f (458 f g+289 e h) d^2+c^2 f^2 (269 f g+454 e h) d+28 c^3 f^3 h\right ) a^2+b^3 \left (12 e^3 (53 f g+8 e h) d^4-2 c e^2 f (487 f g+234 e h) d^3+c^2 e f^2 (58 f g+611 e h) d^2+4 c^3 f^3 (40 f g-31 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (5 f^3 (7 f g-6 e h) c^4+4 d e f^2 (5 f g-3 e h) c^3-d^2 e^2 f (f g-216 e h) c^2-36 d^3 e^3 (9 f g+4 e h) c+240 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {\int \frac {3 \left (6 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+2 b d^3 f^3 \left (e (3 f g+4 e h) d^2-c f (63 f g-37 e h) d+19 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-2 e^2 (19 f g-60 e h) d^3+c e f (58 f g-115 e h) d^2+32 c^2 f^2 (5 f g-8 e h) d+71 c^3 f^3 h\right ) a^3-b^3 d f \left (6 e^3 (37 f g+28 e h) d^4-c e^2 f (313 f g+258 e h) d^3-2 c^2 e f^2 (29 f g-41 e h) d^2+c^3 f^3 (269 f g-140 e h) d+28 c^4 f^4 h\right ) a^2+b^4 \left (64 e^4 (6 f g+e h) d^5-8 c e^3 f (57 f g+34 e h) d^4-4 c^2 e^2 f^2 (29 f g-70 e h) d^3+c^3 e f^3 (58 f g+17 e h) d^2+4 c^4 f^4 (40 f g-31 e h) d+5 c^5 f^5 h\right ) a-b^5 (d e-c f)^2 \left (5 f^2 (7 f g-6 e h) c^3+18 d e f (5 f g-4 e h) c^2+48 d^2 e^2 (3 f g-2 e h) c+160 d^3 e^3 g\right )+b d f \left (6 d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (e (6 f g+239 e h) d^2-2 c f (63 f g+194 e h) d+269 c^2 f^2 h\right ) a^3-b^2 d f \left (e^2 (467 f g+276 e h) d^3-2 c e f (458 f g+289 e h) d^2+c^2 f^2 (269 f g+454 e h) d+28 c^3 f^3 h\right ) a^2+b^3 \left (12 e^3 (53 f g+8 e h) d^4-2 c e^2 f (487 f g+234 e h) d^3+c^2 e f^2 (58 f g+611 e h) d^2+4 c^3 f^3 (40 f g-31 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (5 f^3 (7 f g-6 e h) c^4+4 d e f^2 (5 f g-3 e h) c^3-d^2 e^2 f (f g-216 e h) c^2-36 d^3 e^3 (9 f g+4 e h) c+240 d^4 e^4 g\right )\right ) x\right )}{(a+b x) (c+d x) (e+f x)^{3/2}}dx}{(b c-a d) (d e-c f)}}{(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)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (17 d f g+4 d e h+c f h) a+b^2 (10 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-3 b d f (59 d f g+32 d e h+8 c f h) a^2+b^2 \left (2 e (111 f g+16 e h) d^2+2 c f (66 f g-47 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (31 f g-24 e h) c+80 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {2 d \left (3 d^2 f^2 (4 d f g+43 d e h-47 c f h) a^3+b d f \left (-e (255 f g+142 e h) d^2+c f (219 f g+152 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (-8 e^2 (41 f g+6 e h) d^3+2 c e f (73 f g+94 e h) d^2+c^2 f^2 (146 f g-109 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (41 f g-30 e h) c^2-8 d^2 e^2 (4 f g+9 e h) c+120 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (6 d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (e (6 f g+239 e h) d^2-2 c f (63 f g+194 e h) d+269 c^2 f^2 h\right ) a^3-b^2 d f \left (e^2 (467 f g+276 e h) d^3-2 c e f (458 f g+289 e h) d^2+c^2 f^2 (269 f g+454 e h) d+28 c^3 f^3 h\right ) a^2+b^3 \left (12 e^3 (53 f g+8 e h) d^4-2 c e^2 f (487 f g+234 e h) d^3+c^2 e f^2 (58 f g+611 e h) d^2+4 c^3 f^3 (40 f g-31 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (5 f^3 (7 f g-6 e h) c^4+4 d e f^2 (5 f g-3 e h) c^3-d^2 e^2 f (f g-216 e h) c^2-36 d^3 e^3 (9 f g+4 e h) c+240 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \int \frac {6 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+2 b d^3 f^3 \left (e (3 f g+4 e h) d^2-c f (63 f g-37 e h) d+19 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-2 e^2 (19 f g-60 e h) d^3+c e f (58 f g-115 e h) d^2+32 c^2 f^2 (5 f g-8 e h) d+71 c^3 f^3 h\right ) a^3-b^3 d f \left (6 e^3 (37 f g+28 e h) d^4-c e^2 f (313 f g+258 e h) d^3-2 c^2 e f^2 (29 f g-41 e h) d^2+c^3 f^3 (269 f g-140 e h) d+28 c^4 f^4 h\right ) a^2+b^4 \left (64 e^4 (6 f g+e h) d^5-8 c e^3 f (57 f g+34 e h) d^4-4 c^2 e^2 f^2 (29 f g-70 e h) d^3+c^3 e f^3 (58 f g+17 e h) d^2+4 c^4 f^4 (40 f g-31 e h) d+5 c^5 f^5 h\right ) a-b^5 (d e-c f)^2 \left (5 f^2 (7 f g-6 e h) c^3+18 d e f (5 f g-4 e h) c^2+48 d^2 e^2 (3 f g-2 e h) c+160 d^3 e^3 g\right )+b d f \left (6 d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (e (6 f g+239 e h) d^2-2 c f (63 f g+194 e h) d+269 c^2 f^2 h\right ) a^3-b^2 d f \left (e^2 (467 f g+276 e h) d^3-2 c e f (458 f g+289 e h) d^2+c^2 f^2 (269 f g+454 e h) d+28 c^3 f^3 h\right ) a^2+b^3 \left (12 e^3 (53 f g+8 e h) d^4-2 c e^2 f (487 f g+234 e h) d^3+c^2 e f^2 (58 f g+611 e h) d^2+4 c^3 f^3 (40 f g-31 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (5 f^3 (7 f g-6 