\(\int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^4 (c+d x)^3} \, dx\) [58]

Optimal result

Integrand size = 29, antiderivative size = 1204 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^4 (c+d x)^3} \, dx =\text {Too large to display} \] Output:

1/24*d*(35*a^3*d^2*f^2*h-a^2*b*d*f*(-82*c*f*h+86*d*e*h+101*d*f*g)-b^3*(120 
*d^2*e^2*g-3*c^2*f*(-2*e*h+f*g)-8*c*d*e*(9*e*h+2*f*g))+a*b^2*(3*c^2*f^2*h+ 
16*d^2*e*(3*e*h+14*f*g)-2*c*d*f*(74*e*h+11*f*g)))*(f*x+e)^(1/2)/b/(-a*d+b* 
c)^4/(-a*f+b*e)^2/(d*x+c)^2-1/3*(-a*h+b*g)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(b*x 
+a)^3/(d*x+c)^2+1/12*(3*a^2*d*f*h-a*b*(-7*c*f*h+4*d*e*h+9*d*f*g)+b^2*(10*d 
*e*g-c*(6*e*h+f*g)))*(f*x+e)^(1/2)/b/(-a*d+b*c)^2/(-a*f+b*e)/(b*x+a)^2/(d* 
x+c)^2+1/24*(21*a^3*d^2*f^2*h-7*a^2*b*d*f*(-8*c*f*h+8*d*e*h+9*d*f*g)-b^3*( 
80*d^2*e^2*g-3*c^2*f*(-2*e*h+f*g)-2*c*d*e*(24*e*h+7*f*g))+a*b^2*(3*c^2*f^2 
*h+2*d^2*e*(16*e*h+73*f*g)-2*c*d*f*(49*e*h+10*f*g)))*(f*x+e)^(1/2)/b/(-a*d 
+b*c)^3/(-a*f+b*e)^2/(b*x+a)/(d*x+c)^2+1/8*d*(a^3*d^2*f^2*(-29*c*f*h+27*d* 
e*h+2*d*f*g)-b^3*(80*d^3*e^3*g+c^3*f^2*(-2*e*h+f*g)-12*c*d^2*e^2*(4*e*h+7* 
f*g)+c^2*d*e*f*(48*e*h+5*f*g))-a*b^2*(c^3*f^3*h-4*d^3*e^2*(8*e*h+39*f*g)+2 
*c*d^2*e*f*(66*e*h+79*f*g)-c^2*d*f^2*(95*e*h+8*f*g))-a^2*b*d*f*(50*c^2*f^2 
*h+d^2*e*(60*e*h+77*f*g)-c*d*f*(116*e*h+71*f*g)))*(f*x+e)^(1/2)/(-a*d+b*c) 
^5/(-a*f+b*e)^2/(-c*f+d*e)/(d*x+c)+1/8*b^(1/2)*(35*a^4*d^3*f^3*h-35*a^3*b* 
d^2*f^2*(-3*c*f*h+4*d*e*h+3*d*f*g)+b^4*(160*d^3*e^3*g-6*c^2*d*e*f*(-4*e*h+ 
f*g)-c^3*f^2*(-2*e*h+f*g)-48*c*d^2*e^2*(2*e*h+f*g))-a*b^3*(c^3*f^3*h-3*c^2 
*d*f^2*(-16*e*h+3*f*g)+16*d^3*e^2*(4*e*h+27*f*g)-36*c*d^2*e*f*(8*e*h+3*f*g 
))+21*a^2*b^2*d*f*(c^2*f^2*h+2*d^2*e*(4*e*h+9*f*g)-c*d*f*(14*e*h+3*f*g)))* 
arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/(-a*d+b*c)^6/(-a*f+b*e)...
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(13191\) vs. \(2(1204)=2408\).

