\(\int \frac {(a+b x)^3 (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx\) [141]

Optimal result

Integrand size = 29, antiderivative size = 390 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx=\frac {2 b^2 (3 a d f h+b (d f g-d e h-3 c f h)) \sqrt {e+f x}}{d^4 f^2}+\frac {(b c-a d)^3 (d g-c h) \sqrt {e+f x}}{2 d^4 (d e-c f) (c+d x)^2}+\frac {(b c-a d)^2 \left (a d (3 d f g-4 d e h+c f h)-b \left (12 d^2 e g+13 c^2 f h-c d (9 f g+16 e h)\right )\right ) \sqrt {e+f x}}{4 d^4 (d e-c f)^2 (c+d x)}+\frac {2 b^3 h (e+f x)^{3/2}}{3 d^3 f^2}+\frac {(b c-a d) \left (a^2 d^2 f (3 d f g-4 d e h+c f h)+2 a b d \left (5 c^2 f^2 h+c d f (3 f g-14 e h)-6 d^2 e (f g-2 e h)\right )+b^2 \left (24 d^3 e^2 g-35 c^3 f^2 h-12 c d^2 e (3 f g+4 e h)+5 c^2 d f (3 f g+16 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 d^{9/2} (d e-c f)^{5/2}} \] Output:

2*b^2*(3*a*d*f*h+b*(-3*c*f*h-d*e*h+d*f*g))*(f*x+e)^(1/2)/d^4/f^2+1/2*(-a*d 
+b*c)^3*(-c*h+d*g)*(f*x+e)^(1/2)/d^4/(-c*f+d*e)/(d*x+c)^2+1/4*(-a*d+b*c)^2 
*(a*d*(c*f*h-4*d*e*h+3*d*f*g)-b*(12*d^2*e*g+13*c^2*f*h-c*d*(16*e*h+9*f*g)) 
)*(f*x+e)^(1/2)/d^4/(-c*f+d*e)^2/(d*x+c)+2/3*b^3*h*(f*x+e)^(3/2)/d^3/f^2+1 
/4*(-a*d+b*c)*(a^2*d^2*f*(c*f*h-4*d*e*h+3*d*f*g)+2*a*b*d*(5*c^2*f^2*h+c*d* 
f*(-14*e*h+3*f*g)-6*d^2*e*(-2*e*h+f*g))+b^2*(24*d^3*e^2*g-35*c^3*f^2*h-12* 
c*d^2*e*(4*e*h+3*f*g)+5*c^2*d*f*(16*e*h+3*f*g)))*arctanh(d^(1/2)*(f*x+e)^( 
1/2)/(-c*f+d*e)^(1/2))/d^(9/2)/(-c*f+d*e)^(5/2)
 

Mathematica [A] (verified)

Time = 1.62 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.65 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx=-\frac {\sqrt {e+f x} \left (3 a^3 d^3 f^2 \left (c^2 f h+d^2 (2 e g-3 f g x+4 e h x)-c d (5 f g-2 e h+f h x)\right )+9 a^2 b d^2 f^2 \left (3 c^3 f h+4 d^3 e g x+c d^2 (2 e g-f g x-8 e h x)+c^2 d (f g-6 e h+5 f h x)\right )-9 a b^2 d f \left (15 c^4 f^2 h+8 d^4 e^2 h x^2+c^3 d f (-3 f g-26 e h+25 f h x)+8 c d^3 e x (2 e h+f (g-2 h x))+c^2 d^2 \left (8 e^2 h+e f (6 g-44 h x)+f^2 x (-5 g+8 h x)\right )\right )+b^3 \left (105 c^5 f^3 h-5 c^4 d f^2 (9 f g+34 e h-35 f h x)-8 d^5 e^2 x^2 (3 f g-2 e h+f h x)+4 c^2 d^3 \left (4 e^3 h+3 e f^2 x (11 g-8 h x)-6 e^2 f (g-3 h x)-2 f^3 x^2 (3 g+h x)\right )+8 c d^4 e x \left (4 e^2 h+2 f^2 x (3 g+h x)+e f (-6 g+3 h x)\right )+c^3 d^2 f \left (40 e^2 h+6 e f (13 g-48 h x)+f^2 x (-75 g+56 h x)\right )\right )\right )}{12 d^4 f^2 (d e-c f)^2 (c+d x)^2}+\frac {(-b c+a d) \left (a^2 d^2 f (3 d f g-4 d e h+c f h)+2 a b d \left (5 c^2 f^2 h+c d f (3 f g-14 e h)+6 d^2 e (-f g+2 e h)\right )+b^2 \left (24 d^3 e^2 g-35 c^3 f^2 h-12 c d^2 e (3 f g+4 e h)+5 c^2 d f (3 f g+16 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{4 d^{9/2} (-d e+c f)^{5/2}} \] Input:

