\(\int \frac {g+h x}{(a+b x)^4 (c+d x)^3 \sqrt {e+f x}} \, dx\) [148]

Optimal result

Integrand size = 29, antiderivative size = 1417 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^3 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Output:

1/24*d*(a^3*d^2*f^2*(-101*c*f*h+89*d*e*h+12*d*f*g)-b^3*(120*d^3*e^3*g-c^2* 
d*e*f*(-42*e*h+29*f*g)-3*c^3*f^2*(-6*e*h+5*f*g)-8*c*d^2*e^2*(9*e*h+8*f*g)) 
+a*b^2*(3*c^3*f^3*h-c^2*d*f^2*(-109*e*h+74*f*g)+8*d^3*e^2*(6*e*h+37*f*g)-2 
*c*d^2*e*f*(98*e*h+93*f*g))-a^2*b*d*f*(22*c^2*f^2*h+d^2*e*(122*e*h+203*f*g 
)-c*d*f*(180*e*h+167*f*g)))*(f*x+e)^(1/2)/(-a*d+b*c)^4/(-a*f+b*e)^3/(-c*f+ 
d*e)/(d*x+c)^2-1/3*(-a*h+b*g)*(f*x+e)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^ 
3/(d*x+c)^2+1/12*(9*a^2*d*f*h+b^2*(-6*c*e*h+5*c*f*g+10*d*e*g)-a*b*(-c*f*h+ 
4*d*e*h+15*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)^2+1/24*(63*a^3*d^2*f^2*h-a^2*b*d*f*(-20*c*f*h+80*d*e*h+129*d*f*g)-b^3*( 
80*d^2*e^2*g+2*c*d*e*(-24*e*h+17*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*(-49*e*h+32*f*g)-2*d^2*e*(16*e*h+97*f*g)))*(f*x+e)^(1/2)/(- 
a*d+b*c)^3/(-a*f+b*e)^3/(b*x+a)/(d*x+c)^2+1/8*d*(2*a^4*d^3*f^3*(c*f*h-4*d* 
e*h+3*d*f*g)+a^3*b*d^2*f^2*(71*c^2*f^2*h-2*c*d*f*(56*e*h+19*f*g)+d^2*e*(65 
*e*h+14*f*g))-b^4*(80*d^4*e^4*g+c^4*f^3*(-6*e*h+5*f*g)+4*c^3*d*e*f^2*(-3*e 
*h+2*f*g)-4*c*d^3*e^3*(12*e*h+29*f*g)+c^2*d^2*e^2*f*(72*e*h+17*f*g))+a^2*b 
^2*d*f*(8*c^3*f^3*h-d^3*e^2*(84*e*h+149*f*g)+2*c*d^2*e*f*(103*e*h+128*f*g) 
-c^2*d*f^2*(166*e*h+71*f*g))-a*b^3*(c^4*f^4*h-4*c^3*d*f^3*(-10*e*h+7*f*g)- 
4*d^4*e^3*(8*e*h+51*f*g)+2*c*d^3*e^2*f*(82*e*h+157*f*g)-c^2*d^2*e*f^2*(197 
*e*h+58*f*g)))*(f*x+e)^(1/2)/(-a*d+b*c)^5/(-a*f+b*e)^3/(-c*f+d*e)^2/(d*x+c 
)+1/8*b^(3/2)*(105*a^4*d^3*f^3*h-21*a^3*b*d^2*f^2*(-3*c*f*h+12*d*e*h+11...
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(9721\) vs. \(2(1417)=2834\).

Time = 16.34 (sec) , antiderivative size = 9721, normalized size of antiderivative = 6.86 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 3.65 (sec) , antiderivative size = 1526, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {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.

