\(\int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^3} \, dx\) [103]

Optimal result

Integrand size = 29, antiderivative size = 1108 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^3} \, dx=\frac {d \left (5 a^3 d^2 f^2 h-5 a^2 b d f (7 d f g+10 d e h-14 c f h)-b^3 \left (120 d^2 e^2 g-8 c d e (11 f g+9 e h)+3 c^2 f (f g+14 e h)\right )+a b^2 \left (45 c^2 f^2 h+8 d^2 e (19 f g+6 e h)-2 c d f (41 f g+74 e h)\right )\right ) \sqrt {e+f x}}{24 b^2 (b c-a d)^4 (b e-a f) (c+d x)^2}+\frac {\left (a^2 d f h-a b (7 d f g+4 d e h-9 c f h)+b^2 (10 d e g-3 c (f g+2 e h))\right ) \sqrt {e+f x}}{12 b^2 (b c-a d)^2 (a+b x)^2 (c+d x)^2}+\frac {\left (3 a^3 d^2 f^2 h-a^2 b d f (21 d f g+32 d e h-44 c f h)+a b^2 \left (33 c^2 f^2 h-14 c d f (4 f g+7 e h)+2 d^2 e (49 f g+16 e h)\right )-b^3 \left (80 d^2 e^2 g+3 c^2 f (f g+10 e h)-2 c d e (31 f g+24 e h)\right )\right ) \sqrt {e+f x}}{24 b^2 (b c-a d)^3 (b e-a f) (a+b x) (c+d x)^2}+\frac {d \left (5 a^3 d^2 f^2 h-a^2 b d f (29 d f g+36 d e h-50 c f h)+a b^2 \left (25 c^2 f^2 h-50 c d f (f g+2 e h)+4 d^2 e (27 f g+8 e h)\right )-b^3 \left (80 d^2 e^2 g-4 c d e (13 f g+12 e h)+c^2 f (f g+24 e h)\right )\right ) \sqrt {e+f x}}{8 b (b c-a d)^5 (b e-a f) (c+d x)}-\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}+\frac {\left (5 a^4 d^3 f^3 h-5 a^3 b d^2 f^2 (7 d f g+12 d e h-15 c f h)+b^4 \left (160 d^3 e^3 g+c^3 f^2 (f g-6 e h)-48 c d^2 e^2 (3 f g+2 e h)+18 c^2 d e f (f g+4 e h)\right )+15 a^2 b^2 d f \left (5 c^2 f^2 h+2 d^2 e (7 f g+4 e h)-c d f (7 f g+18 e h)\right )+a b^3 \left (5 c^3 f^3 h-16 d^3 e^2 (21 f g+4 e h)+36 c d^2 e f (7 f g+8 e h)-3 c^2 d f^2 (7 f g+48 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{8 \sqrt {b} (b c-a d)^6 (b e-a f)^{3/2}}-\frac {\sqrt {d} \left (3 a^2 d^2 f (d f g+4 d e h-5 c f h)+b^2 \left (80 d^3 e^2 g-15 c^3 f^2 h-16 c d^2 e (7 f g+3 e h)+5 c^2 d f (7 f g+12 e h)\right )-2 a b d \left (25 c^2 f^2 h+8 d^2 e (3 f g+2 e h)-c d f (21 f g+44 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 (b c-a d)^6 \sqrt {d e-c f}} \] Output:

1/24*d*(5*a^3*d^2*f^2*h-5*a^2*b*d*f*(-14*c*f*h+10*d*e*h+7*d*f*g)-b^3*(120* 
d^2*e^2*g-8*c*d*e*(9*e*h+11*f*g)+3*c^2*f*(14*e*h+f*g))+a*b^2*(45*c^2*f^2*h 
+8*d^2*e*(6*e*h+19*f*g)-2*c*d*f*(74*e*h+41*f*g)))*(f*x+e)^(1/2)/b^2/(-a*d+ 
b*c)^4/(-a*f+b*e)/(d*x+c)^2+1/12*(a^2*d*f*h-a*b*(-9*c*f*h+4*d*e*h+7*d*f*g) 
+b^2*(10*d*e*g-3*c*(2*e*h+f*g)))*(f*x+e)^(1/2)/b^2/(-a*d+b*c)^2/(b*x+a)^2/ 
(d*x+c)^2+1/24*(3*a^3*d^2*f^2*h-a^2*b*d*f*(-44*c*f*h+32*d*e*h+21*d*f*g)+a* 
b^2*(33*c^2*f^2*h-14*c*d*f*(7*e*h+4*f*g)+2*d^2*e*(16*e*h+49*f*g))-b^3*(80* 
d^2*e^2*g+3*c^2*f*(10*e*h+f*g)-2*c*d*e*(24*e*h+31*f*g)))*(f*x+e)^(1/2)/b^2 
/(-a*d+b*c)^3/(-a*f+b*e)/(b*x+a)/(d*x+c)^2+1/8*d*(5*a^3*d^2*f^2*h-a^2*b*d* 
f*(-50*c*f*h+36*d*e*h+29*d*f*g)+a*b^2*(25*c^2*f^2*h-50*c*d*f*(2*e*h+f*g)+4 
*d^2*e*(8*e*h+27*f*g))-b^3*(80*d^2*e^2*g-4*c*d*e*(12*e*h+13*f*g)+c^2*f*(24 
*e*h+f*g)))*(f*x+e)^(1/2)/b/(-a*d+b*c)^5/(-a*f+b*e)/(d*x+c)-1/3*(-a*h+b*g) 
*(f*x+e)^(3/2)/b/(-a*d+b*c)/(b*x+a)^3/(d*x+c)^2+1/8*(5*a^4*d^3*f^3*h-5*a^3 
*b*d^2*f^2*(-15*c*f*h+12*d*e*h+7*d*f*g)+b^4*(160*d^3*e^3*g+c^3*f^2*(-6*e*h 
+f*g)-48*c*d^2*e^2*(2*e*h+3*f*g)+18*c^2*d*e*f*(4*e*h+f*g))+15*a^2*b^2*d*f* 
(5*c^2*f^2*h+2*d^2*e*(4*e*h+7*f*g)-c*d*f*(18*e*h+7*f*g))+a*b^3*(5*c^3*f^3* 
h-16*d^3*e^2*(4*e*h+21*f*g)+36*c*d^2*e*f*(8*e*h+7*f*g)-3*c^2*d*f^2*(48*e*h 
+7*f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^(1/2)/(-a*d+b* 
c)^6/(-a*f+b*e)^(3/2)-1/4*d^(1/2)*(3*a^2*d^2*f*(-5*c*f*h+4*d*e*h+d*f*g)+b^ 
2*(80*d^3*e^2*g-15*c^3*f^2*h-16*c*d^2*e*(3*e*h+7*f*g)+5*c^2*d*f*(12*e*h...
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(17142\) vs. \(2(1108)=2216\).

Time = 16.65 (sec) , antiderivative size = 17142, normalized size of antiderivative = 15.47 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.77 (sec) , antiderivative size = 1165, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {166, 27, 166, 27, 168, 27, 168, 27, 168, 25, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 166