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

Output:

-1/12*(Sqrt[e + f*x]*(3*a^3*d^3*f^2*(c^2*f*h + d^2*(2*e*g - 3*f*g*x + 4*e* 
h*x) - c*d*(5*f*g - 2*e*h + f*h*x)) + 9*a^2*b*d^2*f^2*(3*c^3*f*h + 4*d^3*e 
*g*x + c*d^2*(2*e*g - f*g*x - 8*e*h*x) + c^2*d*(f*g - 6*e*h + 5*f*h*x)) - 
9*a*b^2*d*f*(15*c^4*f^2*h + 8*d^4*e^2*h*x^2 + c^3*d*f*(-3*f*g - 26*e*h + 2 
5*f*h*x) + 8*c*d^3*e*x*(2*e*h + f*(g - 2*h*x)) + c^2*d^2*(8*e^2*h + e*f*(6 
*g - 44*h*x) + f^2*x*(-5*g + 8*h*x))) + b^3*(105*c^5*f^3*h - 5*c^4*d*f^2*( 
9*f*g + 34*e*h - 35*f*h*x) - 8*d^5*e^2*x^2*(3*f*g - 2*e*h + f*h*x) + 4*c^2 
*d^3*(4*e^3*h + 3*e*f^2*x*(11*g - 8*h*x) - 6*e^2*f*(g - 3*h*x) - 2*f^3*x^2 
*(3*g + h*x)) + 8*c*d^4*e*x*(4*e^2*h + 2*f^2*x*(3*g + h*x) + e*f*(-6*g + 3 
*h*x)) + c^3*d^2*f*(40*e^2*h + 6*e*f*(13*g - 48*h*x) + f^2*x*(-75*g + 56*h 
*x)))))/(d^4*f^2*(d*e - c*f)^2*(c + d*x)^2) + ((-(b*c) + a*d)*(a^2*d^2*f*( 
3*d*f*g - 4*d*e*h + c*f*h) + 2*a*b*d*(5*c^2*f^2*h + c*d*f*(3*f*g - 14*e*h) 
 + 6*d^2*e*(-(f*g) + 2*e*h)) + b^2*(24*d^3*e^2*g - 35*c^3*f^2*h - 12*c*d^2 
*e*(3*f*g + 4*e*h) + 5*c^2*d*f*(3*f*g + 16*e*h)))*ArcTan[(Sqrt[d]*Sqrt[e + 
 f*x])/Sqrt[-(d*e) + c*f]])/(4*d^(9/2)*(-(d*e) + c*f)^(5/2))
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.47, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {166, 27, 166, 27, 164, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\frac {\int -\frac {(a+b x) \left (2 a d (b e-a f) (3 d f g-4 d e h+c f h)+(4 b e+a f) \left (a d (3 d f g-4 d e h+c f h)-b \left (7 f h c^2-d (3 f g+10 e h) c+6 d^2 e g\right )\right )+b \left (3 a d f (3 d f g-4 d e h+c f h)-b \left (8 e (3 f g+e h) d^2-c f (15 f g+52 e h) d+35 c^2 f^2 h\right )\right ) x\right )}{2 (c+d x) \sqrt {e+f x}}dx}{d (d e-c f)}+\frac {(a+b x)^2 \sqrt {e+f x} \left (a d (c f h-4 d e h+3 d f g)-b \left (7 c^2 f h-c d (10 e h+3 f g)+6 d^2 e g\right )\right )}{d (c+d x) (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^3 \sqrt {e+f x} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {(a+b x)^2 \sqrt {e+f x} \left (a d (c f h-4 d e h+3 d f g)-b \left (7 c^2 f h-c d (10 e h+3 f g)+6 d^2 e g\right )\right )}{d (c+d x) (d e-c f)}-\frac {\int \frac {(a+b x) \left (2 a d (b e-a f) (3 d f g-4 d e h+c f h)+(4 b e+a f) \left (a d (3 d f g-4 d e h+c f h)-b \left (7 f h c^2-d (3 f g+10 e h) c+6 d^2 e g\right )\right )+b \left (3 a d f (3 d f g-4 d e h+c f h)-b \left (8 e (3 f g+e h) d^2-c f (15 f g+52 e h) d+35 c^2 f^2 h\right )\right ) x\right )}{(c+d x) \sqrt {e+f x}}dx}{2 d (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^3 \sqrt {e+f x} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {\frac {(a+b x)^2 \sqrt {e+f x} \left (a d (c f h-4 d e h+3 d f g)-b \left (7 c^2 f h-c d (10 e h+3 f g)+6 d^2 e g\right )\right )}{d (c+d x) (d e-c f)}-\frac {\frac {(b c-a d) \left (a^2 d^2 f (c f h-4 d e h+3 d f g)+2 a b d \left (5 c^2 f^2 h+c d f (3 f g-14 e h)-6 d^2 e (f g-2 e h)\right )+b^2 \left (-35 c^3 f^2 h+5 c^2 d f (16 e h+3 f g)-12 c d^2 e (4 e h+3 f g)+24 d^3 e^2 g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{d^2}+\frac {2 b \sqrt {e+f x} \left (6 a^2 d^2 f^2 (c f h-4 d e h+3 d f g)+b d f x \left (3 a d f (c f h-4 d e h+3 d f g)-b \left (35 c^2 f^2 h-c d f (52 e h+15 f g)+8 d^2 e (e h+3 f g)\right )\right )-9 a b d f \left (15 c^2 f^2 h-c d f (26 e h+3 f g)+2 d^2 e (4 e h+3 f g)\right )+b^2 \left (105 c^3 f^3 h-5 c^2 d f^2 (34 e h+9 f g)+2 c d^2 e f (20 e h+39 f g)-8 d^3 e^2 (3 f g-2 e h)\right )\right )}{3 d^2 f^2}}{2 d (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^3 \sqrt {e+f x} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {(a+b x)^2 \sqrt {e+f x} \left (a d (c f h-4 d e h+3 d f g)-b \left (7 c^2 f h-c d (10 e h+3 f g)+6 d^2 e g\right )\right )}{d (c+d x) (d e-c f)}-\frac {\frac {2 (b c-a d) \left (a^2 d^2 f (c f h-4 d e h+3 d f g)+2 a b d \left (5 c^2 f^2 h+c d f (3 f g-14 e h)-6 d^2 e (f g-2 e h)\right )+b^2 \left (-35 c^3 f^2 h+5 c^2 d f (16 e h+3 f g)-12 c d^2 e (4 e h+3 f g)+24 d^3 e^2 g\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{d^2 f}+\frac {2 b \sqrt {e+f x} \left (6 a^2 d^2 f^2 (c f h-4 d e h+3 d f g)+b d f x \left (3 a d f (c f h-4 d e h+3 d f g)-b \left (35 c^2 f^2 h-c d f (52 e h+15 f g)+8 d^2 e (e h+3 f g)\right )\right )-9 a b d f \left (15 c^2 f^2 h-c d f (26 e h+3 f g)+2 d^2 e (4 e h+3 f g)\right )+b^2 \left (105 c^3 f^3 h-5 c^2 d f^2 (34 e h+9 f g)+2 c d^2 e f (20 e h+39 f g)-8 d^3 e^2 (3 f g-2 e h)\right )\right )}{3 d^2 f^2}}{2 d (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^3 \sqrt {e+f x} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {(a+b x)^2 \sqrt {e+f x} \left (a d (c f h-4 d e h+3 d f g)-b \left (7 c^2 f h-c d (10 e h+3 f g)+6 d^2 e g\right )\right )}{d (c+d x) (d e-c f)}-\frac {\frac {2 b \sqrt {e+f x} \left (6 a^2 d^2 f^2 (c f h-4 d e h+3 d f g)+b d f x \left (3 a d f (c f h-4 d e h+3 d f g)-b \left (35 c^2 f^2 h-c d f (52 e h+15 f g)+8 d^2 e (e h+3 f g)\right )\right )-9 a b d f \left (15 c^2 f^2 h-c d f (26 e h+3 f g)+2 d^2 e (4 e h+3 f g)\right )+b^2 \left (105 c^3 f^3 h-5 c^2 d f^2 (34 e h+9 f g)+2 c d^2 e f (20 e h+39 f g)-8 d^3 e^2 (3 f g-2 e h)\right )\right )}{3 d^2 f^2}-\frac {2 (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) \left (a^2 d^2 f (c f h-4 d e h+3 d f g)+2 a b d \left (5 c^2 f^2 h+c d f (3 f g-14 e h)-6 d^2 e (f g-2 e h)\right )+b^2 \left (-35 c^3 f^2 h+5 c^2 d f (16 e h+3 f g)-12 c d^2 e (4 e h+3 f g)+24 d^3 e^2 g\right )\right )}{d^{5/2} \sqrt {d e-c f}}}{2 d (d e-c f)}}{4 d (d e-c f)}-\frac {(a+b x)^3 \sqrt {e+f x} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