Time = 16.50 (sec) , antiderivative size = 13191, normalized size of antiderivative = 10.96 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^4 (c+d x)^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.93 (sec) , antiderivative size = 1285, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {166, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-7 c f h+4 d e h+9 d f g)+b^2 (-6 c e h-c f g+10 d e g)\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}-\frac {\int -\frac {7 d f (4 d e-c f) h a^2-b \left (4 e (19 f g+8 e h) d^2-c f (13 f g+56 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (-3 f (f g-2 e h) c^2-2 d e (7 f g+24 e h) c+80 d^2 e^2 g\right )+7 d f \left (3 d f h a^2-b (9 d f g+4 d e h-7 c f h) a+b^2 (10 d e g-c (f g+6 e h))\right ) x}{2 (a+b x)^2 (c+d x)^3 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {7 d f (4 d e-c f) h a^2-b \left (4 e (19 f g+8 e h) d^2-c f (13 f g+56 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (-3 f (f g-2 e h) c^2-2 d e (7 f g+24 e h) c+80 d^2 e^2 g\right )+7 d f \left (3 d f h a^2-b (9 d f g+4 d e h-7 c f h) a+b^2 (10 d e g-c (f g+6 e h))\right ) x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}}dx}{4 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-7 c f h+4 d e h+9 d f g)+b^2 (-6 c e h-c f g+10 d e g)\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {\sqrt {e+f x} \left (21 a^3 d^2 f^2 h-7 a^2 b d f (-8 c f h+8 d e h+9 d f g)+a b^2 \left (3 c^2 f^2 h-2 c d f (49 e h+10 f g)+2 d^2 e (16 e h+73 f g)\right )-b^3 \left (-3 c^2 f (f g-2 e h)-2 c d e (24 e h+7 f g)+80 d^2 e^2 g\right )\right )}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}-\frac {\int -\frac {35 d^2 f^2 (4 d e-c f) h a^3-b d f \left (4 e (101 f g+86 e h) d^2-c f (89 f g+392 e h) d+48 c^2 f^2 h\right ) a^2+b^2 \left (64 e^2 (14 f g+3 e h) d^3-2 c e f (127 f g+312 e h) d^2-6 c^2 f^2 (2 f g-19 e h) d+3 c^3 f^3 h\right ) a-3 b^3 \left (-f^2 (f g-2 e h) c^3-6 d e f (f g-4 e h) c^2-48 d^2 e^2 (f g+2 e h) c+160 d^3 e^3 g\right )+5 d f \left (21 d^2 f^2 h a^3-7 b d f (9 d f g+8 d e h-8 c f h) a^2+b^2 \left (2 e (73 f g+16 e h) d^2-2 c f (10 f g+49 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (7 f g+24 e h) c+80 d^2 e^2 g\right )\right ) x}{2 (a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-7 c f h+4 d e h+9 d f g)+b^2 (-6 c e h-c f g+10 d e g)\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {35 d^2 f^2 (4 d e-c f) h a^3-b d f \left (4 e (101 f g+86 e h) d^2-c f (89 f g+392 e h) d+48 c^2 f^2 h\right ) a^2+b^2 \left (64 e^2 (14 f g+3 e h) d^3-2 c e f (127 f g+312 e h) d^2-6 c^2 f^2 (2 f g-19 e h) d+3 c^3 f^3 h\right ) a-3 b^3 \left (-f^2 (f g-2 e h) c^3-6 d e f (f g-4 e h) c^2-48 d^2 e^2 (f g+2 e h) c+160 d^3 e^3 g\right )+5 d f \left (21 d^2 f^2 h a^3-7 b d f (9 d f g+8 d e h-8 c f h) a^2+b^2 \left (2 e (73 f g+16 e h) d^2-2 c f (10 f g+49 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (7 f g+24 e h) c+80 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (21 a^3 d^2 f^2 h-7 a^2 b d f (-8 c f h+8 d e h+9 d f g)+a b^2 \left (3 c^2 f^2 h-2 c d f (49 e h+10 f g)+2 d^2 e (16 e h+73 f g)\right )-b^3 \left (-3 c^2 f (f g-2 e h)-2 c d e (24 e h+7 f g)+80 d^2 e^2 g\right )\right )}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-7 c f h+4 d e h+9 d f g)+b^2 (-6 c e h-c f g+10 d e