e h) c^4+4 d e f^2 (5 f g-3 e h) c^3-d^2 e^2 f (f g-216 e h) c^2-36 d^3 e^3 (9 f g+4 e h) c+240 d^4 e^4 g\right )\right ) x}{(a+b x) (c+d x) (e+f x)^{3/2}}dx}{(b c-a d) (d e-c f)}}{(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)}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (17 d f g+4 d e h+c f h) a+b^2 (10 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-3 b d f (59 d f g+32 d e h+8 c f h) a^2+b^2 \left (2 e (111 f g+16 e h) d^2+2 c f (66 f g-47 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (31 f g-24 e h) c+80 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {2 d \left (3 d^2 f^2 (4 d f g+43 d e h-47 c f h) a^3+b d f \left (-e (255 f g+142 e h) d^2+c f (219 f g+152 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (-8 e^2 (41 f g+6 e h) d^3+2 c e f (73 f g+94 e h) d^2+c^2 f^2 (146 f g-109 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (41 f g-30 e h) c^2-8 d^2 e^2 (4 f g+9 e h) c+120 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (6 d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (e (6 f g+239 e h) d^2-2 c f (63 f g+194 e h) d+269 c^2 f^2 h\right ) a^3-b^2 d f \left (e^2 (467 f g+276 e h) d^3-2 c e f (458 f g+289 e h) d^2+c^2 f^2 (269 f g+454 e h) d+28 c^3 f^3 h\right ) a^2+b^3 \left (12 e^3 (53 f g+8 e h) d^4-2 c e^2 f (487 f g+234 e h) d^3+c^2 e f^2 (58 f g+611 e h) d^2+4 c^3 f^3 (40 f g-31 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (5 f^3 (7 f g-6 e h) c^4+4 d e f^2 (5 f g-3 e h) c^3-d^2 e^2 f (f g-216 e h) c^2-36 d^3 e^3 (9 f g+4 e h) c+240 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (6 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+2 b d^3 f^3 \left (-4 e (3 f g-4 e h) d^2-c f (63 f g-40 e h) d+19 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-e^2 (44 f g+119 e h) d^3+c e f (184 f g+273 e h) d^2+5 c^2 f^2 (32 f g-105 e h) d+71 c^3 f^3 h\right ) a^3-b^3 d f \left (-e^3 (245 f g+108 e h) d^4+c e^2 f (603 f g+320 e h) d^3-3 c^2 e f^2 (109 f g+124 e h) d^2+c^3 f^3 (269 f g-168 e h) d+28 c^4 f^4 h\right ) a^2+b^4 \left (-4 e^4 (63 f g+8 e h) d^5+14 c e^3 f (37 f g+14 e h) d^4-c^2 e^2 f^2 (174 f g+331 e h) d^3-3 c^3 e f^3 (34 f g-47 e h) d^2+c^4 f^4 (160 f g-129 e h) d+5 c^5 f^5 h\right ) a+b^5 \left (-5 f^4 (7 f g-6 e h) c^5+3 d e f^3 (5 f g-6 e h) c^4+3 d^2 e^2 f^2 (7 f g-10 e h) c^3+d^3 e^3 f (37 f g+96 e h) c^2-4 d^4 e^4 (37 f g+12 e h) c+80 d^5 e^5 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {2 \int \frac {6 d^5 f^5 (5 d f g-4 d e h-c f h) a^6+2 b d^4 f^4 \left (-4 e (3 f g-4 e h) d^2-c f (63 f g-40 e h) d+19 c^2 f^2 h\right ) a^5-4 b^2 d^3 f^3 \left (e^2 (11 f g-28 e h) d^3-c e f (46 f g-105 e h) d^2-2 c^2 f^2 (20 f g+21 e h) d+40 c^3 f^3 h\right ) a^4+b^3 d^2 f^2 \left (-8 e^3 (23 f g+36 e h) d^4+c e^2 f (684 f g+769 e h) d^3-3 c^2 e f^2 (320 f g+173 e h) d^2+c^3 f^3 (160 f g+267 e h) d+71 c^4 f^4 h\right ) a^3-b^4 d f \left (-2 e^4 (303 f g+116 e h) d^5+c e^3 f (1627 f g+794 e h) d^4-c^2 e^2 f^2 (1113 f g+1088 e h) d^3-3 c^3 e f^3 (109 f g-272 e h) d^2+c^4 f^4 (269 f g-168 e h) d+28 c^5 f^5 h\right ) a^2+b^5 \left (-32 e^5 (17 f g+2 e h) d^6+8 c e^4 f (157 f g+54 e h) d^5-2 c^2 e^3 f^2 (313 f g+408 e h) d^4-c^3 e^2 f^3 (174 f g-461 e h) d^3-3 c^4 e f^4 (34 f g-47 e h) d^2+c^5 f^5 (160 f g-129 e h) d+5 c^6 f^6 h\right ) a+b^6 (d e-c f)^3 \left (5 f^2 (7 f g-6 e h) c^3+18 d e f (5 f g-4 e h) c^2+48 d^2 e^2 (3 f g-2 e h) c+160 d^3 e^3 g\right )+b d f \left (6 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+2 b d^3 f^3 \left (-4 e (3 f g-4 e h) d^2-c f (63 f g-40 e h) d+19 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-e^2 (44 f g+119 e h) d^3+c e f (184 f g+273 e h) d^2+5 c^2 f^2 (32 f g-105 e h) d+71 c^3 f^3 h\right ) a^3-b^3 d f \left (-e^3 (245 f g+108 e h) d^4+c e^2 f (603 f g+320 e h) d^3-3 c^2 e f^2 (109 f g+124 e h) d^2+c^3 f^3 (269 f g-168 e h) d+28 c^4 f^4 h\right ) a^2+b^4 \left (-4 e^4 (63 f g+8 e h) d^5+14 c e^3 f (37 f g+14 e h) d^4-c^2 e^2 f^2 (174 f g+331 e h) d^3-3 c^3 e f^3 (34 f g-47 e h) d^2+c^4 f^4 (160 f g-129 e h) d+5 c^5 f^5 h\right ) a+b^5 \left (-5 f^4 (7 f g-6 e h) c^5+3 d e f^3 (5 f g-6 e h) c^4+3 d^2 e^2 f^2 (7 f g-10 e h) c^3+d^3 e^3 f (37 f g+96 e h) c^2-4 d^4 e^4 (37 f g+12 e h) c+80 d^5 e^5 g\right )\right ) x}{2 (a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}\right )}{(b c-a d) (d e-c f)}}{(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)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (17 