Input:

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

Output:

-1/2*((d*g - c*h)*(a + b*x)^3*Sqrt[e + f*x])/(d*(d*e - c*f)*(c + d*x)^2) + 
 (((a*d*(3*d*f*g - 4*d*e*h + c*f*h) - b*(6*d^2*e*g + 7*c^2*f*h - c*d*(3*f* 
g + 10*e*h)))*(a + b*x)^2*Sqrt[e + f*x])/(d*(d*e - c*f)*(c + d*x)) - ((2*b 
*Sqrt[e + f*x]*(6*a^2*d^2*f^2*(3*d*f*g - 4*d*e*h + c*f*h) - 9*a*b*d*f*(15* 
c^2*f^2*h + 2*d^2*e*(3*f*g + 4*e*h) - c*d*f*(3*f*g + 26*e*h)) + b^2*(105*c 
^3*f^3*h - 8*d^3*e^2*(3*f*g - 2*e*h) + 2*c*d^2*e*f*(39*f*g + 20*e*h) - 5*c 
^2*d*f^2*(9*f*g + 34*e*h)) + b*d*f*(3*a*d*f*(3*d*f*g - 4*d*e*h + c*f*h) - 
b*(35*c^2*f^2*h + 8*d^2*e*(3*f*g + e*h) - c*d*f*(15*f*g + 52*e*h)))*x))/(3 
*d^2*f^2) - (2*(b*c - a*d)*(a^2*d^2*f*(3*d*f*g - 4*d*e*h + c*f*h) + 2*a*b* 
d*(5*c^2*f^2*h + c*d*f*(3*f*g - 14*e*h) - 6*d^2*e*(f*g - 2*e*h)) + b^2*(24 
*d^3*e^2*g - 35*c^3*f^2*h - 12*c*d^2*e*(3*f*g + 4*e*h) + 5*c^2*d*f*(3*f*g 
+ 16*e*h)))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(d^(5/2)*Sqr 
t[d*e - c*f]))/(2*d*(d*e - c*f)))/(4*d*(d*e - c*f))
 

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 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.51