g)\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (3 d f h a^2-b (9 d f g+4 d e h-7 c f h) a+b^2 (10 d e g-c f g-6 c e h)\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (21 d^2 f^2 h a^3-7 b d f (9 d f g+8 d e h-8 c f h) a^2+b^2 \left (2 e (73 f g+16 e h) d^2-2 c f (10 f g+49 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (7 f g+24 e h) c+80 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}+\frac {\frac {2 d \sqrt {e+f x} \left (35 d^2 f^2 h a^3-b d f (101 d f g+86 d e h-82 c f h) a^2+b^2 \left (16 e (14 f g+3 e h) d^2-2 c f (11 f g+74 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-8 d e (2 f g+9 e h) c+120 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {\int \frac {6 b (d e-c f) \left (d^2 f^2 (4 d f g+54 d e h-23 c f h) a^3-b d f \left (2 e (77 f g+60 e h) d^2-c f (41 f g+146 e h) d+18 c^2 f^2 h\right ) a^2+b^2 \left (8 e^2 (39 f g+8 e h) d^3-4 c e f (23 f g+54 e h) d^2-6 c^2 f^2 (f g-7 e h) d+c^3 f^3 h\right ) a-b^3 \left (-f^2 (f g-2 e h) c^3-6 d e f (f g-4 e h) c^2-48 d^2 e^2 (f g+2 e h) c+160 d^3 e^3 g\right )+d f \left (35 d^2 f^2 h a^3-b d f (101 d f g+86 d e h-82 c f h) a^2+b^2 \left (16 e (14 f g+3 e h) d^2-2 c f (11 f g+74 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-8 d e (2 f g+9 e h) c+120 d^2 e^2 g\right )\right ) x\right )}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (3 d f h a^2-b (9 d f g+4 d e h-7 c f h) a+b^2 (10 d e g-c f g-6 c e h)\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (21 d^2 f^2 h a^3-7 b d f (9 d f g+8 d e h-8 c f h) a^2+b^2 \left (2 e (73 f g+16 e h) d^2-2 c f (10 f g+49 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (7 f g+24 e h) c+80 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}+\frac {\frac {2 d \sqrt {e+f x} \left (35 d^2 f^2 h a^3-b d f (101 d f g+86 d e h-82 c f h) a^2+b^2 \left (16 e (14 f g+3 e h) d^2-2 c f (11 f g+74 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-8 d e (2 f g+9 e h) c+120 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \int \frac {d^2 f^2 (4 d f g+54 d e h-23 c f h) a^3-b d f \left (2 e (77 f g+60 e h) d^2-c f (41 f g+146 e h) d+18 c^2 f^2 h\right ) a^2+b^2 \left (8 e^2 (39 f g+8 e h) d^3-4 c e f (23 f g+54 e h) d^2-6 c^2 f^2 (f g-7 e h) d+c^3 f^3 h\right ) a-b^3 \left (-f^2 (f g-2 e h) c^3-6 d e f (f g-4 e h) c^2-48 d^2 e^2 (f g+2 e h) c+160 d^3 e^3 g\right )+d f \left (35 d^2 f^2 h a^3-b d f (101 d f g+86 d e h-82 c f h) a^2+b^2 \left (16 e (14 f g+3 e h) d^2-2 c f (11 f g+74 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-8 d e (2 f g+9 e h) c+120 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{b c-a d}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (3 d f h a^2-b (9 d f g+4 d e h-7 c f h) a+b^2 (10 d e g-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 {\sqrt {e+f x} \left (21 d^2 f^2 h a^3-7 b d f (9 d f g+8 d e h-8 c f h) a^2+b^2 \left (2 e (73 f g+16 e h) d^2-2 c f (10 f g+49 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (7 f g+24 e h) c+80 d^2 e^2 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 (35 d^2 f^2 h a^3-b d f (101 d f g+86 d e h-82 c f h) a^2+b^2 \left (16 e (14 f g+3 e h) d^2-2 c f (11 f g+74 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-8 d e (2 f g+9 e h) c+120 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \sqrt {e+f x} \left (d^2 f^2 (2 d f g+27 d e h-29 c f h) a^3-b d