d f g+4 d e h+c f h) a+b^2 (10 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-3 b d f (59 d f g+32 d e h+8 c f h) a^2+b^2 \left (2 e (111 f g+16 e h) d^2+2 c f (66 f g-47 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (31 f g-24 e h) c+80 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {2 d \left (3 d^2 f^2 (4 d f g+43 d e h-47 c f h) a^3+b d f \left (-e (255 f g+142 e h) d^2+c f (219 f g+152 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (-8 e^2 (41 f g+6 e h) d^3+2 c e f (73 f g+94 e h) d^2+c^2 f^2 (146 f g-109 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (41 f g-30 e h) c^2-8 d^2 e^2 (4 f g+9 e h) c+120 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (6 d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (e (6 f g+239 e h) d^2-2 c f (63 f g+194 e h) d+269 c^2 f^2 h\right ) a^3-b^2 d f \left (e^2 (467 f g+276 e h) d^3-2 c e f (458 f g+289 e h) d^2+c^2 f^2 (269 f g+454 e h) d+28 c^3 f^3 h\right ) a^2+b^3 \left (12 e^3 (53 f g+8 e h) d^4-2 c e^2 f (487 f g+234 e h) d^3+c^2 e f^2 (58 f g+611 e h) d^2+4 c^3 f^3 (40 f g-31 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (5 f^3 (7 f g-6 e h) c^4+4 d e f^2 (5 f g-3 e h) c^3-d^2 e^2 f (f g-216 e h) c^2-36 d^3 e^3 (9 f g+4 e h) c+240 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (6 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+2 b d^3 f^3 \left (-4 e (3 f g-4 e h) d^2-c f (63 f g-40 e h) d+19 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-e^2 (44 f g+119 e h) d^3+c e f (184 f g+273 e h) d^2+5 c^2 f^2 (32 f g-105 e h) d+71 c^3 f^3 h\right ) a^3-b^3 d f \left (-e^3 (245 f g+108 e h) d^4+c e^2 f (603 f g+320 e h) d^3-3 c^2 e f^2 (109 f g+124 e h) d^2+c^3 f^3 (269 f g-168 e h) d+28 c^4 f^4 h\right ) a^2+b^4 \left (-4 e^4 (63 f g+8 e h) d^5+14 c e^3 f (37 f g+14 e h) d^4-c^2 e^2 f^2 (174 f g+331 e h) d^3-3 c^3 e f^3 (34 f g-47 e h) d^2+c^4 f^4 (160 f g-129 e h) d+5 c^5 f^5 h\right ) a+b^5 \left (-5 f^4 (7 f g-6 e h) c^5+3 d e f^3 (5 f g-6 e h) c^4+3 d^2 e^2 f^2 (7 f g-10 e h) c^3+d^3 e^3 f (37 f g+96 e h) c^2-4 d^4 e^4 (37 f g+12 e h) c+80 d^5 e^5 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\int \frac {6 d^5 f^5 (5 d f g-4 d e h-c f h) a^6+2 b d^4 f^4 \left (-4 e (3 f g-4 e h) d^2-c f (63 f g-40 e h) d+19 c^2 f^2 h\right ) a^5-4 b^2 d^3 f^3 \left (e^2 (11 f g-28 e h) d^3-c e f (46 f g-105 e h) d^2-2 c^2 f^2 (20 f g+21 e h) d+40 c^3 f^3 h\right ) a^4+b^3 d^2 f^2 \left (-8 e^3 (23 f g+36 e h) d^4+c e^2 f (684 f g+769 e h) d^3-3 c^2 e f^2 (320 f g+173 e h) d^2+c^3 f^3 (160 f g+267 e h) d+71 c^4 f^4 h\right ) a^3-b^4 d f \left (-2 e^4 (303 f g+116 e h) d^5+c e^3 f (1627 f g+794 e h) d^4-c^2 e^2 f^2 (1113 f g+1088 e h) d^3-3 c^3 e f^3 (109 f g-272 e h) d^2+c^4 f^4 (269 f g-168 e h) d+28 c^5 f^5 h\right ) a^2+b^5 \left (-32 e^5 (17 f g+2 e h) d^6+8 c e^4 f (157 f g+54 e h) d^5-2 c^2 e^3 f^2 (313 f g+408 e h) d^4-c^3 e^2 f^3 (174 f g-461 e h) d^3-3 c^4 e f^4 (34 f g-47 e h) d^2+c^5 f^5 (160 f g-129 e h) d+5 c^6 f^6 h\right ) a+b^6 (d e-c f)^3 \left (5 f^2 (7 f g-6 e h) c^3+18 d e f (5 f g-4 e h) c^2+48 d^2 e^2 (3 f g-2 e h) c+160 d^3 e^3 g\right )+b d f \left (6 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+2 b d^3 f^3 \left (-4 e (3 f g-4 e h) d^2-c f (63 f g-40 e h) d+19 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-e^2 (44 f g+119 e h) d^3+c e f (184 f g+273 e h) d^2+5 c^2 f^2 (32 f g-105 e h) d+71 c^3 f^3 h\right ) a^3-b^3 d f \left (-e^3 (245 f g+108 e h) d^4+c e^2 f (603 f g+320 e h) d^3-3 c^2 e f^2 (109 f g+124 e h) d^2+c^3 f^3 (269 f g-168 e h) d+28 c^4 f^4 h\right ) a^2+b^4 \left (-4 e^4 (63 f g+8 e h) d^5+14 c e^3 f (37 f g+14 e h) d^4-c^2 e^2 f^2 (174 f g+331 e h) d^3-3 c^3 e f^3 (34 f g-47 e h) d^2+c^4 f^4 (160 f g-129 e h) d+5 c^5 f^5 h\right ) a+b^5 \left (-5 f^4 (7 f g-6 e h) c^5+3 d e f^3 (5 f g-6 e h) c^4+3 d^2 e^2 f^2 (7 f g-10 e h) c^3+d^3 e^3 f (37 f g+96 e h) c^2-4 d^4 e^4 (37 f g+12 e h) c+80 d^5 e^5 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}\right )}{(b c-a d) (d e-c f)}}{(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)}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (17 d f g+4 d e h+c f h) a+b^2 (10 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-3 b d f (59 d f g+32 d e h+8 c f h) a^2+b^2 \left (2 e (111 f g+16 e h) d^2+2 c f (66 f g-47 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (31 f g-24 e h) c+80 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {2 d \left (3 d^2 f^2 (4 d f g+43 d e h-47 c f h) a^3+b d f \left (-e (255 f g+142 e h) d^2+c f (219 f g+152 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (-8 e^2 (41 f g+6 e h) d^3+2 c e f (73 f g+94 e h) d^2+c^2 f^2 (146 f g-109 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (41 f g-30 e h) c^2-8 d^2 e^2 (4 f g+9 e h) c+120 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (6 d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (e (6 f g+239 e h) d^2-2 c f (63 f g+194 e h) d+269 c^2 f^2 h\right ) a^3-b^2 d f \left (e^2 (467 f g+276 e h) d^3-2 c e f (458 f g+289 e h) d^2+c^2 f^2 (269 f g+454 e h) d+28 c^3 f^3 h\right ) a^2+b^3 \left (12 e^3 (53 f g+8 e h) d^4-2 c e^2 f (487 f g+234 e h) d^3+c^2 e f^2 (58 f g+611 e h) d^2+4 c^3 f^3 (40 f g-31 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (5 f^3 (7 f g-6 e h) c^4+4 d e f^2 (5 f g-3 e h) c^3-d^2 e^2 f (f g-216 e h) c^2-36 d^3 e^3 (9 f g+4 e h) c+240 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (6 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+2 b d^3 f^3 \left (-4 e (3 f g-4 e h) d^2-c f (63 f g-40 e h) d+19 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-e^2 (44 f g+119 e h) d^3+c e f (184 f g+273 e h) d^2+5 c^2 f^2 (32 f g-105 e h) d+71 c^3 f^3 h\right ) a^3-b^3 d f \left (-e^3 (245 f g+108 e h) d^4+c e^2 f (603 f g+320 e h) d^3-3 c^2 e f^2 (109 f g+124 e h) d^2+c^3 f^3 (269 f g-168 e h) d+28 c^4 f^4 h\right ) a^2+b^4 \left (-4 e^4 (63 f g+8 e h) d^5+14 c e^3 f (37 f g+14 e h) d^4-c^2 e^2 f^2 (174 f g+331 e h) d^3-3 c^3 e f^3 (34 f g-47 e h) d^2+c^4 f^4 (160 f g-129 e h) d+5 c^5 f^5 h\right ) a+b^5 \left (-5 f^4 (7 f g-6 e h) c^5+3 d e f^3 (5 f g-6 e h) c^4+3 d^2 e^2 f^2 (7 f g-10 e h) c^3+d^3 e^3 f (37 f g+96 e h) c^2-4 d^4 e^4 (37 f g+12 e h) c+80 d^5 e^5 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {b^3 (d e-c f)^3 \left (231 d^3 f^3 h a^4-33 b d^2 f^2 (13 d f g+12 d e h+3 c f h) a^3+33 b^2 d f \left (2 e (13 f g+4 e h) d^2+c f (13 f g-6 e h) d+c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (39 f g+4 e h) d^3+36 c e f (13 f g-8 e h) d^2+3 c^2 f^2 (65 f g-48 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (5 f^2 (7 f g-6 e h) c^3+18 d e f (5 f g-4 e h) c^2+48 d^2 e^2 (3 f g-2 e h) c+160 d^3 e^3 g\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}-\frac {2 d^4 (b e-a f)^4 \left (\left (-99 f^2 h c^3+11 d f (13 f g+12 e h) c^2-16 d^2 e (13 f g+3 e h) c+80 d^3 e^2 g\right ) b^2+2 a d \left (8 e (3 f g-2 e h) d^2-c f (39 f g-20 e h) d+11 c^2 f^2 h\right ) b+3 a^2 d^2 f (5 d f g-4 d e h-c f h)\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}}{(b e-a f) (d e-c f)}\right )}{(b c-a d) (d e-c f)}}{(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)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (17 d f g+4 d e h+c f h) a+b^2 (10 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-3 b d f (59 d f g+32 d e h+8 c f h) a^2+b^2 \left (2 e (111 f g+16 e h) d^2+2 c f (66 f g-47 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (31 f g-24 e h) c+80 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {2 d \left (3 d^2 f^2 (4 d f g+43 d e h-47 c f h) a^3+b d f \left (-e (255 f g+142 e h) d^2+c f (219 f g+152 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (-8 e^2 (41 f g+6 e h) d^3+2 c e f (73 f g+94 e h) d^2+c^2 f^2 (146 f g-109 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (41 f g-30 e h) c^2-8 d^2 e^2 (4 f g+9 e h) c+120 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (6 d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (e (6 f g+239 e h) d^2-2 c f (63 f g+194 e h) d+269 c^2 f^2 h\right ) a^3-b^2 d f \left (e^2 (467 f g+276 e h) d^3-2 c e f (458 f g+289 e h) d^2+c^2 f^2 (269 f g+454 e h) d+28 c^3 f^3 h\right ) a^2+b^3 \left (12 e^3 (53 f g+8 e h) d^4-2 c e^2 f (487 f g+234 e h) d^3+c^2 e f^2 (58 f g+611 e h) d^2+4 c^3 f^3 (40 f g-31 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (5 f^3 (7 f g-6 e h) c^4+4 d e f^2 (5 f g-3 e h) c^3-d^2 e^2 f (f g-216 e h) c^2-36 d^3 e^3 (9 f g+4 e h) c+240 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (6 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+2 b d^3 f^3 \left (-4 e (3 f g-4 e h) d^2-c f (63 f g-40 e h) d+19 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-e^2 (44 f g+119 e h) d^3+c e f (184 f g+273 e h) d^2+5 c^2 f^2 (32 f g-105 e h) d+71 c^3 f^3 h\right ) a^3-b^3 d f \left (-e^3 (245 f g+108 e h) d^4+c e^2 f (603 f g+320 e h) d^3-3 c^2 e