method	result	size
risch	\(\frac {2 b^{2} \left (h b d f x +9 a d f h -9 b c f h -2 b d e h +3 b g d f \right ) \sqrt {f x +e}}{3 f^{2} d^{4}}+\frac {\left (2 a d -2 b c \right ) \left (\frac {\frac {d f \left (a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h +3 a^{2} d^{3} f g -14 a b \,c^{2} d f h +20 a b c \,d^{2} e h +6 a b c \,d^{2} f g -12 a b \,d^{3} e g +13 b^{2} c^{3} f h -16 b^{2} c^{2} d e h -9 b^{2} c^{2} d f g +12 b^{2} c \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 c^{2} f^{2}-16 c d e f +8 d^{2} e^{2}}-\frac {\left (a^{2} c \,d^{2} f h +4 a^{2} d^{3} e h -5 a^{2} d^{3} f g +10 a b \,c^{2} d f h -20 a b c \,d^{2} e h -2 a b c \,d^{2} f g +12 a b \,d^{3} e g -11 b^{2} c^{3} f h +16 b^{2} c^{2} d e h +7 b^{2} c^{2} d f g -12 b^{2} c \,d^{2} e g \right ) f \sqrt {f x +e}}{8 \left (c f -d e \right )}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (a^{2} c \,d^{2} f^{2} h -4 a^{2} d^{3} e f h +3 a^{2} d^{3} f^{2} g +10 a b \,c^{2} d \,f^{2} h -28 a b c \,d^{2} e f h +6 a b c \,d^{2} f^{2} g +24 e^{2} h b a \,d^{3}-12 a b \,d^{3} e f g -35 b^{2} c^{3} f^{2} h +80 b^{2} c^{2} d e f h +15 b^{2} c^{2} d \,f^{2} g -48 b^{2} c \,d^{2} e^{2} h -36 b^{2} c \,d^{2} e f g +24 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {\left (c f -d e \right ) d}}\right )}{d^{4}}\)	\(589\)
pseudoelliptic	\(-\frac {-\left (a d -b c \right ) \left (x d +c \right )^{2} f^{2} \left (\left (3 a^{2} f^{2} g -4 a e \left (a h +3 b g \right ) f +24 b \,e^{2} \left (a h +b g \right )\right ) d^{3}+c \left (\left (a^{2} h +6 g a b \right ) f^{2}-28 \left (a h +\frac {9 b g}{7}\right ) b e f -48 b^{2} e^{2} h \right ) d^{2}+10 \left (\left (a h +\frac {3 b g}{2}\right ) f +8 e h b \right ) c^{2} b f d -35 b^{2} c^{3} f^{2} h \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\sqrt {\left (c f -d e \right ) d}\, \left (\left (-3 a^{3} f^{3} g x +2 a^{2} \left (6 x g b +a \left (2 h x +g \right )\right ) e \,f^{2}-24 \left (\frac {\left (\frac {h x}{3}+g \right ) b}{3}+a h \right ) x^{2} b^{2} e^{2} f +\frac {16 b^{3} e^{3} h \,x^{2}}{3}\right ) d^{5}+2 c \left (-\frac {5 a^{2} \left (\frac {3 x g b}{5}+a \left (\frac {h x}{5}+g \right )\right ) f^{3}}{2}+\left (8 \left (\frac {1}{3} h \,x^{3}+g \,x^{2}\right ) b^{3}-12 a x \left (-2 h x +g \right ) b^{2}+3 a^{2} \left (-4 h x +g \right ) b +h \,a^{3}\right ) e \,f^{2}-24 \left (\frac {\left (-\frac {h x}{2}+g \right ) b}{3}+a h \right ) x \,b^{2} e^{2} f +\frac {16 b^{3} e^{3} h x}{3}\right ) d^{4}+c^{2} \left (\left (8 \left (-\frac {1}{3} h \,x^{3}-g \,x^{2}\right ) b^{3}+15 a x \left (-\frac {8 h x}{5}+g \right ) b^{2}+3 a^{2} \left (5 h x +g \right ) b +h \,a^{3}\right ) f^{3}-18 \left (-\frac {22 x \left (-\frac {8 h x}{11}+g \right ) b^{2}}{9}+a \left (-\frac {22 h x}{3}+g \right ) b +a^{2} h \right ) b e \,f^{2}-24 \left (\left (-h x +\frac {g}{3}\right ) b +a h \right ) b^{2} e^{2} f +\frac {16 b^{3} e^{3} h}{3}\right ) d^{3}+9 c^{3} b f \left (\left (-\frac {25 x \left (-\frac {56 h x}{75}+g \right ) b^{2}}{9}+a \left (-\frac {25 h x}{3}+g \right ) b +a^{2} h \right ) f^{2}+\frac {26 \left (\left (-\frac {16 h x}{13}+\frac {g}{3}\right ) b +a h \right ) b e f}{3}+\frac {40 b^{2} e^{2} h}{27}\right ) d^{2}-45 c^{4} \left (\left (\frac {\left (-\frac {35 h x}{9}+g \right ) b}{3}+a h \right ) f +\frac {34 e h b}{27}\right ) b^{2} f^{2} d +35 b^{3} c^{5} f^{3} h \right ) \sqrt {f x +e}}{4 \sqrt {\left (c f -d e \right ) d}\, \left (c f -d e \right )^{2} d^{4} \left (x d +c \right )^{2} f^{2}}\)	\(654\)