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

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\int \frac {b (10 d e g+5 c f g-6 c e h)-a (6 d f g+4 d e h-c f h)+9 d f (b g-a h) x}{2 (a+b x)^3 (c+d x)^3 \sqrt {e+f x}}dx}{3 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {b (10 d e g+5 c f g-6 c e h)-a (6 d f g+4 d e h-c f h)+9 d f (b g-a h) x}{(a+b x)^3 (c+d x)^3 \sqrt {e+f x}}dx}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\int \frac {d f (24 d f g+52 d e h-13 c f h) a^2+b \left (-4 e (31 f g+8 e h) d^2-c f (29 f g-56 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (3 f (5 f g-6 e h) c^2+2 d e (17 f g-24 e h) c+80 d^2 e^2 g\right )+7 d f \left (9 d f h a^2-b (15 d f g+4 d e h-c f h) a+b^2 (10 d e g+5 c f g-6 c e h)\right ) x}{2 (a+b x)^2 (c+d x)^3 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (9 a^2 d f h-a b (-c f h+4 d e h+15 d f g)+b^2 (-6 c e h+5 c f g+10 d e g)\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {d f (24 d f g+52 d e h-13 c f h) a^2+b \left (-4 e (31 f g+8 e h) d^2-c f (29 f g-56 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (3 f (5 f g-6 e h) c^2+2 d e (17 f g-24 e h) c+80 d^2 e^2 g\right )+7 d f \left (9 d f h a^2-b (15 d f g+4 d e h-c f h) a+b^2 (10 d e g+5 c f g-6 c e h)\right ) x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}}dx}{4 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (9 a^2 d f h-a b (-c f h+4 d e h+15 d f g)+b^2 (-6 c e h+5 c f g+10 d e g)\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\frac {\sqrt {e+f x} \left (63 a^3 d^2 f^2 h-a^2 b d f (-20 c f h+80 d e h+129 d f g)-a b^2 \left (3 c^2 f^2 h-2 c d f (32 f g-49 e h)-2 d^2 e (16 e h+97 f g)\right )-b^3 \left (3 c^2 f (5 f g-6 e h)+2 c d e (17 f g-24 e h)+80 d^2 e^2 g\right )\right )}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}-\frac {\int \frac {d^2 f^2 (89 c f h-4 d (12 f g+89 e h)) a^3-b d f \left (-4 e (203 f g+122 e h) d^2+c f (23 f g+320 e h) d+12 c^2 f^2 h\right ) a^2+b^2 \left (-32 e^2 (37 f g+6 e h) d^3-2 c e f (113 f g-312 e h) d^2-6 c^2 f^2 (4 f g-9 e h) d+3 c^3 f^3 h\right ) a+3 b^3 \left (f^2 (5 f g-6 e h) c^3+6 d e f (3 f g-4 e h) c^2+48 d^2 e^2 (f g-2 e h) c+160 d^3 e^3 g\right )-5 d f \left (63 d^2 f^2 h a^3-b d f (129 d f g+80 d e h-20 c f h) a^2-b^2 \left (-2 e (97 f g+16 e h) d^2-2 c f (32 f g-49 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (3 f (5 f g-6 e h) c^2+2 d e (17 f g-24 e h) c+80 d^2 e^2 g\right )\right ) x}{2 (a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (9 a^2 d f h-a b (-c f h+4 d e h+15 d f g)+b^2 (-6 c e h+5 c f g+10 d e g)\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\sqrt {e+f x} \left (63 a^3 d^2 f^2 h-a^2 b d f (-20 c f h+80 d e h+129 d f g)-a b^2 \left (3 c^2 f^2 h-2 c d f (32 f g-49 e h)-2 d^2 e (16 e h+97 f g)\right )-b^3 \left (3 c^2 f (5 f g-6 e h)+2 c d e (17 f g-24 e h)+80 d^2 e^2 g\right )\right )}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}-\frac {\int \frac {d^2 f^2 (89 c f h-4 d (12 f g+89 e h)) a^3-b d f \left (-4 e (203 f g+122 e h) d^2+c f (23 f g+320 e h) d+12 c^2 f^2 h\right ) a^2+b^2 \left (-32 e^2 (37 f g+6 e h) d^3-2 c e f (113 f g-312 e h) d^2-6 c^2 f^2 (4 f g-9 e h) d+3 c^3 f^3 h\right ) a+3 b^3 \left (f^2 (5 f g-6 e h) c^3+6 d e f (3 f g-4 e h) c^2+48 d^2 e^2 (f g-2 e h) c+160 d^3 e^3 g\right )-5 d f \left (63 d^2 f^2 h a^3-b d f (129 d f g+80 d e h-20 c f h) a^2-b^2 \left (-2 e (97 f g+16 e h) d^2-2 c f (32 f g-49 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (3 f (5 f g-6 e h) c^2+2 d e (17 f g-24 e h) c+80 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (9 a^2 d f h-a b (-c f h+4 d e h+15 d f g)+b^2 (-6 c e h+5 c f g+10 d e g)\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 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)^2}-\frac {-\frac {\sqrt {e+f x} \left (9 d f h a^2-b (15 d f g+4 d e h-c f h) a+b^2 (10 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)^2}-\frac {\frac {\left (63 d^2 f^2 h a^3-b d f (129 d f g+80 d e h-20 c f h) a^2-b^2 \left (-2 e (97 f g+16 e h) d^2-2 c f (32 f g-49 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 (17 f g-24 e h) c+80 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)^2}-\frac {\frac {\int -\frac {6 \left (4 d^3 f^3 (3 d f g-4 d e h+c f h) a^4+b d^2 f^2 \left (2 e (14 f g+65 e h) d^2-c f (64 f g+135 e h) d+41 c^2 f^2 h\right ) a^3-b^2 d f \left (2 e^2 (149 f g+84 e h) d^3-c e f (309 f g+290 e h) d^2-c^2 f^2 (25 f g-152 e h) d+6 c^3 f^3 h\right ) a^2+b^3 \left (8 e^3 (51 f g+8 e h) d^4-4 c e^2 f (83 f g+70 e h) d^3-2 c^2 e f^2 (35 f g-99 e h) d^2-c^3 f^3 (18 f g-29 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+6 d e f (3 f g-4 e h) c^2+48 d^2 e^2 (f g-2 e h) c+160 d^3 e^3 g\right )+b d f \left (d^2 f^2 (12 d f g+89 d e h-101 c f h) a^3-b d f \left (e (203 f g+122 e h) d^2-c f (167 f g+180 e h) d+22 c^2 f^2 h\right ) a^2+b^2 \left (8 e^2 (37 f g+6 e h) d^3-2 c e f (93 f g+98 e h) d^2-c^2 f^2 (74 f g-109 e h) d+3 c^3 f^3 h\right ) a-b^3 \left (-3 f^2 (5 f g-6 e h) c^3-d e f (29 f g-42 e h) c^2-8 d^2 e^2 (8 f g+9 e h) c+120 d^3 e^3 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)}-\frac {2 d \left (d^2 f^2 (12 d f g+89 d e h-101 c f h) a^3-b d f \left (e (203 f g+122 e h) d^2-c f (167 f g+180 e h) d+22 c^2 f^2 h\right ) a^2+b^2 \left (8 e^2 (37 f g+6 e h) d^3-2 c e f (93 f g+98 e h) d^2-c^2 f^2 (74 f g-109 e h) d+3 c^3 f^3 h\right ) a-b^3 \left (-3 f^2 (5 f g-6 e h) c^3-d e f (29 f g-42 e h) c^2-8 d^2 e^2 (8 f g+9 e h) c+120 d^3 e^3 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (d e-c f) (c+d x)^2}}{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 {\sqrt {e+f x} (b g-a h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (9 d f h a^2-b (15 d f g+4 d e h-c f h) a+b^2 (10 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)^2}-\frac {\frac {\left (63 d^2 f^2 h a^3-b d f (129 d f g+80 d e h-20 c f h) a^2-b^2 \left (-2 e (97 f g+16 e h) d^2-2 c f (32 f g-49 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 (17 f g-24 e h) c+80 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)^2}-\frac {-\frac {2 d \sqrt {e+f x} \left (d^2 f^2 (12 d f g+89 d e h-101 c f h) a^3-b d f \left (e (203 f g+122 e h) d^2-c f (167 f g+180 e h) d+22 c^2 f^2 h\right ) a^2+b^2 \left (8 e^2 (37 f g+6 e h) d^3-2 c e f (93 f g+98 e h) d^2-c^2 f^2 (74 f g-109 e h) d+3 c^3 f^3 h\right ) a-b^3 \left (-3 f^2 (5 f g-6 e h) c^3-d e f (29 f g-42 e h) c^2-8 d^2 e^2 (8 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}-\frac {3 \int \frac {4 d^3 f^3 (3 d f g-4 d e h+c f h) a^4+b d^2 f^2 \left (2 e (14 f g+65 e h) d^2-c f (64 f g+135 e h) d+41 c^2 f^2 h\right ) a^3-b^2 d f \left (2 e^2 (149 f g+84 e h) d^3-c e f (309 f g+290 e h) d^2-c^2 f^2 (25 f g-152 e h) d+6 c^3 f^3 h\right ) a^2+b^3 \left (8 e^3 (51 f g+8 e h) d^4-4 c e^2 f (83 f g+70 e h) d^3-2 c^2 e f^2 (35 f g-99 e h) d^2-c^3 f^3 (18 f g-29 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+6 d e f (3 f g-4 e h) c^2+48 d^2 e^2 (f g-2 e h) c+160 d^3 e^3 g\right )+b d f \left (d^2 f^2 (12 d f g+89 d e h-101 c f h) a^3-b d f \left (e (203 f g+122 e h) d^2-c f (167 f g+180 e h) d+22 c^2 f^2 h\right ) a^2+b^2 \left (8 e^2 (37 f g+6 e h) d^3-2 c e f (93 f g+98 e h) d^2-c^2 f^2 (74 f g-109 e h) d+3 c^3 f^3 h\right ) a-b^3 \left (-3 f^2 (5 f g-6 e h) c^3-d e f (29 f g-42 e h) c^2-8 d^2 e^2 (8 f g+9 e h) c+120 d^3 e^3 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)}\)