f^2 (109 f g+124 e h) d^2+c^3 f^3 (269 f g-168 e h) d+28 c^4 f^4 h\right ) a^2+b^4 \left (-4 e^4 (63 f g+8 e h) d^5+14 c e^3 f (37 f g+14 e h) d^4-c^2 e^2 f^2 (174 f g+331 e h) d^3-3 c^3 e f^3 (34 f g-47 e h) d^2+c^4 f^4 (160 f g-129 e h) d+5 c^5 f^5 h\right ) a+b^5 \left (-5 f^4 (7 f g-6 e h) c^5+3 d e f^3 (5 f g-6 e h) c^4+3 d^2 e^2 f^2 (7 f g-10 e h) c^3+d^3 e^3 f (37 f g+96 e h) c^2-4 d^4 e^4 (37 f g+12 e h) c+80 d^5 e^5 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {2 b^3 (d e-c f)^3 \left (231 d^3 f^3 h a^4-33 b d^2 f^2 (13 d f g+12 d e h+3 c f h) a^3+33 b^2 d f \left (2 e (13 f g+4 e h) d^2+c f (13 f g-6 e h) d+c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (39 f g+4 e h) d^3+36 c e f (13 f g-8 e h) d^2+3 c^2 f^2 (65 f g-48 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (5 f^2 (7 f g-6 e h) c^3+18 d e f (5 f g-4 e h) c^2+48 d^2 e^2 (3 f g-2 e h) c+160 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}-\frac {4 d^4 (b e-a f)^4 \left (\left (-99 f^2 h c^3+11 d f (13 f g+12 e h) c^2-16 d^2 e (13 f g+3 e h) c+80 d^3 e^2 g\right ) b^2+2 a d \left (8 e (3 f g-2 e h) d^2-c f (39 f g-20 e h) d+11 c^2 f^2 h\right ) b+3 a^2 d^2 f (5 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 c-a d) f}}{(b e-a f) (d e-c f)}\right )}{(b c-a d) (d e-c f)}}{(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)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (17 d f g+4 d e h+c f h) a+b^2 (10 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-3 b d f (59 d f g+32 d e h+8 c f h) a^2+b^2 \left (2 e (111 f g+16 e h) d^2+2 c f (66 f g-47 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (31 f g-24 e h) c+80 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {2 d \left (3 d^2 f^2 (4 d f g+43 d e h-47 c f h) a^3+b d f \left (-e (255 f g+142 e h) d^2+c f (219 f g+152 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (-8 e^2 (41 f g+6 e h) d^3+2 c e f (73 f g+94 e h) d^2+c^2 f^2 (146 f g-109 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (41 f g-30 e h) c^2-8 d^2 e^2 (4 f g+9 e h) c+120 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (6 d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (e (6 f g+239 e h) d^2-2 c f (63 f g+194 e h) d+269 c^2 f^2 h\right ) a^3-b^2 d f \left (e^2 (467 f g+276 e h) d^3-2 c e f (458 f g+289 e h) d^2+c^2 f^2 (269 f g+454 e h) d+28 c^3 f^3 h\right ) a^2+b^3 \left (12 e^3 (53 f g+8 e h) d^4-2 c e^2 f (487 f g+234 e h) d^3+c^2 e f^2 (58 f g+611 e h) d^2+4 c^3 f^3 (40 f g-31 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (5 f^3 (7 f g-6 e h) c^4+4 d e f^2 (5 f g-3 e h) c^3-d^2 e^2 f (f g-216 e h) c^2-36 d^3 e^3 (9 f g+4 e h) c+240 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (6 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+2 b d^3 f^3 \left (-4 e (3 f g-4 e h) d^2-c f (63 f g-40 e h) d+19 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-e^2 (44 f g+119 e h) d^3+c e f (184 f g+273 e h) d^2+5 c^2 f^2 (32 f g-105 e h) d+71 c^3 f^3 h\right ) a^3-b^3 d f \left (-e^3 (245 f g+108 e h) d^4+c e^2 f (603 f g+320 e h) d^3-3 c^2 e f^2 (109 f g+124 e h) d^2+c^3 f^3 (269 f g-168 e h) d+28 c^4 f^4 h\right ) a^2+b^4 \left (-4 e^4 (63 f g+8 e h) d^5+14 c e^3 f (37 f g+14 e h) d^4-c^2 e^2 f^2 (174 f g+331 e h) d^3-3 c^3 e f^3 (34 f g-47 e h) d^2+c^4 f^4 (160 f g-129 e h) d+5 c^5 f^5 h\right ) a+b^5 \left (-5 f^4 (7 f g-6 e h) c^5+3 d e f^3 (5 f g-6 e h) c^4+3 d^2 e^2 f^2 (7 f g-10 e h) c^3+d^3 e^3 f (37 f g+96 e h) c^2-4 d^4 e^4 (37 f g+12 e h) c+80 d^5 e^5 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {4 d^{7/2} (b e-a f)^4 \left (\left (-99 f^2 h c^3+11 d f (13 f g+12 e h) c^2-16 d^2 e (13 f g+3 e h) c+80 d^3 e^2 g\right ) b^2+2 a d \left (8 e (3 f g-2 e h) d^2-c f (39 f g-20 e h) d+11 c^2 f^2 h\right ) b+3 a^2 d^2 f (5 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 c-a d) \sqrt {d e-c f}}-\frac {2 b^{5/2} (d e-c f)^3 \left (231 d^3 f^3 h a^4-33 b d^2 f^2 (13 d f g+12 d e h+3 c f h) a^3+33 b^2 d f \left (2 e (13 f g+4 e h) d^2+c f (13 f g-6 e h) d+c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (39 f g+4 e h) d^3+36 c e f (13 f g-8 e h) d^2+3 c^2 f^2 (65 f g-48 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (5 f^2 (7 f g-6 e h) c^3+18 d e f (5 f g-4 e h) c^2+48 d^2 e^2 (3 f g-2 e h) c+160 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}}}{(b e-a f) (d e-c f)}\right )}{(b c-a d) (d e-c f)}}{(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)}\)