derivativedivides	\(\frac {\frac {2 b^{2} \left (\frac {d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+3 a d f h \sqrt {f x +e}-3 b c f h \sqrt {f x +e}-b d e h \sqrt {f x +e}+b d f g \sqrt {f x +e}\right )}{d^{4}}-\frac {2 f^{2} \left (\frac {-\frac {d f \left (a^{3} c \,d^{3} f h -4 a^{3} d^{4} e h +3 a^{3} d^{4} f g -15 a^{2} b \,c^{2} d^{2} f h +24 a^{2} b c \,d^{3} e h +3 a^{2} b c \,d^{3} f g -12 b e \,a^{2} d^{4} g +27 a \,b^{2} c^{3} d f h -36 h \,c^{2} e \,b^{2} a \,d^{2}-15 a \,b^{2} c^{2} d^{2} f g +24 a \,b^{2} c \,d^{3} e g -13 b^{3} c^{4} f h +16 b^{3} c^{3} d e h +9 b^{3} c^{3} d f g -12 b^{3} c^{2} d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}+\frac {\left (a^{3} c \,d^{3} f h +4 a^{3} d^{4} e h -5 a^{3} d^{4} f g +9 a^{2} b \,c^{2} d^{2} f h -24 a^{2} b c \,d^{3} e h +3 a^{2} b c \,d^{3} f g +12 b e \,a^{2} d^{4} g -21 a \,b^{2} c^{3} d f h +36 h \,c^{2} e \,b^{2} a \,d^{2}+9 a \,b^{2} c^{2} d^{2} f g -24 a \,b^{2} c \,d^{3} e g +11 b^{3} c^{4} f h -16 b^{3} c^{3} d e h -7 b^{3} c^{3} d f g +12 b^{3} c^{2} d^{2} e g \right ) f \sqrt {f x +e}}{8 c f -8 d e}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}-\frac {\left (a^{3} c \,d^{3} f^{2} h -4 a^{3} d^{4} e f h +3 a^{3} d^{4} f^{2} g +9 a^{2} b \,c^{2} d^{2} f^{2} h -24 a^{2} b c \,d^{3} e f h +3 a^{2} b c \,d^{3} f^{2} g +24 a^{2} b \,d^{4} e^{2} h -12 a^{2} b \,d^{4} e f g -45 a \,b^{2} c^{3} d \,f^{2} h +108 a \,b^{2} c^{2} d^{2} e f h +9 a \,b^{2} c^{2} d^{2} f^{2} g -72 a \,b^{2} c \,d^{3} e^{2} h -24 a \,b^{2} c \,d^{3} e f g +24 a \,b^{2} d^{4} e^{2} g +35 b^{3} c^{4} f^{2} h -80 b^{3} c^{3} d e f h -15 b^{3} c^{3} d \,f^{2} g +48 b^{3} c^{2} d^{2} e^{2} h +36 b^{3} c^{2} d^{2} e f g -24 b^{3} c \,d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {\left (c f -d e \right ) d}}\right )}{d^{4}}}{f^{2}}\)	\(841\)
default	\(\frac {\frac {2 b^{2} \left (\frac {d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+3 a d f h \sqrt {f x +e}-3 b c f h \sqrt {f x +e}-b d e h \sqrt {f x +e}+b d f g \sqrt {f x +e}\right )}{d^{4}}-\frac {2 f^{2} \left (\frac {-\frac {d f \left (a^{3} c \,d^{3} f h -4 a^{3} d^{4} e h +3 a^{3} d^{4} f g -15 a^{2} b \,c^{2} d^{2} f h +24 a^{2} b c \,d^{3} e h +3 a^{2} b c \,d^{3} f g -12 b e \,a^{2} d^{4} g +27 a \,b^{2} c^{3} d f h -36 h \,c^{2} e \,b^{2} a \,d^{2}-15 a \,b^{2} c^{2} d^{2} f g +24 a \,b^{2} c \,d^{3} e g -13 b^{3} c^{4} f h +16 b^{3} c^{3} d e h +9 b^{3} c^{3} d f g -12 b^{3} c^{2} d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}+\frac {\left (a^{3} c \,d^{3} f h +4 a^{3} d^{4} e h -5 a^{3} d^{4} f g +9 a^{2} b \,c^{2} d^{2} f h -24 a^{2} b c \,d^{3} e h +3 a^{2} b c \,d^{3} f g +12 b e \,a^{2} d^{4} g -21 a \,b^{2} c^{3} d f h +36 h \,c^{2} e \,b^{2} a \,d^{2}+9 a \,b^{2} c^{2} d^{2} f g -24 a \,b^{2} c \,d^{3} e g +11 b^{3} c^{4} f h -16 b^{3} c^{3} d e h -7 b^{3} c^{3} d f g +12 b^{3} c^{2} d^{2} e g \right ) f \sqrt {f x +e}}{8 c f -8 d e}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}-\frac {\left (a^{3} c \,d^{3} f^{2} h -4 a^{3} d^{4} e f h +3 a^{3} d^{4} f^{2} g +9 a^{2} b \,c^{2} d^{2} f^{2} h -24 a^{2} b c \,d^{3} e f h +3 a^{2} b c \,d^{3} f^{2} g +24 a^{2} b \,d^{4} e^{2} h -12 a^{2} b \,d^{4} e f g -45 a \,b^{2} c^{3} d \,f^{2} h +108 a \,b^{2} c^{2} d^{2} e f h +9 a \,b^{2} c^{2} d^{2} f^{2} g -72 a \,b^{2} c \,d^{3} e^{2} h -24 a \,b^{2} c \,d^{3} e f g +24 a \,b^{2} d^{4} e^{2} g +35 b^{3} c^{4} f^{2} h -80 b^{3} c^{3} d e f h -15 b^{3} c^{3} d \,f^{2} g +48 b^{3} c^{2} d^{2} e^{2} h +36 b^{3} c^{2} d^{2} e f g -24 b^{3} c \,d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {\left (c f -d e \right ) d}}\right )}{d^{4}}}{f^{2}}\)	\(841\)