\(\Big \downarrow \) 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)^2}-\frac {-\frac {\sqrt {e+f x} \left (9 d f h a^2-b (15 d f g+4 d e h-c f h) a+b^2 (10 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)^2}-\frac {\frac {\left (63 d^2 f^2 h a^3-b d f (129 d f g+80 d e h-20 c f h) a^2-b^2 \left (-2 e (97 f g+16 e h) d^2-2 c f (32 f g-49 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 (17 f g-24 e h) c+80 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)^2}-\frac {-\frac {2 d \sqrt {e+f x} \left (d^2 f^2 (12 d f g+89 d e h-101 c f h) a^3-b d f \left (e (203 f g+122 e h) d^2-c f (167 f g+180 e h) d+22 c^2 f^2 h\right ) a^2+b^2 \left (8 e^2 (37 f g+6 e h) d^3-2 c e f (93 f g+98 e h) d^2-c^2 f^2 (74 f g-109 e h) d+3 c^3 f^3 h\right ) a-b^3 \left (-3 f^2 (5 f g-6 e h) c^3-d e f (29 f g-42 e h) c^2-8 d^2 e^2 (8 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}-\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (2 d^3 f^3 (3 d f g-4 d e h+c f h) a^4+b d^2 f^2 \left (e (14 f g+65 e h) d^2-2 c f (19 f g+56 e h) d+71 c^2 f^2 h\right ) a^3+b^2 d f \left (-e^2 (149 f g+84 e h) d^3+2 c e f (128 f g+103 e h) d^2-c^2 f^2 (71 f g+166 e h) d+8 c^3 f^3 h\right ) a^2-b^3 \left (-4 e^3 (51 f g+8 e h) d^4+2 c e^2 f (157 f g+82 e h) d^3-c^2 e f^2 (58 f g+197 e h) d^2-4 c^3 f^3 (7 f g-10 e h) d+c^4 f^4 h\right ) a-b^4 \left (f^3 (5 f g-6 e h) c^4+4 d e f^2 (2 f g-3 e h) c^3+d^2 e^2 f (17 f g+72 e h) c^2-4 d^3 e^3 (29 f g+12 e h) c+80 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\int \frac {2 d^4 f^4 (3 d f g-4 d e h+c f h) a^5-2 b d^3 f^3 \left (-e (7 f g-20 e h) d^2+c f (19 f g-49 e h) d+17 c^2 f^2 h\right ) a^4-b^2 d^2 f^2 \left (-2 e^2 (41 f g+84 e h) d^3+c e f (206 f g+361 e h) d^2-4 c^2 f^2 (40 f g+53 e h) d+55 c^3 f^3 h\right ) a^3+b^3 d f \left (-2 e^3 (195 f g+92 e h) d^4+c e^2 f (775 f g+538 e h) d^3-2 c^2 e f^2 (169 f g+275 e h) d^2-c^3 f^3 (71 f g-212 e h) d+8 c^4 f^4 h\right ) a^2-b^4 \left (-64 e^4 (7 f g+e h) d^5+8 c e^3 f (101 f g+46 e h) d^4-40 c^2 e^2 f^2 (7 f g+13 e h) d^3-c^3 e f^3 (58 f g-181 e h) d^2-4 c^4 f^4 (7 f g-10 e h) d+c^5 f^5 h\right ) a-b^5 (d e-c f)^2 \left (f^2 (5 f g-6 e h) c^3+6 d e f (3 f g-4 e h) c^2+48 d^2 e^2 (f g-2 e h) c+160 d^3 e^3 g\right )+b d f \left (2 d^3 f^3 (3 d f g-4 d e h+c f h) a^4+b d^2 f^2 \left (e (14 f g+65 e h) d^2-2 c f (19 f g+56 e h) d+71 c^2 f^2 h\right ) a^3+b^2 d f \left (-e^2 (149 f g+84 e h) d^3+2 c e f (128 f g+103 e h) d^2-c^2 f^2 (71 f g+166 e h) d+8 c^3 f^3 h\right ) a^2-b^3 \left (-4 e^3 (51 f g+8 e h) d^4+2 c e^2 f (157 f g+82 e h) d^3-c^2 e f^2 (58 f g+197 e h) d^2-4 c^3 f^3 (7 f g-10 e h) d+c^4 f^4 h\right ) a-b^4 \left (f^3 (5 f g-6 e h) c^4+4 d e f^2 (2 f g-3 e h) c^3+d^2 e^2 f (17 f g+72 e h) c^2-4 d^3 e^3 (29 f g+12 e h) c+80 d^4 e^4 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)}\)