\(\displaystyle -\frac {\frac {\int -\frac {8 b e (d e-c f) (5 b d g-3 b c h-2 a d h)+(4 d e-c f) \left (d f h a^2-b (7 d f g+4 d e h-9 c f h) a+b^2 (10 d e g-3 c f g-6 c e h)\right )+f \left (\left (24 f h c^2-7 d (7 f g+6 e h) c+70 d^2 e g\right ) b^2+a d (43 c f h-7 d (3 f g+4 e h)) b+3 a^2 d^2 f h\right ) x}{2 (a+b x)^2 (c+d x)^3 \sqrt {e+f x}}dx}{2 b (b c-a d)}-\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-9 c f h+4 d e h+7 d f g)+b^2 (10 d e g-3 c (2 e h+f g))\right )}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}}{6 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{3 b (a+b x)^3 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {8 b e (d e-c f) (5 b d g-3 b c h-2 a d h)+(4 d e-c f) \left (d f h a^2-b (7 d f g+4 d e h-9 c f h) a+b^2 (10 d e g-3 c (f g+2 e h))\right )+f \left (\left (24 f h c^2-7 d (7 f g+6 e h) c+70 d^2 e g\right ) b^2+a d (43 c f h-7 d (3 f g+4 e h)) b+3 a^2 d^2 f h\right ) x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}}dx}{4 b (b c-a d)}-\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-9 c f h+4 d e h+7 d f g)+b^2 (10 d e g-3 c (2 e h+f g))\right )}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}}{6 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{3 b (a+b x)^3 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\frac {\sqrt {e+f x} \left (3 a^3 d^2 f^2 h-a^2 b d f (-44 c f h+32 d e h+21 d f g)+a b^2 \left (33 c^2 f^2 h-14 c d f (7 e h+4 f g)+2 d^2 e (16 e h+49 f g)\right )-b^3 \left (3 c^2 f (10 e h+f g)-2 c d e (24 e h+31 f g)+80 d^2 e^2 g\right )\right )}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}-\frac {\int -\frac {5 d^2 f^2 (4 d e-c f) h a^3-5 b d f \left (4 e (7 f g+10 e h) d^2-c f (7 f g+64 e h) d+12 c^2 f^2 h\right ) a^2-b^2 \left (-32 e^2 (19 f g+6 e h) d^3+2 c e f (223 f g+312 e h) d^2-6 c^2 f^2 (8 f g+47 e h) d+15 c^3 f^3 h\right ) a-3 b^3 \left (f^2 (f g-6 e h) c^3+18 d e f (f g+4 e h) c^2-48 d^2 e^2 (3 f g+2 e h) c+160 d^3 e^3 g\right )+5 d f \left (3 d^2 f^2 h a^3-b d f (21 d f g+32 d e h-44 c f h) a^2+b^2 \left (2 e (49 f g+16 e h) d^2-14 c f (4 f g+7 e h) d+33 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+10 e h) c^2-2 d e (31 f g+24 e h) c+80 d^2 e^2 g\right )\right ) x}{2 (a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{4 b (b c-a d)}-\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-9 c f h+4 d e h+7 d f g)+b^2 (10 d e g-3 c (2 e h+f g))\right )}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}}{6 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{3 b (a+b x)^3 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\int \frac {5 d^2 f^2 (4 d e-c f) h a^3-5 b d f \left (4 e (7 f g+10 e h) d^2-c f (7 f g+64 e h) d+12 c^2 f^2 h\right ) a^2-b^2 \left (-32 e^2 (19 f g+6 e h) d^3+2 c e f (223 f g+312 e h) d^2-6 c^2 f^2 (8 f g+47 e h) d+15 c^3 f^3 h\right ) a-3 b^3 \left (f^2 (f g-6 e h) c^3+18 d e f (f g+4 e h) c^2-48 d^2 e^2 (3 f g+2 e h) c+160 d^3 e^3 g\right )+5 d f \left (3 d^2 f^2 h a^3-b d f (21 d f g+32 d e h-44 c f h) a^2+b^2 \left (2 e (49 f g+16 e h) d^2-14 c f (4 f g+7 e h) d+33 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+10 e h) c^2-2 d e (31 f g+24 e h) c+80 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (3 a^3 d^2 f^2 h-a^2 b d f (-44 c f h+32 d e h+21 d f g)+a b^2 \left (33 c^2 f^2 h-14 c d f (7 e h+4 f g)+2 d^2 e (16 e h+49 f g)\right )-b^3 \left (3 c^2 f (10 e h+f g)-2 c d e (24 e h+31 f g)+80 d^2 e^2 g\right )\right )}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}-\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-9 c f h+4 d e h+7 d f g)+b^2 (10 d e g-3 c (2 e h+f g))\right )}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}}{6 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{3 b (a+b x)^3 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+4 d e h-9 c f h) a+b^2 (10 d e g-3 c (f g+2 e h))\right )}{2 b (b c-a d) (a+b x)^2 (c+d x)^2}-\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (21 d f g+32 d e h-44 c f h) a^2+b^2 \left (2 e (49 f g+16 e h) d^2-14 c f (4 f g+7 e h) d+33 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+10 e h) c^2-2 d e (31 f g+24 e h) c+80 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}+\frac {\frac {2 d \sqrt {e+f x} \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+10 d e h-14 c f h) a^2+b^2 \left (8 e (19 f g+6 e h) d^2-2 c f (41 f g+74 e h) d+45 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+14 e h) c^2-8 d e (11 f g+9 e h) c+120 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {\int \frac {6 b (d e-c f) \left (5 d^2 f^2 (2 d e-c f) h a^3-b d f \left (2 e (29 f g+36 e h) d^2-c f (23 f g+122 e h) d+30 c^2 f^2 h\right ) a^2-b^2 \left (-8 e^2 (27 f g+8 e h) d^3+4 c e f (41 f g+54 e h) d^2-6 c^2 f^2 (3 f g+17 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (f^2 (f g-6 e h) c^3+18 d e f (f g+4 e h) c^2-48 d^2 e^2 (3 f g+2 e h) c+160 d^3 e^3 g\right )+d f \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+10 d e h-14 c f h) a^2+b^2 \left (8 e (19 f g+6 e h) d^2-2 c f (41 f g+74 e h) d+45 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+14 e h) c^2-8 d e (11 f g+9 e h) c+120 d^2 e^2 g\right )\right ) x\right )}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+4 d e h-9 c f h) a+b^2 (10 d e g-3 c (f g+2 e h))\right )}{2 b (b c-a d) (a+b x)^2 (c+d x)^2}-\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (21 d f g+32 d e h-44 c f h) a^2+b^2 \left (2 e (49 f g+16 e h) d^2-14 c f (4 f g+7 e h) d+33 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+10 e h) c^2-2 d e (31 f g+24 e h) c+80 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}+\frac {\frac {2 d \sqrt {e+f x} \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+10 d e h-14 c f h) a^2+b^2 \left (8 e (19 f g+6 e h) d^2-2 c f (41 f g+74 e h) d+45 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+14 e h) c^2-8 d e (11 f g+9 e h) c+120 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \int \frac {5 d^2 f^2 (2 d e-c f) h a^3-b d f \left (2 e (29 f g+36 e h) d^2-c f (23 f g+122 e h) d+30 c^2 f^2 h\right ) a^2-b^2 \left (-8 e^2 (27 f g+8 e h) d^3+4 c e f (41 f g+54 e h) d^2-6 c^2 f^2 (3 f g+17 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (f^2 (f g-6 e h) c^3+18 d e f (f g+4 e h) c^2-48 d^2 e^2 (3 f g+2 e h) c+160 d^3 e^3 g\right )+d f \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+10 d e h-14 c f h) a^2+b^2 \left (8 e (19 f g+6 e h) d^2-2 c f (41 f g+74 e h) d+45 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+14 e h) c^2-8 d e (11 f g+9 e h) c+120 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{b c-a d}}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+4 d e h-9 c f h) a+b^2 (10 d e g-3 c (f g+2 e h))\right )}{2 b (b c-a d) (a+b x)^2 (c+d x)^2}-\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (21 d f g+32 d e h-44 c f h) a^2+b^2 \left (2 e (49 f g+16 e h) d^2-14 c f (4 f g+7 e h) d+33 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+10 e h) c^2-2 d e (31 f g+24 e h) c+80 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}+\frac {\frac {2 d \sqrt {e+f x} \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+10 d e h-14 c f h) a^2+b^2 \left (8 e (19 f g+6 e h) d^2-2 c f (41 f g+74 e h) d+45 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+14 e h) c^2-8 d e (11 f g+9 e h) c+120 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \sqrt {e+f x} \left (5 d^2 f^2 h a^3-b d f (29 d f g+36 d e h-50 c f h) a^2+b^2 \left (4 e (27 f g+8 e h) d^2-50 c f (f g+2 e h) d+25 c^2 f^2 h\right ) a-b^3 \left (f (f g+24 e h) c^2-4 d e (13 f g+12 e h) c+80 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)}+\frac {\int -\frac {b (d e-c f) \left (d^2 f^2 (25 c f h-6 d (f g+4 e h)) a^3+b d f \left (2 e (51 f g+44 e h) d^2-5 c f (11 f g+34 e h) d+50 c^2 f^2 h\right ) a^2+b^2 \left (-64 e^2 (4 f g+e h) d^3+40 c e f (5 f g+6 e h) d^2-20 c^2 f^2 (f g+6 e h) d+5 c^3 f^3 h\right ) a+b^3 \left (f^2 (f g-6 e h) c^3+18 d e f (f g+4 e h) c^2-48 d^2 e^2 (3 f g+2 e h) c+160 d^3 e^3 g\right )-d f \left (5 d^2 f^2 h a^3-b d f (29 d f g+36 d e h-50 c f h) a^2+b^2 \left (4 e (27 f g+8 e h) d^2-50 c f (f g+2 e h) d+25 c^2 f^2 h\right ) a-b^3 \left (f (f g+24 e h) c^2-4 d e (13 f g+12 e h) c+80 d^2 e^2 g\right )\right ) x\right )}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}\right )}{b c-a d}}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+4 d e h-9 c f h) a+b^2 (10 d e g-3 c (f g+2 e h))\right )}{2 b (b c-a d) (a+b x)^2 (c+d x)^2}-\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (21 d f g+32 d e h-44 c f h) a^2+b^2 \left (2 e (49 f g+16 e h) d^2-14 c f (4 f g+7 e h) d+33 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+10 e h) c^2-2 d e (31 f g+24 e h) c+80 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}+\frac {\frac {2 d \sqrt {e+f x} \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+10 d e h-14 c f h) a^2+b^2 \left (8 e (19 f g+6 e h) d^2-2 c f (41 f g+74 e h) d+45 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+14 e h) c^2-8 d e (11 f g+9 e h) c+120 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \left (5 d^2 f^2 h a^3-b d f (29 d f g+36 d e h-50 c f h) a^2+b^2 \left (4 e (27 f g+8 e h) d^2-50 c f (f g+2 e h) d+25 c^2 f^2 h\right ) a-b^3 \left (f (f g+24 e h) c^2-4 d e (13 f g+12 e h) c+80 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {\int \frac {b (d e-c f) \left (d^2 f^2 (25 c f h-6 d (f g+4 e h)) a^3+b d f \left (2 e (51 f g+44 e h) d^2-5 c f (11 f g+34 e h) d+50 c^2 f^2 h\right ) a^2+b^2 \left (-64 e^2 (4 f g+e h) d^3+40 c e f (5 f g+6 e h) d^2-20 c^2 f^2 (f g+6 e h) d+5 c^3 f^3 h\right ) a+b^3 \left (f^2 (f g-6 e h) c^3+18 d e f (f g+4 e h) c^2-48 d^2 e^2 (3 f g+2 e h) c+160 d^3 e^3 g\right )-d f \left (5 d^2 f^2 h a^3-b d f (29 d f g+36 d e h-50 c f h) a^2+b^2 \left (4 e (27 f g+8 e h) d^2-50 c f (f g+2 e h) d+25 c^2 f^2 h\right ) a-b^3 \left (f (f g+24 e h) c^2-4 d e (13 f g+12 e h) c+80 d^2 e^2 g\right )\right ) x\right )}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}\right )}{b c-a d}}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+4 d e h-9 c f h) a+b^2 (10 d e g-3 c (f g+2 e h))\right )}{2 b (b c-a d) (a+b x)^2 (c+d x)^2}-\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (21 d f g+32 d e h-44 c f h) a^2+b^2 \left (2 e (49 f g+16 e h) d^2-14 c f (4 f g+7 e h) d+33 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+10 e h) c^2-2 d e (31 f g+24 e h) c+80 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}+\frac {\frac {2 d \sqrt {e+f x} \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+10 d e h-14 c f h) a^2+b^2 \left (8 e (19 f g+6 e h) d^2-2 c f (41 f g+74 e h) d+45 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+14 e h) c^2-8 d e (11 f g+9 e h) c+120 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \left (5 d^2 f^2 h a^3-b d f (29 d f g+36 d e h-50 c f h) a^2+b^2 \left (4 e (27 f g+8 e h) d^2-50 c f (f g+2 e h) d+25 c^2 f^2 h\right ) a-b^3 \left (f (f g+24 e h) c^2-4 d e (13 f g+12 e h) c+80 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \int \frac {d^2 f^2 (25 c f h-6 d (f g+4 e h)) a^3+b d f \left (2 e (51 f g+44 e h) d^2-5 c f (11 f g+34 e h) d+50 c^2 f^2 h\right ) a^2+b^2 \left (-64 e^2 (4 f g+e h) d^3+40 c e f (5 f g+6 e h) d^2-20 c^2 f^2 (f g+6 e h) d+5 c^3 f^3 h\right ) a+b^3 \left (f^2 (f g-6 e h) c^3+18 d e f (f g+4 e h) c^2-48 d^2 e^2 (3 f g+2 e h) c+160 d^3 e^3 g\right )-d f \left (5 d^2 f^2 h a^3-b d f (29 d f g+36 d e h-50 c f h) a^2+b^2 \left (4 e (27 f g+8 e h) d^2-50 c f (f g+2 e h) d+25 c^2 f^2 h\right ) a-b^3 \left (f (f g+24 e h) c^2-4 d e (13 f g+12 e h) c+80 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{b c-a d}\right )}{b c-a d}}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+4 d e h-9 c f h) a+b^2 (10 d e g-3 c (f g+2 e h))\right )}{2 b (b c-a d) (a+b x)^2 (c+d x)^2}-\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (21 d f g+32 d e h-44 c f h) a^2+b^2 \left (2 e (49 f g+16 e h) d^2-14 c f (4 f g+7 e h) d+33 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+10 e h) c^2-2 d e (31 f g+24 e h) c+80 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}+\frac {\frac {2 d \sqrt {e+f x} \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+10 d e h-14 c f h) a^2+b^2 \left (8 e (19 f g+6 e h) d^2-2 c f (41 f g+74 e h) d+45 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+14 e h) c^2-8 d e (11 f g+9 e h) c+120 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \left (5 d^2 f^2 h a^3-b d f (29 d f g+36 d e h-50 c f h) a^2+b^2 \left (4 e (27 f g+8 e h) d^2-50 c f (f g+2 e h) d+25 c^2 f^2 h\right ) a-b^3 \left (f (f g+24 e h) c^2-4 d e (13 f g+12 e h) c+80 