Input:

Int[(g + h*x)/((a + b*x)^4*(c + d*x)^3*(e + f*x)^(3/2)),x]
 

Output:

-1/3*(b*g - a*h)/((b*c - a*d)*(b*e - a*f)*(a + b*x)^3*(c + d*x)^2*Sqrt[e + 
 f*x]) - (-1/2*(11*a^2*d*f*h + b^2*(10*d*e*g + 7*c*f*g - 6*c*e*h) - a*b*(1 
7*d*f*g + 4*d*e*h + c*f*h))/((b*c - a*d)*(b*e - a*f)*(a + b*x)^2*(c + d*x) 
^2*Sqrt[e + f*x]) - ((99*a^3*d^2*f^2*h - 3*a^2*b*d*f*(59*d*f*g + 32*d*e*h 
+ 8*c*f*h) - b^3*(80*d^2*e^2*g + 2*c*d*e*(31*f*g - 24*e*h) + 5*c^2*f*(7*f* 
g - 6*e*h)) + a*b^2*(5*c^2*f^2*h + 2*c*d*f*(66*f*g - 47*e*h) + 2*d^2*e*(11 
1*f*g + 16*e*h)))/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x)^2*Sqrt[e + 
f*x]) - ((-2*d*(3*a^3*d^2*f^2*(4*d*f*g + 43*d*e*h - 47*c*f*h) - b^3*(120*d 
^3*e^3*g - c^2*d*e*f*(41*f*g - 30*e*h) - 5*c^3*f^2*(7*f*g - 6*e*h) - 8*c*d 
^2*e^2*(4*f*g + 9*e*h)) - a*b^2*(5*c^3*f^3*h + c^2*d*f^2*(146*f*g - 109*e* 
h) - 8*d^3*e^2*(41*f*g + 6*e*h) + 2*c*d^2*e*f*(73*f*g + 94*e*h)) + a^2*b*d 
*f*(26*c^2*f^2*h - d^2*e*(255*f*g + 142*e*h) + c*d*f*(219*f*g + 152*e*h))) 
)/((b*c - a*d)*(d*e - c*f)*(c + d*x)^2*Sqrt[e + f*x]) - ((2*d*(6*a^4*d^3*f 
^3*(5*d*f*g - 4*d*e*h - c*f*h) - b^4*(240*d^4*e^4*g - c^2*d^2*e^2*f*(f*g - 
 216*e*h) + 5*c^4*f^3*(7*f*g - 6*e*h) + 4*c^3*d*e*f^2*(5*f*g - 3*e*h) - 36 
*c*d^3*e^3*(9*f*g + 4*e*h)) + a^3*b*d^2*f^2*(269*c^2*f^2*h - 2*c*d*f*(63*f 
*g + 194*e*h) + d^2*e*(6*f*g + 239*e*h)) - a^2*b^2*d*f*(28*c^3*f^3*h + d^3 
*e^2*(467*f*g + 276*e*h) - 2*c*d^2*e*f*(458*f*g + 289*e*h) + c^2*d*f^2*(26 
9*f*g + 454*e*h)) + a*b^3*(5*c^4*f^4*h + 4*c^3*d*f^3*(40*f*g - 31*e*h) + 1 
2*d^4*e^3*(53*f*g + 8*e*h) - 2*c*d^3*e^2*f*(487*f*g + 234*e*h) + c^2*d^...
 