\(\Big \downarrow \) 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)^2}-\frac {-\frac {\sqrt {e+f x} \left (9 d f h a^2-b (15 d f g+4 d e h-c f h) a+b^2 (10 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)^2}-\frac {\frac {\left (63 d^2 f^2 h a^3-b d f (129 d f g+80 d e h-20 c f h) a^2-b^2 \left (-2 e (97 f g+16 e h) d^2-2 c f (32 f g-49 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 (17 f g-24 e h) c+80 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)^2}-\frac {-\frac {2 d \sqrt {e+f x} \left (d^2 f^2 (12 d f g+89 d e h-101 c f h) a^3-b d f \left (e (203 f g+122 e h) d^2-c f (167 f g+180 e h) d+22 c^2 f^2 h\right ) a^2+b^2 \left (8 e^2 (37 f g+6 e h) d^3-2 c e f (93 f g+98 e h) d^2-c^2 f^2 (74 f g-109 e h) d+3 c^3 f^3 h\right ) a-b^3 \left (-3 f^2 (5 f g-6 e h) c^3-d e f (29 f g-42 e h) c^2-8 d^2 e^2 (8 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}-\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (2 d^3 f^3 (3 d f g-4 d e h+c f h) a^4+b d^2 f^2 \left (e (14 f g+65 e h) d^2-2 c f (19 f g+56 e h) d+71 c^2 f^2 h\right ) a^3+b^2 d f \left (-e^2 (149 f g+84 e h) d^3+2 c e f (128 f g+103 e h) d^2-c^2 f^2 (71 f g+166 e h) d+8 c^3 f^3 h\right ) a^2-b^3 \left (-4 e^3 (51 f g+8 e h) d^4+2 c e^2 f (157 f g+82 e h) d^3-c^2 e f^2 (58 f g+197 e h) d^2-4 c^3 f^3 (7 f g-10 e h) d+c^4 f^4 h\right ) a-b^4 \left (f^3 (5 f g-6 e h) c^4+4 d e f^2 (2 f g-3 e h) c^3+d^2 e^2 f (17 f g+72 e h) c^2-4 d^3 e^3 (29 f g+12 e h) c+80 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\frac {2 d^3 (b e-a f)^3 \left (\left (-63 f^2 h c^3+9 d f (11 f g+12 e h) c^2-16 d^2 e (11 f g+3 e h) c+80 d^3 e^2 g\right ) b^2-2 a d \left (-8 e (f g-2 e h) d^2+c f (11 f g-28 e h) d+9 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 c-a d}-\frac {b^2 (d e-c f)^2 \left (105 d^3 f^3 h a^4-21 b d^2 f^2 (11 d f g+12 d e h-3 c f h) a^3-9 b^2 d f \left (-6 e (11 f g+4 e h) d^2-c f (11 f g-30 e h) d+c^2 f^2 h\right ) a^2+b^3 \left (-16 e^2 (33 f g+4 e h) d^3-12 c e f (11 f g-24 e h) d^2-3 c^2 f^2 (11 f g-16 e h) d+c^3 f^3 h\right ) a+b^4 \left (f^2 (5 f g-6 e h) c^3+6 d e f (3 f g-4 e h) c^2+48 d^2 e^2 (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}}{(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)}\)