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \left (\frac {\left (5 d^3 f^3 h a^4-5 b d^2 f^2 (7 d f g+12 d e h-15 c f h) a^3+15 b^2 d f \left (2 e (7 f g+4 e h) d^2-c f (7 f g+18 e h) d+5 c^2 f^2 h\right ) a^2+b^3 \left (-16 e^2 (21 f g+4 e h) d^3+36 c e f (7 f g+8 e h) d^2-3 c^2 f^2 (7 f g+48 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (f^2 (f g-6 e h) c^3+18 d e f (f g+4 e h) c^2-48 d^2 e^2 (3 f g+2 e h) c+160 d^3 e^3 g\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}-\frac {2 d (b e-a f) \left (\left (-15 f^2 h c^3+5 d f (7 f g+12 e h) c^2-16 d^2 e (7 f g+3 e h) c+80 d^3 e^2 g\right ) b^2-2 a d \left (8 e (3 f g+2 e h) d^2-c f (21 f g+44 e h) d+25 c^2 f^2 h\right ) b+3 a^2 d^2 f (d f g+4 d e h-5 c f h)\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}\right )}{b c-a d}\right )}{b c-a d}}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+4 d e h-9 c f h) a+b^2 (10 d e g-3 c (f g+2 e h))\right )}{2 b (b c-a d) (a+b x)^2 (c+d x)^2}-\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (21 d f g+32 d e h-44 c f h) a^2+b^2 \left (2 e (49 f g+16 e h) d^2-14 c f (4 f g+7 e h) d+33 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+10 e h) c^2-2 d e (31 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 (5 d^2 f^2 h a^3-5 b d f (7 d f g+10 d e h-14 c f h) a^2+b^2 \left (8 e (19 f g+6 e h) d^2-2 c f (41 f g+74 e h) d+45 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+14 e h) c^2-8 d e (11 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 \left (5 d^2 f^2 h a^3-b d f (29 d f g+36 d e h-50 c f h) a^2+b^2 \left (4 e (27 f g+8 e h) d^2-50 c f (f g+2 e h) d+25 c^2 f^2 h\right ) a-b^3 \left (f (f g+24 e h) c^2-4 d e (13 f g+12 e h) c+80 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \left (\frac {2 \left (5 d^3 f^3 h a^4-5 b d^2 f^2 (7 d f g+12 d e h-15 c f h) a^3+15 b^2 d f \left (2 e (7 f g+4 e h) d^2-c f (7 f g+18 e h) d+5 c^2 f^2 h\right ) a^2+b^3 \left (-16 e^2 (21 f g+4 e h) d^3+36 c e f (7 f g+8 e h) d^2-3 c^2 f^2 (7 f g+48 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (f^2 (f g-6 e h) c^3+18 d e f (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 (b e-a f) \left (\left (-15 f^2 h c^3+5 d f (7 f g+12 e h) c^2-16 d^2 e (7 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 (21 f g+44 e h) d+25 c^2 f^2 h\right ) b+3 a^2 d^2 f (d f g+4 d e h-5 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}\right )}{b c-a d}\right )}{b c-a d}}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {(b g-a h) (e+f x)^{3/2}}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+4 d e h-9 c f h) a+b^2 (10 d e g-3 c (f g+2 e h))\right )}{2 b (b c-a d) (a+b x)^2 (c+d x)^2}-\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (21 d f g+32 d e h-44 c f h) a^2+b^2 \left (2 e (49 f g+16 e h) d^2-14 c f (4 f g+7 e h) d+33 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+10 e h) c^2-2 d e (31 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 (5 d^2 f^2 h a^3-5 b d f (7 d f g+10 d e h-14 c f h) a^2+b^2 \left (8 e (19 f g+6 e h) d^2-2 c f (41 f g+74 e h) d+45 c^2 f^2 h\right ) a-b^3 \left (3 f (f g+14 e h) c^2-8 d e (11 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 \left (5 d^2 f^2 h a^3-b d f (29 d f g+36 d e h-50 c f h) a^2+b^2 \left (4 e (27 f g+8 e h) d^2-50 c f (f g+2 e h) d+25 c^2 f^2 h\right ) a-b^3 \left (f (f g+24 e h) c^2-4 d e (13 f g+12 e h) c+80 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \left (\frac {4 \sqrt {d} (b e-a f) \left (\left (-15 f^2 h c^3+5 d f (7 f g+12 e h) c^2-16 d^2 e (7 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 (21 f g+44 e h) d+25 c^2 f^2 h\right ) b+3 a^2 d^2 f (d f g+4 d e h-5 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 \left (5 d^3 f^3 h a^4-5 b d^2 f^2 (7 d f g+12 d e h-15 c f h) a^3+15 b^2 d f \left (2 e (7 f g+4 e h) d^2-c f (7 f g+18 e h) d+5 c^2 f^2 h\right ) a^2+b^3 \left (-16 e^2 (21 f g+4 e h) d^3+36 c e f (7 f g+8 e h) d^2-3 c^2 f^2 (7 f g+48 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (f^2 (f g-6 e h) c^3+18 d e f (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 )}{\sqrt {b} (b c-a d) \sqrt {b e-a f}}\right )}{b c-a d}\right )}{b c-a d}}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}}{6 b (b c-a d)}\)