f \left (e (77 f g+60 e h) d^2-c f (71 f g+116 e h) d+50 c^2 f^2 h\right ) a^2-b^2 \left (-4 e^2 (39 f g+8 e h) d^3+2 c e f (79 f g+66 e h) d^2-c^2 f^2 (8 f g+95 e h) d+c^3 f^3 h\right ) a-b^3 \left (f^2 (f g-2 e h) c^3+d e f (5 f g+48 e h) c^2-12 d^2 e^2 (7 f g+4 e h) c+80 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\int \frac {2 d^3 f^3 (d f g-4 d e h+3 c f h) a^4+b d^2 f^2 \left (4 e (7 f g+20 e h) d^2-c f (34 f g+129 e h) d+55 c^2 f^2 h\right ) a^3+b^2 d f \left (-2 e^2 (111 f g+68 e h) d^3+c e f (283 f g+330 e h) d^2-55 c^2 f^2 (f g+4 e h) d+20 c^3 f^3 h\right ) a^2-b^3 \left (-32 e^3 (11 f g+2 e h) d^4+152 c e^2 f (3 f g+2 e h) d^3-2 c^2 e f^2 (47 f g+144 e h) d^2-c^3 f^3 (8 f g-45 e h) d+c^4 f^4 h\right ) a-b^4 (d e-c f) \left (-f^2 (f g-2 e h) c^3-6 d e f (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 (2 d f g+27 d e h-29 c f h) a^3-b d f \left (e (77 f g+60 e h) d^2-c f (71 f g+116 e h) d+50 c^2 f^2 h\right ) a^2-b^2 \left (-4 e^2 (39 f g+8 e h) d^3+2 c e f (79 f g+66 e h) d^2-c^2 f^2 (8 f g+95 e h) d+c^3 f^3 h\right ) a-b^3 \left (f^2 (f g-2 e h) c^3+d e f (5 f g+48 e h) c^2-12 d^2 e^2 (7 f g+4 e h) c+80 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}\right )}{b c-a d}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (3 d f h a^2-b (9 d f g+4 d e h-7 c f h) a+b^2 (10 d e g-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 {\sqrt {e+f x} \left (21 d^2 f^2 h a^3-7 b d f (9 d f g+8 d e h-8 c f h) a^2+b^2 \left (2 e (73 f g+16 e h) d^2-2 c f (10 f g+49 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (7 f g+24 e h) c+80 d^2 e^2 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 (35 d^2 f^2 h a^3-b d f (101 d f g+86 d e h-82 c f h) a^2+b^2 \left (16 e (14 f g+3 e h) d^2-2 c f (11 f g+74 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-8 d e (2 f g+9 e h) c+120 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \sqrt {e+f x} \left (d^2 f^2 (2 d f g+27 d e h-29 c f h) a^3-b d f \left (e (77 f g+60 e h) d^2-c f (71 f g+116 e h) d+50 c^2 f^2 h\right ) a^2-b^2 \left (-4 e^2 (39 f g+8 e h) d^3+2 c e f (79 f g+66 e h) d^2-c^2 f^2 (8 f g+95 e h) d+c^3 f^3 h\right ) a-b^3 \left (f^2 (f g-2 e h) c^3+d e f (5 f g+48 e h) c^2-12 d^2 e^2 (7 f g+4 e h) c+80 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {-\frac {2 d^2 \left (-\left (\left (-35 f^2 h c^3+21 d f (3 f g+4 e h) c^2-48 d^2 e (3 f g+e h) c+80 d^3 e^2 g\right ) b^2\right )+2 a d \left (8 e (f g+2 e h) d^2-9 c f (f g+4 e h) d+21 c^2 f^2 h\right ) b+a^2 d^2 f (d f g-4 d e h+3 c f h)\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx (b e-a f)^2}{b c-a d}-\frac {b (d e-c f) \left (35 d^3 f^3 h a^4-35 b d^2 f^2 (3 d f g+4 d e h-3 c f h) a^3+21 b^2 d f \left (2 e (9 f g+4 e h) d^2-c f (3 f g+14 e h) d+c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (27 f g+4 e h) d^3-36 c e f (3 f g+8 e h) d^2-3 c^2 f^2 (3 f g-16 e h) d+c^3 f^3 h\right ) a+b^4 \left (-f^2 (f g-2 e h) c^3-6 d e f (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}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (3 d f h a^2-b (9 d f g+4 d e h-7 c f h) a+b^2 (10 d e g-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 {\sqrt {e+f x} \left (21 d^2 f^2 h a^3-7 b d f (9 d f g+8 d e h-8 c f h) a^2+b^2 \left (2 e (73 f g+16 e h) d^2-2 c f (10 f g+49 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (7 