\(\Big \downarrow \) 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)^2}-\frac {-\frac {\sqrt {e+f x} \left (9 d f h a^2-b (15 d f g+4 d e h-c f h) a+b^2 (10 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)^2}-\frac {\frac {\left (63 d^2 f^2 h a^3-b d f (129 d f g+80 d e h-20 c f h) a^2-b^2 \left (-2 e (97 f g+16 e h) d^2-2 c f (32 f g-49 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 (17 f g-24 e h) c+80 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)^2}-\frac {-\frac {2 d \sqrt {e+f x} \left (d^2 f^2 (12 d f g+89 d e h-101 c f h) a^3-b d f \left (e (203 f g+122 e h) d^2-c f (167 f g+180 e h) d+22 c^2 f^2 h\right ) a^2+b^2 \left (8 e^2 (37 f g+6 e h) d^3-2 c e f (93 f g+98 e h) d^2-c^2 f^2 (74 f g-109 e h) d+3 c^3 f^3 h\right ) a-b^3 \left (-3 f^2 (5 f g-6 e h) c^3-d e f (29 f g-42 e h) c^2-8 d^2 e^2 (8 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}-\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (2 d^3 f^3 (3 d f g-4 d e h+c f h) a^4+b d^2 f^2 \left (e (14 f g+65 e h) d^2-2 c f (19 f g+56 e h) d+71 c^2 f^2 h\right ) a^3+b^2 d f \left (-e^2 (149 f g+84 e h) d^3+2 c e f (128 f g+103 e h) d^2-c^2 f^2 (71 f g+166 e h) d+8 c^3 f^3 h\right ) a^2-b^3 \left (-4 e^3 (51 f g+8 e h) d^4+2 c e^2 f (157 f g+82 e h) d^3-c^2 e f^2 (58 f g+197 e h) d^2-4 c^3 f^3 (7 f g-10 e h) d+c^4 f^4 h\right ) a-b^4 \left (f^3 (5 f g-6 e h) c^4+4 d e f^2 (2 f g-3 e h) c^3+d^2 e^2 f (17 f g+72 e h) c^2-4 d^3 e^3 (29 f g+12 e h) c+80 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\frac {4 d^3 (b e-a f)^3 \left (\left (-63 f^2 h c^3+9 d f (11 f g+12 e h) c^2-16 d^2 e (11 f g+3 e h) c+80 d^3 e^2 g\right ) b^2-2 a d \left (-8 e (f g-2 e h) d^2+c f (11 f g-28 e h) d+9 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 c-a d) f}-\frac {2 b^2 (d e-c f)^2 \left (105 d^3 f^3 h a^4-21 b d^2 f^2 (11 d f g+12 d e h-3 c f h) a^3-9 b^2 d f \left (-6 e (11 f g+4 e h) d^2-c f (11 f g-30 e h) d+c^2 f^2 h\right ) a^2+b^3 \left (-16 e^2 (33 f g+4 e h) d^3-12 c e f (11 f g-24 e h) d^2-3 c^2 f^2 (11 f g-16 e h) d+c^3 f^3 h\right ) a+b^4 \left (f^2 (5 f g-6 e h) c^3+6 d e f (3 f g-4 e h) c^2+48 d^2 e^2 (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}}{(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)}\)

\(\Big \downarrow \) 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)^2}-\frac {-\frac {\sqrt {e+f x} \left (9 d f h a^2-b (15 d f g+4 d e h-c f h) a+b^2 (10 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)^2}-\frac {\frac {\left (63 d^2 f^2 h a^3-b d f (129 d f g+80 d e h-20 c f h) a^2-b^2 \left (-2 e (97 f g+16 e h) d^2-2 c f (32 f g-49 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 (17 f g-24 e h) c+80 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)^2}-\frac {-\frac {2 d \sqrt {e+f x} \left (d^2 f^2 (12 d f g+89 d e h-101 c f h) a^3-b d f \left (e (203 f g+122 e h) d^2-c f (167 f g+180 e h) d+22 c^2 f^2 h\right ) a^2+b^2 \left (8 e^2 (37 f g+6 e h) d^3-2 c e f (93 f g+98 e h) d^2-c^2 f^2 (74 f g-109 e h) d+3 c^3 f^3 h\right ) a-b^3 \left (-3 f^2 (5 f g-6 e h) c^3-d e f (29 f g-42 e h) c^2-8 d^2 e^2 (8 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}-\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (2 d^3 f^3 (3 d f g-4 d e h+c f h) a^4+b d^2 f^2 \left (e (14 f g+65 e h) d^2-2 c f (19 f g+56 e h) d+71 c^2 f^2 h\right ) a^3+b^2 d f \left (-e^2 (149 f g+84 e h) d^3+2 c e f (128 f g+103 e h) d^2-c^2 f^2 (71 f g+166 e h) d+8 c^3 f^3 h\right ) a^2-b^3 \left (-4 e^3 (51 f g+8 e h) d^4+2 c e^2 f (157 f g+82 e h) d^3-c^2 e f^2 (58 f g+197 e h) d^2-4 c^3 f^3 (7 f g-10 e h) d+c^4 f^4 h\right ) a-b^4 \left (f^3 (5 f g-6 e h) c^4+4 d e f^2 (2 f g-3 e h) c^3+d^2 e^2 f (17 f g+72 e h) c^2-4 d^3 e^3 (29 f g+12 e h) c+80 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\frac {2 b^{3/2} (d e-c f)^2 \left (105 d^3 f^3 h a^4-21 b d^2 f^2 (11 d f g+12 d e h-3 c f h) a^3-9 b^2 d f \left (-6 e (11 f g+4 e h) d^2-c f (11 f g-30 e h) d+c^2 f^2 h\right ) a^2+b^3 \left (-16 e^2 (33 f g+4 e h) d^3-12 c e f (11 f g-24 e h) d^2-3 c^2 f^2 (11 f g-16 e h) d+c^3 f^3 h\right ) a+b^4 \left (f^2 (5 f g-6 e h) c^3+6 d e f (3 f g-4 e h) c^2+48 d^2 e^2 (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}}-\frac {4 d^{5/2} (b e-a f)^3 \left (\left (-63 f^2 h c^3+9 d f (11 f g+12 e h) c^2-16 d^2 e (11 f g+3 e h) c+80 d^3 e^2 g\right ) b^2-2 a d \left (-8 e (f g-2 e h) d^2+c f (11 f g-28 e h) d+9 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 c-a d) \sqrt {d e-c 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)}\)