Input:

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

Output:

2/3*b^2*(b*d*f*h*x+9*a*d*f*h-9*b*c*f*h-2*b*d*e*h+3*b*d*f*g)*(f*x+e)^(1/2)/ 
f^2/d^4+1/d^4*(2*a*d-2*b*c)*((1/8*d*f*(a^2*c*d^2*f*h-4*a^2*d^3*e*h+3*a^2*d 
^3*f*g-14*a*b*c^2*d*f*h+20*a*b*c*d^2*e*h+6*a*b*c*d^2*f*g-12*a*b*d^3*e*g+13 
*b^2*c^3*f*h-16*b^2*c^2*d*e*h-9*b^2*c^2*d*f*g+12*b^2*c*d^2*e*g)/(c^2*f^2-2 
*c*d*e*f+d^2*e^2)*(f*x+e)^(3/2)-1/8*(a^2*c*d^2*f*h+4*a^2*d^3*e*h-5*a^2*d^3 
*f*g+10*a*b*c^2*d*f*h-20*a*b*c*d^2*e*h-2*a*b*c*d^2*f*g+12*a*b*d^3*e*g-11*b 
^2*c^3*f*h+16*b^2*c^2*d*e*h+7*b^2*c^2*d*f*g-12*b^2*c*d^2*e*g)*f/(c*f-d*e)* 
(f*x+e)^(1/2))/((f*x+e)*d+c*f-d*e)^2+1/8*(a^2*c*d^2*f^2*h-4*a^2*d^3*e*f*h+ 
3*a^2*d^3*f^2*g+10*a*b*c^2*d*f^2*h-28*a*b*c*d^2*e*f*h+6*a*b*c*d^2*f^2*g+24 
*a*b*d^3*e^2*h-12*a*b*d^3*e*f*g-35*b^2*c^3*f^2*h+80*b^2*c^2*d*e*f*h+15*b^2 
*c^2*d*f^2*g-48*b^2*c*d^2*e^2*h-36*b^2*c*d^2*e*f*g+24*b^2*d^3*e^2*g)/(c^2* 
f^2-2*c*d*e*f+d^2*e^2)/((c*f-d*e)*d)^(1/2)*arctan(d*(f*x+e)^(1/2)/((c*f-d* 
e)*d)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1983 vs. \(2 (364) = 728\).