Input:

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

Output:

-1/3*((b*g - a*h)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(a + b*x)^3*(c + d*x)^2) 
 - (-1/2*((a^2*d*f*h - a*b*(7*d*f*g + 4*d*e*h - 9*c*f*h) + b^2*(10*d*e*g - 
 3*c*(f*g + 2*e*h)))*Sqrt[e + f*x])/(b*(b*c - a*d)*(a + b*x)^2*(c + d*x)^2 
) - (((3*a^3*d^2*f^2*h - a^2*b*d*f*(21*d*f*g + 32*d*e*h - 44*c*f*h) + a*b^ 
2*(33*c^2*f^2*h - 14*c*d*f*(4*f*g + 7*e*h) + 2*d^2*e*(49*f*g + 16*e*h)) - 
b^3*(80*d^2*e^2*g + 3*c^2*f*(f*g + 10*e*h) - 2*c*d*e*(31*f*g + 24*e*h)))*S 
qrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x)^2) + ((2*d*(5*a 
^3*d^2*f^2*h - 5*a^2*b*d*f*(7*d*f*g + 10*d*e*h - 14*c*f*h) - b^3*(120*d^2* 
e^2*g - 8*c*d*e*(11*f*g + 9*e*h) + 3*c^2*f*(f*g + 14*e*h)) + a*b^2*(45*c^2 
*f^2*h + 8*d^2*e*(19*f*g + 6*e*h) - 2*c*d*f*(41*f*g + 74*e*h)))*Sqrt[e + f 
*x])/((b*c - a*d)*(c + d*x)^2) + (3*b*((2*d*(5*a^3*d^2*f^2*h - a^2*b*d*f*( 
29*d*f*g + 36*d*e*h - 50*c*f*h) + a*b^2*(25*c^2*f^2*h - 50*c*d*f*(f*g + 2* 
e*h) + 4*d^2*e*(27*f*g + 8*e*h)) - b^3*(80*d^2*e^2*g - 4*c*d*e*(13*f*g + 1 
2*e*h) + c^2*f*(f*g + 24*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(c + d*x)) - ( 
b*((-2*(5*a^4*d^3*f^3*h - 5*a^3*b*d^2*f^2*(7*d*f*g + 12*d*e*h - 15*c*f*h) 
+ b^4*(160*d^3*e^3*g + c^3*f^2*(f*g - 6*e*h) - 48*c*d^2*e^2*(3*f*g + 2*e*h 
) + 18*c^2*d*e*f*(f*g + 4*e*h)) + 15*a^2*b^2*d*f*(5*c^2*f^2*h + 2*d^2*e*(7 
*f*g + 4*e*h) - c*d*f*(7*f*g + 18*e*h)) + a*b^3*(5*c^3*f^3*h - 16*d^3*e^2* 
(21*f*g + 4*e*h) + 36*c*d^2*e*f*(7*f*g + 8*e*h) - 3*c^2*d*f^2*(7*f*g + 48* 
e*h)))*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(Sqrt[b]*(b*c ...
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 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 = 52.52 (sec) , antiderivative size = 1520, normalized size of antiderivative = 1.37

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(1520\)
derivativedivides	\(\text {Expression too large to display}\)	\(1973\)
default	\(\text {Expression too large to display}\)	\(1973\)

Input:

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

Output:

5/8*(((c*f-d*e)*d)^(1/2)*(d*x+c)^2*(b*x+a)^3*((32*d^3*e^3*g-96/5*c*(e*h+3/ 
2*f*g)*e^2*d^2+72/5*c^2*(e*h+1/4*f*g)*f*e*d-6/5*c^3*f^2*(e*h-1/6*f*g))*b^4 
+a*((-64/5*e^3*h-336/5*g*f*e^2)*d^3+288/5*c*(e*h+7/8*f*g)*f*e*d^2-144/5*(e 
*h+7/48*f*g)*c^2*f^2*d+c^3*f^3*h)*b^3+15*a^2*((8/5*e^2*h+14/5*e*f*g)*d^2-1 
8/5*(e*h+7/18*f*g)*c*f*d+c^2*f^2*h)*d*f*b^2+15*a^3*((-4/5*e*h-7/15*f*g)*d+ 
c*f*h)*d^2*f^2*b+a^4*d^3*f^3*h)*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2) 
)+5*(-6/5*d*(d*x+c)^2*(b*x+a)^3*((-16/3*d^3*e^2*g+16/5*c*(e*h+7/3*f*g)*e*d 
^2-4*c^2*f*(e*h+7/12*f*g)*d+c^3*f^2*h)*b^2+10/3*a*d*((16/25*e^2*h+24/25*e* 
f*g)*d^2-44/25*(e*h+21/44*f*g)*c*f*d+c^2*f^2*h)*b+a^2*d^2*((-4/5*e*h-1/5*f 
*g)*d+c*f*h)*f)*(a*f-b*e)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))+((-1 
6/5*d^4*e^2*g*x^4-24/5*x^3*c*((-2/5*e*h-13/30*f*g)*x+g*e)*e*d^3-16/15*x^2* 
c^2*((9/10*e*f*h+3/80*f^2*g)*x^2+(-27/10*e^2*h-61/20*e*f*g)*x+e^2*g)*d^2+4 
/15*x*c^3*((-57/10*e*f*h-3/10*f^2*g)*x^2+(12/5*e^2*h+31/10*e*f*g)*x+e^2*g) 
*d-8/75*c^4*((15/4*e*f*h+3/8*f^2*g)*x^2+(3/2*e^2*h+7/4*e*f*g)*x+e^2*g))*b^ 
5-4/75*a*(((-24*e^2*h-81*e*f*g)*x^4+150*e^2*x^3*g)*d^4+230*x^2*c*((15/46*e 
*f*h+15/92*f^2*g)*x^2+(-63/115*e^2*h-107/115*e*f*g)*x+e^2*g)*d^3+55*x*c^2* 
(-15/44*f^2*h*x^3+(311/110*e*f*h+119/110*f^2*g)*x^2+(-146/55*e^2*h-353/110 
*e*f*g)*x+e^2*g)*d^2-13*c^3*(30/13*f^2*h*x^3+(-95/13*e*f*h-17/13*f^2*g)*x^ 
2+(31/13*e^2*h+35/13*e*f*g)*x+e^2*g)*d+c^4*(-33/4*f^2*h*x^2+(11/2*e*f*h-2* 
f^2*g)*x+e*(e*h-1/2*f*g)))*b^4-8/75*a^2*(((27/2*e*f*h+87/8*f^2*g)*x^4+(...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(3/2)*(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. 2698 vs. \(2 (1064) = 2128\).