f g+24 e h) c+80 d^2 e^2 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 (35 d^2 f^2 h a^3-b d f (101 d f g+86 d e h-82 c f h) a^2+b^2 \left (16 e (14 f g+3 e h) d^2-2 c f (11 f g+74 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-8 d e (2 f g+9 e h) c+120 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \sqrt {e+f x} \left (d^2 f^2 (2 d f g+27 d e h-29 c f h) a^3-b d f \left (e (77 f g+60 e h) d^2-c f (71 f g+116 e h) d+50 c^2 f^2 h\right ) a^2-b^2 \left (-4 e^2 (39 f g+8 e h) d^3+2 c e f (79 f g+66 e h) d^2-c^2 f^2 (8 f g+95 e h) d+c^3 f^3 h\right ) a-b^3 \left (f^2 (f g-2 e h) c^3+d e f (5 f g+48 e h) c^2-12 d^2 e^2 (7 f g+4 e h) c+80 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {-\frac {4 d^2 \left (-\left (\left (-35 f^2 h c^3+21 d f (3 f g+4 e h) c^2-48 d^2 e (3 f g+e h) c+80 d^3 e^2 g\right ) b^2\right )+2 a d \left (8 e (f g+2 e h) d^2-9 c f (f g+4 e h) d+21 c^2 f^2 h\right ) b+a^2 d^2 f (d f g-4 d e h+3 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)^2}{(b c-a d) f}-\frac {2 b (d e-c f) \left (35 d^3 f^3 h a^4-35 b d^2 f^2 (3 d f g+4 d e h-3 c f h) a^3+21 b^2 d f \left (2 e (9 f g+4 e h) d^2-c f (3 f g+14 e h) d+c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (27 f g+4 e h) d^3-36 c e f (3 f g+8 e h) d^2-3 c^2 f^2 (3 f g-16 e h) d+c^3 f^3 h\right ) a+b^4 \left (-f^2 (f g-2 e h) c^3-6 d e f (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}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (3 d f h a^2-b (9 d f g+4 d e h-7 c f h) a+b^2 (10 d e g-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 {\sqrt {e+f x} \left (21 d^2 f^2 h a^3-7 b d f (9 d f g+8 d e h-8 c f h) a^2+b^2 \left (2 e (73 f g+16 e h) d^2-2 c f (10 f g+49 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (7 f g+24 e h) c+80 d^2 e^2 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 (35 d^2 f^2 h a^3-b d f (101 d f g+86 d e h-82 c f h) a^2+b^2 \left (16 e (14 f g+3 e h) d^2-2 c f (11 f g+74 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-8 d e (2 f g+9 e h) c+120 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \sqrt {e+f x} \left (d^2 f^2 (2 d f g+27 d e h-29 c f h) a^3-b d f \left (e (77 f g+60 e h) d^2-c f (71 f g+116 e h) d+50 c^2 f^2 h\right ) a^2-b^2 \left (-4 e^2 (39 f g+8 e h) d^3+2 c e f (79 f g+66 e h) d^2-c^2 f^2 (8 f g+95 e h) d+c^3 f^3 h\right ) a-b^3 \left (f^2 (f g-2 e h) c^3+d e f (5 f g+48 e h) c^2-12 d^2 e^2 (7 f g+4 e h) c+80 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\frac {4 d^{3/2} \left (-\left (\left (-35 f^2 h c^3+21 d f (3 f g+4 e h) c^2-48 d^2 e (3 f g+e h) c+80 d^3 e^2 g\right ) b^2\right )+2 a d \left (8 e (f g+2 e h) d^2-9 c f (f g+4 e h) d+21 c^2 f^2 h\right ) b+a^2 d^2 f (d f g-4 d e h+3 c f h)\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) (b e-a f)^2}{(b c-a d) \sqrt {d e-c f}}+\frac {2 \sqrt {b} (d e-c f) \left (35 d^3 f^3 h a^4-35 b d^2 f^2 (3 d f g+4 d e h-3 c f h) a^3+21 b^2 d f \left (2 e (9 f g+4 e h) d^2-c f (3 f g+14 e h) d+c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (27 f g+4 e h) d^3-36 c e f (3 f g+8 e h) d^2-3 c^2 f^2 (3 f g-16 e h) d+c^3 f^3 h\right ) a+b^4 \left (-f^2 (f g-2 e h) c^3-6 d e f (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}}}{(b c-a d) (d e-c f)}\right )}{b c-a d}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}\)