Time = 0.34 (sec) , antiderivative size = 3979, normalized size of antiderivative = 10.20 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((b*x+a)^3*(h*x+g)/(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. 1306 vs. \(2 (364) = 728\).

Time = 0.15 (sec) , antiderivative size = 1306, normalized size of antiderivative = 3.35 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/4*(24*b^3*c*d^3*e^2*g - 24*a*b^2*d^4*e^2*g - 36*b^3*c^2*d^2*e*f*g + 24* 
a*b^2*c*d^3*e*f*g + 12*a^2*b*d^4*e*f*g + 15*b^3*c^3*d*f^2*g - 9*a*b^2*c^2* 
d^2*f^2*g - 3*a^2*b*c*d^3*f^2*g - 3*a^3*d^4*f^2*g - 48*b^3*c^2*d^2*e^2*h + 
 72*a*b^2*c*d^3*e^2*h - 24*a^2*b*d^4*e^2*h + 80*b^3*c^3*d*e*f*h - 108*a*b^ 
2*c^2*d^2*e*f*h + 24*a^2*b*c*d^3*e*f*h + 4*a^3*d^4*e*f*h - 35*b^3*c^4*f^2* 
h + 45*a*b^2*c^3*d*f^2*h - 9*a^2*b*c^2*d^2*f^2*h - a^3*c*d^3*f^2*h)*arctan 
(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((d^6*e^2 - 2*c*d^5*e*f + c^2*d^4*f 
^2)*sqrt(-d^2*e + c*d*f)) - 1/4*(12*(f*x + e)^(3/2)*b^3*c^2*d^3*e*f*g - 24 
*(f*x + e)^(3/2)*a*b^2*c*d^4*e*f*g + 12*(f*x + e)^(3/2)*a^2*b*d^5*e*f*g - 
12*sqrt(f*x + e)*b^3*c^2*d^3*e^2*f*g + 24*sqrt(f*x + e)*a*b^2*c*d^4*e^2*f* 
g - 12*sqrt(f*x + e)*a^2*b*d^5*e^2*f*g - 9*(f*x + e)^(3/2)*b^3*c^3*d^2*f^2 
*g + 15*(f*x + e)^(3/2)*a*b^2*c^2*d^3*f^2*g - 3*(f*x + e)^(3/2)*a^2*b*c*d^ 
4*f^2*g - 3*(f*x + e)^(3/2)*a^3*d^5*f^2*g + 19*sqrt(f*x + e)*b^3*c^3*d^2*e 
*f^2*g - 33*sqrt(f*x + e)*a*b^2*c^2*d^3*e*f^2*g + 9*sqrt(f*x + e)*a^2*b*c* 
d^4*e*f^2*g + 5*sqrt(f*x + e)*a^3*d^5*e*f^2*g - 7*sqrt(f*x + e)*b^3*c^4*d* 
f^3*g + 9*sqrt(f*x + e)*a*b^2*c^3*d^2*f^3*g + 3*sqrt(f*x + e)*a^2*b*c^2*d^ 
3*f^3*g - 5*sqrt(f*x + e)*a^3*c*d^4*f^3*g - 16*(f*x + e)^(3/2)*b^3*c^3*d^2 
*e*f*h + 36*(f*x + e)^(3/2)*a*b^2*c^2*d^3*e*f*h - 24*(f*x + e)^(3/2)*a^2*b 
*c*d^4*e*f*h + 4*(f*x + e)^(3/2)*a^3*d^5*e*f*h + 16*sqrt(f*x + e)*b^3*c^3* 
d^2*e^2*f*h - 36*sqrt(f*x + e)*a*b^2*c^2*d^3*e^2*f*h + 24*sqrt(f*x + e)...
 