Input:

Int[(g + h*x)/((a + b*x)^4*(c + d*x)^3*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)^2) - (-1/2*((9*a^2*d*f*h + b^2*(10*d*e*g + 5*c*f*g - 6*c*e*h) - a*b* 
(15*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)^2*(c + d*x)^2) - (((63*a^3*d^2*f^2*h - a^2*b*d*f*(129*d*f*g + 80*d*e 
*h - 20*c*f*h) - b^3*(80*d^2*e^2*g + 2*c*d*e*(17*f*g - 24*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*(32*f*g - 49*e*h) - 2*d^2*e 
*(97*f*g + 16*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c 
+ d*x)^2) - ((-2*d*(a^3*d^2*f^2*(12*d*f*g + 89*d*e*h - 101*c*f*h) - b^3*(1 
20*d^3*e^3*g - c^2*d*e*f*(29*f*g - 42*e*h) - 3*c^3*f^2*(5*f*g - 6*e*h) - 8 
*c*d^2*e^2*(8*f*g + 9*e*h)) + a*b^2*(3*c^3*f^3*h - c^2*d*f^2*(74*f*g - 109 
*e*h) + 8*d^3*e^2*(37*f*g + 6*e*h) - 2*c*d^2*e*f*(93*f*g + 98*e*h)) - a^2* 
b*d*f*(22*c^2*f^2*h + d^2*e*(203*f*g + 122*e*h) - c*d*f*(167*f*g + 180*e*h 
)))*Sqrt[e + f*x])/((b*c - a*d)*(d*e - c*f)*(c + d*x)^2) - (3*((2*d*(2*a^4 
*d^3*f^3*(3*d*f*g - 4*d*e*h + c*f*h) + a^3*b*d^2*f^2*(71*c^2*f^2*h - 2*c*d 
*f*(19*f*g + 56*e*h) + d^2*e*(14*f*g + 65*e*h)) - b^4*(80*d^4*e^4*g + c^4* 
f^3*(5*f*g - 6*e*h) + 4*c^3*d*e*f^2*(2*f*g - 3*e*h) - 4*c*d^3*e^3*(29*f*g 
+ 12*e*h) + c^2*d^2*e^2*f*(17*f*g + 72*e*h)) + a^2*b^2*d*f*(8*c^3*f^3*h - 
d^3*e^2*(149*f*g + 84*e*h) + 2*c*d^2*e*f*(128*f*g + 103*e*h) - c^2*d*f^2*( 
71*f*g + 166*e*h)) - a*b^3*(c^4*f^4*h - 4*c^3*d*f^3*(7*f*g - 10*e*h) - 4*d 
^4*e^3*(51*f*g + 8*e*h) + 2*c*d^3*e^2*f*(157*f*g + 82*e*h) - c^2*d^2*e*...
 

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 = 293.42 (sec) , antiderivative size = 1854, normalized size of antiderivative = 1.31

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1854\)
default	\(\text {Expression too large to display}\)	\(1854\)
pseudoelliptic	\(\text {Expression too large to display}\)	\(2594\)

Input:

int((h*x+g)/(b*x+a)^4/(d*x+c)^3/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2*f^5*(b^2/f^5/(a*d-b*c)^6*((1/16*b^2*f*(41*a^4*d^3*f^2*h-33*a^3*b*c*d^2*f 
^2*h-60*a^3*b*d^3*e*f*h-71*a^3*b*d^3*f^2*g-9*a^2*b^2*c^2*d*f^2*h+18*a^2*b^ 
2*c*d^2*e*f*h+99*a^2*b^2*c*d^2*f^2*g+24*a^2*b^2*d^3*e^2*h+114*a^2*b^2*d^3* 
e*f*g+a*b^3*c^3*f^2*h+48*a*b^3*c^2*d*e*f*h-33*a*b^3*c^2*d*f^2*g-132*a*b^3* 
c*d^2*e*f*g-48*a*b^3*d^3*e^2*g-6*b^4*c^3*e*f*h+5*b^4*c^3*f^2*g-24*b^4*c^2* 
d*e^2*h+18*b^4*c^2*d*e*f*g+48*b^4*c*d^2*e^2*g)/(a^3*f^3-3*a^2*b*e*f^2+3*a* 
b^2*e^2*f-b^3*e^3)*(f*x+e)^(5/2)+1/6*(35*a^4*d^3*f^2*h-33*a^3*b*c*d^2*f^2* 
h-48*a^3*b*d^3*e*f*h-59*a^3*b*d^3*f^2*g-3*a^2*b^2*c^2*d*f^2*h+18*a^2*b^2*c 
*d^2*e*f*h+87*a^2*b^2*c*d^2*f^2*g+18*a^2*b^2*d^3*e^2*h+90*a^2*b^2*d^3*e*f* 
g+a*b^3*c^3*f^2*h+36*a*b^3*c^2*d*e*f*h-33*a*b^3*c^2*d*f^2*g-108*a*b^3*c*d^ 
2*e*f*g-36*a*b^3*d^3*e^2*g-6*b^4*c^3*e*f*h+5*b^4*c^3*f^2*g-18*b^4*c^2*d*e^ 
2*h+18*b^4*c^2*d*e*f*g+36*b^4*c*d^2*e^2*g)*b*f/(a^2*f^2-2*a*b*e*f+b^2*e^2) 
*(f*x+e)^(3/2)+1/16*(55*a^4*d^3*f^2*h-63*a^3*b*c*d^2*f^2*h-68*a^3*b*d^3*e* 
f*h-89*a^3*b*d^3*f^2*g+9*a^2*b^2*c^2*d*f^2*h+30*a^2*b^2*c*d^2*e*f*h+141*a^ 
2*b^2*c*d^2*f^2*g+24*a^2*b^2*d^3*e^2*h+126*a^2*b^2*d^3*e*f*g-a*b^3*c^3*f^2 
*h+48*a*b^3*c^2*d*e*f*h-63*a*b^3*c^2*d*f^2*g-156*a*b^3*c*d^2*e*f*g-48*a*b^ 
3*d^3*e^2*g-10*b^4*c^3*e*f*h+11*b^4*c^3*f^2*g-24*b^4*c^2*d*e^2*h+30*b^4*c^ 
2*d*e*f*g+48*b^4*c*d^2*e^2*g)*f/(a*f-b*e)*(f*x+e)^(1/2))/((f*x+e)*b+a*f-b* 
e)^3+1/16*(105*a^4*d^3*f^3*h+63*a^3*b*c*d^2*f^3*h-252*a^3*b*d^3*e*f^2*h-23 
1*a^3*b*d^3*f^3*g-9*a^2*b^2*c^2*d*f^3*h-270*a^2*b^2*c*d^2*e*f^2*h+99*a^...
 

Fricas [F(-1)]

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

integrate((h*x+g)/(b*x+a)^4/(d*x+c)^3/(f*x+e)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((h*x+g)/(b*x+a)**4/(d*x+c)**3/(f*x+e)**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((h*x+g)/(b*x+a)^4/(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. 3617 vs. \(2 (1373) = 2746\).

Time = 0.40 (sec) , antiderivative size = 3617, normalized size of antiderivative = 2.55 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)/(b*x+a)^4/(d*x+c)^3/(f*x+e)^(1/2),x, algorithm="giac")
 

Output:

-1/8*(160*b^6*d^3*e^3*g + 48*b^6*c*d^2*e^2*f*g - 528*a*b^5*d^3*e^2*f*g + 1 
8*b^6*c^2*d*e*f^2*g - 132*a*b^5*c*d^2*e*f^2*g + 594*a^2*b^4*d^3*e*f^2*g + 
5*b^6*c^3*f^3*g - 33*a*b^5*c^2*d*f^3*g + 99*a^2*b^4*c*d^2*f^3*g - 231*a^3* 
b^3*d^3*f^3*g - 96*b^6*c*d^2*e^3*h - 64*a*b^5*d^3*e^3*h - 24*b^6*c^2*d*e^2 
*f*h + 288*a*b^5*c*d^2*e^2*f*h + 216*a^2*b^4*d^3*e^2*f*h - 6*b^6*c^3*e*f^2 
*h + 48*a*b^5*c^2*d*e*f^2*h - 270*a^2*b^4*c*d^2*e*f^2*h - 252*a^3*b^3*d^3* 
e*f^2*h + a*b^5*c^3*f^3*h - 9*a^2*b^4*c^2*d*f^3*h + 63*a^3*b^3*c*d^2*f^3*h 
 + 105*a^4*b^2*d^3*f^3*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b 
^9*c^6*e^3 - 6*a*b^8*c^5*d*e^3 + 15*a^2*b^7*c^4*d^2*e^3 - 20*a^3*b^6*c^3*d 
^3*e^3 + 15*a^4*b^5*c^2*d^4*e^3 - 6*a^5*b^4*c*d^5*e^3 + a^6*b^3*d^6*e^3 - 
3*a*b^8*c^6*e^2*f + 18*a^2*b^7*c^5*d*e^2*f - 45*a^3*b^6*c^4*d^2*e^2*f + 60 
*a^4*b^5*c^3*d^3*e^2*f - 45*a^5*b^4*c^2*d^4*e^2*f + 18*a^6*b^3*c*d^5*e^2*f 
 - 3*a^7*b^2*d^6*e^2*f + 3*a^2*b^7*c^6*e*f^2 - 18*a^3*b^6*c^5*d*e*f^2 + 45 
*a^4*b^5*c^4*d^2*e*f^2 - 60*a^5*b^4*c^3*d^3*e*f^2 + 45*a^6*b^3*c^2*d^4*e*f 
^2 - 18*a^7*b^2*c*d^5*e*f^2 + 3*a^8*b*d^6*e*f^2 - a^3*b^6*c^6*f^3 + 6*a^4* 
b^5*c^5*d*f^3 - 15*a^5*b^4*c^4*d^2*f^3 + 20*a^6*b^3*c^3*d^3*f^3 - 15*a^7*b 
^2*c^2*d^4*f^3 + 6*a^8*b*c*d^5*f^3 - a^9*d^6*f^3)*sqrt(-b^2*e + a*b*f)) + 
1/4*(80*b^2*d^6*e^2*g - 176*b^2*c*d^5*e*f*g + 16*a*b*d^6*e*f*g + 99*b^2*c^ 
2*d^4*f^2*g - 22*a*b*c*d^5*f^2*g + 3*a^2*d^6*f^2*g - 48*b^2*c*d^5*e^2*h - 
32*a*b*d^6*e^2*h + 108*b^2*c^2*d^4*e*f*h + 56*a*b*c*d^5*e*f*h - 4*a^2*d...
 