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 = 0.12 (sec) , antiderivative size = 2348, normalized size of antiderivative = 1.21

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

Input:

int((h*x+g)/(b*x+a)^4/(d*x+c)^3/(f*x+e)^(3/2),x)
 

Output:

2*f^5*(-b^3/(a*d-b*c)^6/f^5/(a*f-b*e)^4*(((-19/4*a^3*b^3*d^3*e*f^2*h+189/1 
6*a^2*b^4*c*d^2*f^3*g+3/2*a^2*b^4*d^3*e^2*f*h+69/8*a^2*b^4*d^3*e*f^2*g-99/ 
16*a*b^5*c^2*d*f^3*g-3*a*b^5*d^3*e^2*f*g-3/2*b^6*c^2*d*e^2*f*h+21/8*b^6*c^ 
2*d*e*f^2*g+3*b^6*c*d^2*e^2*f*g-99/16*b^3*a^3*c*d^2*f^3*h+33/16*b^4*a^2*c^ 
2*d*f^3*h+19/16*b^6*c^3*f^3*g+71/16*a^4*b^2*d^3*f^3*h-109/16*a^3*d^3*f^3*g 
*b^3-7/8*b^6*c^3*e*f^2*h-5/16*b^5*a*c^3*f^3*h+21/8*a^2*b^4*c*d^2*e*f^2*h+3 
*a*b^5*c^2*d*e*f^2*h-45/4*a*b^5*c*d^2*e*f^2*g)*(f*x+e)^(5/2)+1/6*b*f*(59*a 
^5*d^3*f^3*h-87*a^4*b*c*d^2*f^3*h-119*a^4*b*d^3*e*f^2*h-89*a^4*b*d^3*f^3*g 
+33*a^3*b^2*c^2*d*f^3*h+123*a^3*b^2*c*d^2*e*f^2*h+159*a^3*b^2*c*d^2*f^3*g+ 
78*a^3*b^2*d^3*e^2*f*h+197*a^3*b^2*d^3*e*f^2*g-5*a^2*b^3*c^3*f^3*h+3*a^2*b 
^3*c^2*d*e*f^2*h-87*a^2*b^3*c^2*d*f^3*g-36*a^2*b^3*c*d^2*e^2*f*h-303*a^2*b 
^3*c*d^2*e*f^2*g-18*a^2*b^3*d^3*e^3*h-144*a^2*b^3*d^3*e^2*f*g-7*a*b^4*c^3* 
e*f^2*h+17*a*b^4*c^3*f^3*g-54*a*b^4*c^2*d*e^2*f*h+123*a*b^4*c^2*d*e*f^2*g+ 
180*a*b^4*c*d^2*e^2*f*g+36*a*b^4*d^3*e^3*g+12*b^5*c^3*e^2*f*h-17*b^5*c^3*e 
*f^2*g+18*b^5*c^2*d*e^3*h-36*b^5*c^2*d*e^2*f*g-36*b^5*c*d^2*e^3*g)*(f*x+e) 
^(3/2)+(-141/16*a^5*b*c*d^2*f^5*h-131/8*a^5*b*d^3*e*f^4*h+63/16*a^4*b^2*c^ 
2*d*f^5*h+243/16*a^4*b^2*c*d^2*f^5*g+281/16*a^4*b^2*d^3*e^2*f^3*h+103/4*a^ 
4*b^2*d^3*e*f^4*g-141/16*a^3*b^3*c^2*d*f^5*g-479/16*a^3*b^3*d^3*e^2*f^3*g+ 
1/4*a^2*b^4*c^3*e*f^4*h+123/8*a^2*b^4*d^3*e^3*f^2*g+25/16*a*b^5*c^3*e^2*f^ 
3*h+29/16*b^6*c^3*e^2*f^3*g-131/16*a^5*b*d^3*f^5*g-11/16*a^3*b^3*c^3*f^...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)^4/(d*x+c)^3/(f*x+e)^(3/2),x, algorithm="fricas")
 