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 1242, normalized size of antiderivative = 3.18 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

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

Output:

(e + f*x)^(1/2)*((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(d^3*f^2) - (6*b^3* 
h*(c*f - d*e))/(d^4*f^2)) - (((e + f*x)^(1/2)*(11*b^3*c^4*f^2*h - 5*a^3*d^ 
4*f^2*g + 4*a^3*d^4*e*f*h + a^3*c*d^3*f^2*h - 7*b^3*c^3*d*f^2*g + 9*a*b^2* 
c^2*d^2*f^2*g + 9*a^2*b*c^2*d^2*f^2*h + 12*a^2*b*d^4*e*f*g - 16*b^3*c^3*d* 
e*f*h + 3*a^2*b*c*d^3*f^2*g - 21*a*b^2*c^3*d*f^2*h + 12*b^3*c^2*d^2*e*f*g 
+ 36*a*b^2*c^2*d^2*e*f*h - 24*a*b^2*c*d^3*e*f*g - 24*a^2*b*c*d^3*e*f*h))/( 
4*(c*f - d*e)) - ((e + f*x)^(3/2)*(3*a^3*d^5*f^2*g - 4*a^3*d^5*e*f*h + a^3 
*c*d^4*f^2*h - 13*b^3*c^4*d*f^2*h + 9*b^3*c^3*d^2*f^2*g - 15*a*b^2*c^2*d^3 
*f^2*g + 27*a*b^2*c^3*d^2*f^2*h - 15*a^2*b*c^2*d^3*f^2*h - 12*a^2*b*d^5*e* 
f*g + 3*a^2*b*c*d^4*f^2*g - 12*b^3*c^2*d^3*e*f*g + 16*b^3*c^3*d^2*e*f*h - 
36*a*b^2*c^2*d^3*e*f*h + 24*a*b^2*c*d^4*e*f*g + 24*a^2*b*c*d^4*e*f*h))/(4* 
(c*f - d*e)^2))/(d^6*(e + f*x)^2 - (e + f*x)*(2*d^6*e - 2*c*d^5*f) + d^6*e 
^2 + c^2*d^4*f^2 - 2*c*d^5*e*f) + (atan((d^(1/2)*(e + f*x)^(1/2)*(a*d - b* 
c)*(3*a^2*d^3*f^2*g + 24*b^2*d^3*e^2*g - 35*b^2*c^3*f^2*h + 24*a*b*d^3*e^2 
*h - 4*a^2*d^3*e*f*h + a^2*c*d^2*f^2*h - 48*b^2*c*d^2*e^2*h + 15*b^2*c^2*d 
*f^2*g - 12*a*b*d^3*e*f*g + 6*a*b*c*d^2*f^2*g + 10*a*b*c^2*d*f^2*h - 36*b^ 
2*c*d^2*e*f*g + 80*b^2*c^2*d*e*f*h - 28*a*b*c*d^2*e*f*h))/((c*f - d*e)^(1/ 
2)*(3*a^3*d^4*f^2*g + 35*b^3*c^4*f^2*h - 4*a^3*d^4*e*f*h + 24*a*b^2*d^4*e^ 
2*g + 24*a^2*b*d^4*e^2*h - 24*b^3*c*d^3*e^2*g + a^3*c*d^3*f^2*h - 15*b^3*c 
^3*d*f^2*g + 48*b^3*c^2*d^2*e^2*h + 9*a*b^2*c^2*d^2*f^2*g + 9*a^2*b*c^2...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 4932, normalized size of antiderivative = 12.65 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e) 
))*a**3*c**3*d**3*f**4*h - 12*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)* 
d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c**2*d**4*e*f**3*h + 9*sqrt(d)*sqrt(c*f 
 - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c**2*d**4*f 
**4*g + 6*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f 
 - d*e)))*a**3*c**2*d**4*f**4*h*x - 24*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt( 
e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c*d**5*e*f**3*h*x + 18*sqrt(d) 
*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c* 
d**5*f**4*g*x + 3*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)* 
sqrt(c*f - d*e)))*a**3*c*d**5*f**4*h*x**2 - 12*sqrt(d)*sqrt(c*f - d*e)*ata 
n((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*d**6*e*f**3*h*x**2 + 9 
*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e))) 
*a**3*d**6*f**4*g*x**2 + 27*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d) 
/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c**4*d**2*f**4*h - 72*sqrt(d)*sqrt(c*f 
- d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c**3*d**3* 
e*f**3*h + 9*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt( 
c*f - d*e)))*a**2*b*c**3*d**3*f**4*g + 54*sqrt(d)*sqrt(c*f - d*e)*atan((sq 
rt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c**3*d**3*f**4*h*x + 72*s 
qrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a 
**2*b*c**2*d**4*e**2*f**2*h - 36*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e +...