Time = 0.36 (sec) , antiderivative size = 2698, normalized size of antiderivative = 2.44 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*(160*b^4*d^3*e^3*g - 144*b^4*c*d^2*e^2*f*g - 336*a*b^3*d^3*e^2*f*g + 
18*b^4*c^2*d*e*f^2*g + 252*a*b^3*c*d^2*e*f^2*g + 210*a^2*b^2*d^3*e*f^2*g + 
 b^4*c^3*f^3*g - 21*a*b^3*c^2*d*f^3*g - 105*a^2*b^2*c*d^2*f^3*g - 35*a^3*b 
*d^3*f^3*g - 96*b^4*c*d^2*e^3*h - 64*a*b^3*d^3*e^3*h + 72*b^4*c^2*d*e^2*f* 
h + 288*a*b^3*c*d^2*e^2*f*h + 120*a^2*b^2*d^3*e^2*f*h - 6*b^4*c^3*e*f^2*h 
- 144*a*b^3*c^2*d*e*f^2*h - 270*a^2*b^2*c*d^2*e*f^2*h - 60*a^3*b*d^3*e*f^2 
*h + 5*a*b^3*c^3*f^3*h + 75*a^2*b^2*c^2*d*f^3*h + 75*a^3*b*c*d^2*f^3*h + 5 
*a^4*d^3*f^3*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^7*c^6*e - 
 6*a*b^6*c^5*d*e + 15*a^2*b^5*c^4*d^2*e - 20*a^3*b^4*c^3*d^3*e + 15*a^4*b^ 
3*c^2*d^4*e - 6*a^5*b^2*c*d^5*e + a^6*b*d^6*e - a*b^6*c^6*f + 6*a^2*b^5*c^ 
5*d*f - 15*a^3*b^4*c^4*d^2*f + 20*a^4*b^3*c^3*d^3*f - 15*a^5*b^2*c^2*d^4*f 
 + 6*a^6*b*c*d^5*f - a^7*d^6*f)*sqrt(-b^2*e + a*b*f)) + 1/4*(80*b^2*d^4*e^ 
2*g - 112*b^2*c*d^3*e*f*g - 48*a*b*d^4*e*f*g + 35*b^2*c^2*d^2*f^2*g + 42*a 
*b*c*d^3*f^2*g + 3*a^2*d^4*f^2*g - 48*b^2*c*d^3*e^2*h - 32*a*b*d^4*e^2*h + 
 60*b^2*c^2*d^2*e*f*h + 88*a*b*c*d^3*e*f*h + 12*a^2*d^4*e*f*h - 15*b^2*c^3 
*d*f^2*h - 50*a*b*c^2*d^2*f^2*h - 15*a^2*c*d^3*f^2*h)*arctan(sqrt(f*x + e) 
*d/sqrt(-d^2*e + c*d*f))/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 
20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*sqrt(-d 
^2*e + c*d*f)) - 1/4*(16*(f*x + e)^(3/2)*b*d^4*e*f*g - 16*sqrt(f*x + e)*b* 
d^4*e^2*f*g - 11*(f*x + e)^(3/2)*b*c*d^3*f^2*g - 5*(f*x + e)^(3/2)*a*d^...
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [B] (verification not implemented)

Time = 98.25 (sec) , antiderivative size = 36506, normalized size of antiderivative = 32.95 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^4 (c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(15*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e 
)))*a**7*c**3*d**3*f**4*h - 15*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x) 
*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*c**2*d**4*e*f**3*h + 30*sqrt(b)*sqrt(a 
*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*c**2*d**4 
*f**4*h*x - 30*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqr 
t(a*f - b*e)))*a**7*c*d**5*e*f**3*h*x + 15*sqrt(b)*sqrt(a*f - b*e)*atan((s 
qrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*c*d**5*f**4*h*x**2 - 15*sq 
rt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a* 
*7*d**6*e*f**3*h*x**2 + 225*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b) 
/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**4*d**2*f**4*h - 405*sqrt(b)*sqrt(a*f 
 - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**3*d**3 
*e*f**3*h - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sq 
rt(a*f - b*e)))*a**6*b*c**3*d**3*f**4*g + 495*sqrt(b)*sqrt(a*f - b*e)*atan 
((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**3*d**3*f**4*h*x + 
180*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e 
)))*a**6*b*c**2*d**4*e**2*f**2*h + 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt( 
e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**2*d**4*e*f**3*g - 855*sqr 
t(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a** 
6*b*c**2*d**4*e*f**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**6*b*c**2*d**4*f**4*g*x + 315*sqrt(b)*...