Input:

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

Output:

-1/3*((b*g - a*h)*Sqrt[e + f*x])/(b*(b*c - a*d)*(a + b*x)^3*(c + d*x)^2) + 
 (((3*a^2*d*f*h + b^2*(10*d*e*g - c*f*g - 6*c*e*h) - a*b*(9*d*f*g + 4*d*e* 
h - 7*c*f*h))*Sqrt[e + f*x])/(2*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2*(c + d 
*x)^2) + (((21*a^3*d^2*f^2*h - 7*a^2*b*d*f*(9*d*f*g + 8*d*e*h - 8*c*f*h) - 
 b^3*(80*d^2*e^2*g - 3*c^2*f*(f*g - 2*e*h) - 2*c*d*e*(7*f*g + 24*e*h)) + a 
*b^2*(3*c^2*f^2*h + 2*d^2*e*(73*f*g + 16*e*h) - 2*c*d*f*(10*f*g + 49*e*h)) 
)*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x)^2) + ((2*d*( 
35*a^3*d^2*f^2*h - a^2*b*d*f*(101*d*f*g + 86*d*e*h - 82*c*f*h) - b^3*(120* 
d^2*e^2*g - 3*c^2*f*(f*g - 2*e*h) - 8*c*d*e*(2*f*g + 9*e*h)) + a*b^2*(3*c^ 
2*f^2*h + 16*d^2*e*(14*f*g + 3*e*h) - 2*c*d*f*(11*f*g + 74*e*h)))*Sqrt[e + 
 f*x])/((b*c - a*d)*(c + d*x)^2) + (3*b*((2*d*(a^3*d^2*f^2*(2*d*f*g + 27*d 
*e*h - 29*c*f*h) - b^3*(80*d^3*e^3*g + c^3*f^2*(f*g - 2*e*h) - 12*c*d^2*e^ 
2*(7*f*g + 4*e*h) + c^2*d*e*f*(5*f*g + 48*e*h)) - a*b^2*(c^3*f^3*h - 4*d^3 
*e^2*(39*f*g + 8*e*h) + 2*c*d^2*e*f*(79*f*g + 66*e*h) - c^2*d*f^2*(8*f*g + 
 95*e*h)) - a^2*b*d*f*(50*c^2*f^2*h + d^2*e*(77*f*g + 60*e*h) - c*d*f*(71* 
f*g + 116*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(d*e - c*f)*(c + d*x)) + ((2* 
Sqrt[b]*(d*e - c*f)*(35*a^4*d^3*f^3*h - 35*a^3*b*d^2*f^2*(3*d*f*g + 4*d*e* 
h - 3*c*f*h) + b^4*(160*d^3*e^3*g - 6*c^2*d*e*f*(f*g - 4*e*h) - c^3*f^2*(f 
*g - 2*e*h) - 48*c*d^2*e^2*(f*g + 2*e*h)) - a*b^3*(c^3*f^3*h - 3*c^2*d*f^2 
*(3*f*g - 16*e*h) + 16*d^3*e^2*(27*f*g + 4*e*h) - 36*c*d^2*e*f*(3*f*g +...
 

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 166

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*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

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 = 95.63 (sec) , antiderivative size = 1745, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1745\)
default	\(\text {Expression too large to display}\)	\(1745\)
pseudoelliptic	\(\text {Expression too large to display}\)	\(1987\)

Input:

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

Output:

2*f^5*(-d^2/f^5/(a*d-b*c)^6*((1/8*d*f*(5*a^2*c*d^2*f*h-4*a^2*d^3*e*h-a^2*d 
^3*f*g+6*a*b*c^2*d*f*h-8*a*b*c*d^2*e*h-14*a*b*c*d^2*f*g+16*a*b*d^3*e*g-11* 
b^2*c^3*f*h+12*b^2*c^2*d*e*h+15*b^2*c^2*d*f*g-16*b^2*c*d^2*e*g)/(c*f-d*e)* 
(f*x+e)^(3/2)+1/8*(3*a^2*c*d^2*f*h-4*a^2*d^3*e*h+a^2*d^3*f*g+10*a*b*c^2*d* 
f*h-8*a*b*c*d^2*e*h-18*a*b*c*d^2*f*g+16*a*b*d^3*e*g-13*b^2*c^3*f*h+12*b^2* 
c^2*d*e*h+17*b^2*c^2*d*f*g-16*b^2*c*d^2*e*g)*f*(f*x+e)^(1/2))/((f*x+e)*d+c 
*f-d*e)^2-1/8*(3*a^2*c*d^2*f^2*h-4*a^2*d^3*e*f*h+a^2*d^3*f^2*g+42*a*b*c^2* 
d*f^2*h-72*a*b*c*d^2*e*f*h-18*a*b*c*d^2*f^2*g+32*a*b*d^3*e^2*h+16*a*b*d^3* 
e*f*g+35*b^2*c^3*f^2*h-84*b^2*c^2*d*e*f*h-63*b^2*c^2*d*f^2*g+48*b^2*c*d^2* 
e^2*h+144*b^2*c*d^2*e*f*g-80*b^2*d^3*e^2*g)/(c*f-d*e)/((c*f-d*e)*d)^(1/2)* 
arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)))-b/f^5/(a*d-b*c)^6*((1/16*b^2* 
f*(19*a^4*d^3*f^2*h+9*a^3*b*c*d^2*f^2*h-44*a^3*b*d^3*e*f*h-41*a^3*b*d^3*f^ 
2*g-27*a^2*b^2*c^2*d*f^2*h-6*a^2*b^2*c*d^2*e*f*h+33*a^2*b^2*c*d^2*f^2*g+24 
*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+48*a*b^3*c^2*d*e*f 
*h+9*a*b^3*c^2*d*f^2*g-84*a*b^3*c*d^2*e*f*g-48*a*b^3*d^3*e^2*g+2*b^4*c^3*e 
*f*h-b^4*c^3*f^2*g-24*b^4*c^2*d*e^2*h-6*b^4*c^2*d*e*f*g+48*b^4*c*d^2*e^2*g 
)/(a^2*f^2-2*a*b*e*f+b^2*e^2)*(f*x+e)^(5/2)+1/6*(17*a^4*d^3*f^2*h+3*a^3*b* 
c*d^2*f^2*h-36*a^3*b*d^3*e*f*h-35*a^3*b*d^3*f^2*g-21*a^2*b^2*c^2*d*f^2*h+3 
3*a^2*b^2*c*d^2*f^2*g+18*a^2*b^2*d^3*e^2*h+72*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+3*a*b^3*c^2*d*f^2*g-72*a*b^3*c*d^2*e*f*g-36*...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^4/(d*x+c)^3,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. 3092 vs. \(2 (1160) = 2320\).