Output:

Timed out
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)**4/(d*x+c)**3/(f*x+e)**(3/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((h*x+g)/(b*x+a)^4/(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(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. 4418 vs. \(2 (1887) = 3774\).

Time = 1.00 (sec) , antiderivative size = 4418, normalized size of antiderivative = 2.28 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)/(b*x+a)^4/(d*x+c)^3/(f*x+e)^(3/2),x, algorithm="giac")
 

Output:

-1/8*(160*b^7*d^3*e^3*g + 144*b^7*c*d^2*e^2*f*g - 624*a*b^6*d^3*e^2*f*g + 
90*b^7*c^2*d*e*f^2*g - 468*a*b^6*c*d^2*e*f^2*g + 858*a^2*b^5*d^3*e*f^2*g + 
 35*b^7*c^3*f^3*g - 195*a*b^6*c^2*d*f^3*g + 429*a^2*b^5*c*d^2*f^3*g - 429* 
a^3*b^4*d^3*f^3*g - 96*b^7*c*d^2*e^3*h - 64*a*b^6*d^3*e^3*h - 72*b^7*c^2*d 
*e^2*f*h + 288*a*b^6*c*d^2*e^2*f*h + 264*a^2*b^5*d^3*e^2*f*h - 30*b^7*c^3* 
e*f^2*h + 144*a*b^6*c^2*d*e*f^2*h - 198*a^2*b^5*c*d^2*e*f^2*h - 396*a^3*b^ 
4*d^3*e*f^2*h - 5*a*b^6*c^3*f^3*h + 33*a^2*b^5*c^2*d*f^3*h - 99*a^3*b^4*c* 
d^2*f^3*h + 231*a^4*b^3*d^3*f^3*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a* 
b*f))/((b^10*c^6*e^4 - 6*a*b^9*c^5*d*e^4 + 15*a^2*b^8*c^4*d^2*e^4 - 20*a^3 
*b^7*c^3*d^3*e^4 + 15*a^4*b^6*c^2*d^4*e^4 - 6*a^5*b^5*c*d^5*e^4 + a^6*b^4* 
d^6*e^4 - 4*a*b^9*c^6*e^3*f + 24*a^2*b^8*c^5*d*e^3*f - 60*a^3*b^7*c^4*d^2* 
e^3*f + 80*a^4*b^6*c^3*d^3*e^3*f - 60*a^5*b^5*c^2*d^4*e^3*f + 24*a^6*b^4*c 
*d^5*e^3*f - 4*a^7*b^3*d^6*e^3*f + 6*a^2*b^8*c^6*e^2*f^2 - 36*a^3*b^7*c^5* 
d*e^2*f^2 + 90*a^4*b^6*c^4*d^2*e^2*f^2 - 120*a^5*b^5*c^3*d^3*e^2*f^2 + 90* 
a^6*b^4*c^2*d^4*e^2*f^2 - 36*a^7*b^3*c*d^5*e^2*f^2 + 6*a^8*b^2*d^6*e^2*f^2 
 - 4*a^3*b^7*c^6*e*f^3 + 24*a^4*b^6*c^5*d*e*f^3 - 60*a^5*b^5*c^4*d^2*e*f^3 
 + 80*a^6*b^4*c^3*d^3*e*f^3 - 60*a^7*b^3*c^2*d^4*e*f^3 + 24*a^8*b^2*c*d^5* 
e*f^3 - 4*a^9*b*d^6*e*f^3 + a^4*b^6*c^6*f^4 - 6*a^5*b^5*c^5*d*f^4 + 15*a^6 
*b^4*c^4*d^2*f^4 - 20*a^7*b^3*c^3*d^3*f^4 + 15*a^8*b^2*c^2*d^4*f^4 - 6*a^9 
*b*c*d^5*f^4 + a^10*d^6*f^4)*sqrt(-b^2*e + a*b*f)) + 1/4*(80*b^2*d^7*e^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Hanged} \] Input:

int((g + h*x)/((e + f*x)^(3/2)*(a + b*x)^4*(c + d*x)^3),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^3 (e+f x)^{3/2}} \, dx=\int \frac {h x +g}{\left (b x +a \right )^{4} \left (d x +c \right )^{3} \left (f x +e \right )^{\frac {3}{2}}}d x \] Input:

int((h*x+g)/(b*x+a)^4/(d*x+c)^3/(f*x+e)^(3/2),x)
 

Output:

int((h*x+g)/(b*x+a)^4/(d*x+c)^3/(f*x+e)^(3/2),x)