Mupad [B] (verification not implemented)

Time = 56.15 (sec) , antiderivative size = 1590452, normalized size of antiderivative = 1122.41 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

int((g + h*x)/((e + f*x)^(1/2)*(a + b*x)^4*(c + d*x)^3),x)
 

Output:

- atan(((((11264*a^4*b^18*c^18*d^3*f^13*g - 640*a^3*b^19*c^19*d^2*f^13*g - 
 94336*a^5*b^17*c^17*d^4*f^13*g + 506880*a^6*b^16*c^16*d^5*f^13*g - 195571 
2*a^7*b^15*c^15*d^6*f^13*g + 5691136*a^8*b^14*c^14*d^7*f^13*g - 12764288*a 
^9*b^13*c^13*d^8*f^13*g + 22274560*a^10*b^12*c^12*d^9*f^13*g - 30349440*a^ 
11*b^11*c^11*d^10*f^13*g + 32268544*a^12*b^10*c^10*d^11*f^13*g - 26652032* 
a^13*b^9*c^9*d^12*f^13*g + 16943104*a^14*b^8*c^8*d^13*f^13*g - 8164992*a^1 
5*b^7*c^7*d^14*f^13*g + 2914560*a^16*b^6*c^6*d^15*f^13*g - 744832*a^17*b^5 
*c^5*d^16*f^13*g + 129536*a^18*b^4*c^4*d^17*f^13*g - 14080*a^19*b^3*c^3*d^ 
18*f^13*g + 768*a^20*b^2*c^2*d^19*f^13*g - 128*a^4*b^18*c^19*d^2*f^13*h + 
2560*a^5*b^17*c^18*d^3*f^13*h - 27776*a^6*b^16*c^17*d^4*f^13*h + 175872*a^ 
7*b^15*c^16*d^5*f^13*h - 700800*a^8*b^14*c^15*d^6*f^13*h + 1866752*a^9*b^1 
3*c^14*d^7*f^13*h - 3439744*a^10*b^12*c^13*d^8*f^13*h + 4412672*a^11*b^11* 
c^12*d^9*f^13*h - 3805824*a^12*b^10*c^11*d^10*f^13*h + 1886720*a^13*b^9*c^ 
10*d^11*f^13*h - 35200*a^14*b^8*c^9*d^12*f^13*h - 739072*a^15*b^7*c^8*d^13 
*f^13*h + 607104*a^16*b^6*c^7*d^14*f^13*h - 258048*a^17*b^5*c^6*d^15*f^13* 
h + 62080*a^18*b^4*c^5*d^16*f^13*h - 7424*a^19*b^3*c^4*d^17*f^13*h + 256*a 
^20*b^2*c^3*d^18*f^13*h + 10240*a^12*b^10*d^21*e^10*f^3*g - 61952*a^13*b^9 
*d^21*e^9*f^4*g + 155264*a^14*b^8*d^21*e^8*f^5*g - 205184*a^15*b^7*d^21*e^ 
7*f^6*g + 148864*a^16*b^6*d^21*e^6*f^7*g - 54656*a^17*b^5*d^21*e^5*f^8*g + 
 7936*a^18*b^4*d^21*e^4*f^9*g - 1280*a^19*b^3*d^21*e^3*f^10*g + 768*a^2...
 

Reduce [B] (verification not implemented)

Time = 1.74 (sec) , antiderivative size = 60838, normalized size of antiderivative = 42.93 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^3 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

int((h*x+g)/(b*x+a)^4/(d*x+c)^3/(f*x+e)^(1/2),x)
 

Output:

(315*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b* 
e)))*a**7*b*c**5*d**3*f**6*h - 945*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + 
f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*b*c**4*d**4*e*f**5*h + 630*sqrt(b) 
*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*b* 
c**4*d**4*f**6*h*x + 945*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(s 
qrt(b)*sqrt(a*f - b*e)))*a**7*b*c**3*d**5*e**2*f**4*h - 1890*sqrt(b)*sqrt( 
a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*b*c**3*d 
**5*e*f**5*h*x + 315*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt( 
b)*sqrt(a*f - b*e)))*a**7*b*c**3*d**5*f**6*h*x**2 - 315*sqrt(b)*sqrt(a*f - 
 b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*b*c**2*d**6*e 
**3*f**3*h + 1890*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)* 
sqrt(a*f - b*e)))*a**7*b*c**2*d**6*e**2*f**4*h*x - 945*sqrt(b)*sqrt(a*f - 
b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*b*c**2*d**6*e* 
f**5*h*x**2 - 630*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)* 
sqrt(a*f - b*e)))*a**7*b*c*d**7*e**3*f**3*h*x + 945*sqrt(b)*sqrt(a*f - b*e 
)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*b*c*d**7*e**2*f** 
4*h*x**2 - 315*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqr 
t(a*f - b*e)))*a**7*b*d**8*e**3*f**3*h*x**2 + 189*sqrt(b)*sqrt(a*f - b*e)* 
atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b**2*c**6*d**2*f**6 
*h - 1323*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(...