Time = 0.37 (sec) , antiderivative size = 3092, normalized size of antiderivative = 2.57 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^4 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*(160*b^5*d^3*e^3*g - 48*b^5*c*d^2*e^2*f*g - 432*a*b^4*d^3*e^2*f*g - 6 
*b^5*c^2*d*e*f^2*g + 108*a*b^4*c*d^2*e*f^2*g + 378*a^2*b^3*d^3*e*f^2*g - b 
^5*c^3*f^3*g + 9*a*b^4*c^2*d*f^3*g - 63*a^2*b^3*c*d^2*f^3*g - 105*a^3*b^2* 
d^3*f^3*g - 96*b^5*c*d^2*e^3*h - 64*a*b^4*d^3*e^3*h + 24*b^5*c^2*d*e^2*f*h 
 + 288*a*b^4*c*d^2*e^2*f*h + 168*a^2*b^3*d^3*e^2*f*h + 2*b^5*c^3*e*f^2*h - 
 48*a*b^4*c^2*d*e*f^2*h - 294*a^2*b^3*c*d^2*e*f^2*h - 140*a^3*b^2*d^3*e*f^ 
2*h - a*b^4*c^3*f^3*h + 21*a^2*b^3*c^2*d*f^3*h + 105*a^3*b^2*c*d^2*f^3*h + 
 35*a^4*b*d^3*f^3*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^8*c^ 
6*e^2 - 6*a*b^7*c^5*d*e^2 + 15*a^2*b^6*c^4*d^2*e^2 - 20*a^3*b^5*c^3*d^3*e^ 
2 + 15*a^4*b^4*c^2*d^4*e^2 - 6*a^5*b^3*c*d^5*e^2 + a^6*b^2*d^6*e^2 - 2*a*b 
^7*c^6*e*f + 12*a^2*b^6*c^5*d*e*f - 30*a^3*b^5*c^4*d^2*e*f + 40*a^4*b^4*c^ 
3*d^3*e*f - 30*a^5*b^3*c^2*d^4*e*f + 12*a^6*b^2*c*d^5*e*f - 2*a^7*b*d^6*e* 
f + a^2*b^6*c^6*f^2 - 6*a^3*b^5*c^5*d*f^2 + 15*a^4*b^4*c^4*d^2*f^2 - 20*a^ 
5*b^3*c^3*d^3*f^2 + 15*a^6*b^2*c^2*d^4*f^2 - 6*a^7*b*c*d^5*f^2 + a^8*d^6*f 
^2)*sqrt(-b^2*e + a*b*f)) + 1/4*(80*b^2*d^5*e^2*g - 144*b^2*c*d^4*e*f*g - 
16*a*b*d^5*e*f*g + 63*b^2*c^2*d^3*f^2*g + 18*a*b*c*d^4*f^2*g - a^2*d^5*f^2 
*g - 48*b^2*c*d^4*e^2*h - 32*a*b*d^5*e^2*h + 84*b^2*c^2*d^3*e*f*h + 72*a*b 
*c*d^4*e*f*h + 4*a^2*d^5*e*f*h - 35*b^2*c^3*d^2*f^2*h - 42*a*b*c^2*d^3*f^2 
*h - 3*a^2*c*d^4*f^2*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b^6 
*c^6*d*e - 6*a*b^5*c^5*d^2*e + 15*a^2*b^4*c^4*d^3*e - 20*a^3*b^3*c^3*d^...
 

Mupad [B] (verification not implemented)

Time = 48.26 (sec) , antiderivative size = 1218467, normalized size of antiderivative = 1012.02 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^4 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((256*a^18*b^2*c*d^18*f^10*g - 256*a^18*b^2*d^19*e*f^9*g - 128*a^2* 
b^18*c^17*d^2*f^10*g + 2560*a^3*b^17*c^16*d^3*f^10*g - 27776*a^4*b^16*c^15 
*d^4*f^10*g + 175872*a^5*b^15*c^14*d^5*f^10*g - 700800*a^6*b^14*c^13*d^6*f 
^10*g + 1866752*a^7*b^13*c^12*d^7*f^10*g - 3439744*a^8*b^12*c^11*d^8*f^10* 
g + 4412672*a^9*b^11*c^10*d^9*f^10*g - 3805824*a^10*b^10*c^9*d^10*f^10*g + 
 1886720*a^11*b^9*c^8*d^11*f^10*g - 35200*a^12*b^8*c^7*d^12*f^10*g - 73907 
2*a^13*b^7*c^6*d^13*f^10*g + 607104*a^14*b^6*c^5*d^14*f^10*g - 258048*a^15 
*b^5*c^4*d^15*f^10*g + 62080*a^16*b^4*c^3*d^16*f^10*g - 7424*a^17*b^3*c^2* 
d^17*f^10*g - 128*a^3*b^17*c^17*d^2*f^10*h + 4096*a^4*b^16*c^16*d^3*f^10*h 
 - 32128*a^5*b^15*c^15*d^4*f^10*h + 113408*a^6*b^14*c^14*d^5*f^10*h - 1711 
36*a^7*b^13*c^13*d^6*f^10*h - 129536*a^8*b^12*c^12*d^7*f^10*h + 1170048*a^ 
9*b^11*c^11*d^8*f^10*h - 2728704*a^10*b^10*c^10*d^9*f^10*h + 3805824*a^11* 
b^9*c^9*d^10*f^10*h - 3570688*a^12*b^8*c^8*d^11*f^10*h + 2304896*a^13*b^7* 
c^7*d^12*f^10*h - 998144*a^14*b^6*c^6*d^13*f^10*h + 264832*a^15*b^5*c^5*d^ 
14*f^10*h - 31232*a^16*b^4*c^4*d^15*f^10*h - 2176*a^17*b^3*c^3*d^16*f^10*h 
 + 768*a^18*b^2*c^2*d^17*f^10*h + 10240*a^12*b^8*d^19*e^7*f^3*g - 45568*a^ 
13*b^7*d^19*e^6*f^4*g + 78976*a^14*b^6*d^19*e^5*f^5*g - 65536*a^15*b^5*d^1 
9*e^4*f^6*g + 24960*a^16*b^4*d^19*e^3*f^7*g - 2816*a^17*b^3*d^19*e^2*f^8*g 
 - 4096*a^13*b^7*d^19*e^7*f^3*h + 17920*a^14*b^6*d^19*e^6*f^4*h - 30336*a^ 
15*b^5*d^19*e^5*f^5*h + 24320*a^16*b^4*d^19*e^4*f^6*h - 8832*a^17*b^3*d...
 

Reduce [B] (verification not implemented)

Time = 16.47 (sec) , antiderivative size = 48332, normalized size of antiderivative = 40.14 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^4 (c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - 
 b*e)))*a**7*c**4*d**3*f**5*h + 210*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + 
 f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*c**3*d**4*e*f**4*h - 210*sqrt(b)* 
sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*c** 
3*d**4*f**5*h*x - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt 
(b)*sqrt(a*f - b*e)))*a**7*c**2*d**5*e**2*f**3*h + 420*sqrt(b)*sqrt(a*f - 
b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*c**2*d**5*e*f* 
*4*h*x - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt( 
a*f - b*e)))*a**7*c**2*d**5*f**5*h*x**2 - 210*sqrt(b)*sqrt(a*f - b*e)*atan 
((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*c*d**6*e**2*f**3*h*x + 
210*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e 
)))*a**7*c*d**6*e*f**4*h*x**2 - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + 
 f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*d**7*e**2*f**3*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**6* 
b*c**5*d**2*f**5*h + 1050*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/( 
sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**4*d**3*e*f**4*h + 315*sqrt(b)*sqrt(a*f 
 - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**4*d**3 
*f**5*g - 945*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt 
(a*f - b*e)))*a**6*b*c**4*d**3*f**5*h*x - 1155*sqrt(b)*sqrt(a*f - b*e)*ata 
n((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**3*d**4*e**2*f*...