\(\int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx\) [75]

Optimal result

Integrand size = 29, antiderivative size = 422 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx=\frac {2 \left (6 a^2 d^2 f h-3 a b d (d f g+d e h+2 c f h)+b^2 \left (d^2 e g+c^2 f h+2 c d (f g+e h)\right )\right ) \sqrt {e+f x}}{b^5}-\frac {(b c-a d) \left (15 a^2 d f h+b^2 (8 d e g+3 c f g+4 c e h)-a b (11 d f g+12 d e h+7 c f h)\right ) \sqrt {e+f x}}{4 b^5 (a+b x)}+\frac {2 d (b d g+2 b c h-3 a d h) (e+f x)^{3/2}}{3 b^4}-\frac {(b c-a d)^2 (b g-a h) (e+f x)^{3/2}}{2 b^4 (a+b x)^2}+\frac {2 d^2 h (e+f x)^{5/2}}{5 b^3 f}+\frac {\left (63 a^3 d^2 f^2 h-7 a^2 b d f (5 d f g+12 d e h+10 c f h)-b^3 \left (8 d^2 e^2 g+8 c d e (3 f g+2 e h)+3 c^2 f (f g+4 e h)\right )+a b^2 \left (15 c^2 f^2 h+8 d^2 e (5 f g+3 e h)+10 c d f (3 f g+8 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{4 b^{11/2} \sqrt {b e-a f}} \] Output:

2*(6*a^2*d^2*f*h-3*a*b*d*(2*c*f*h+d*e*h+d*f*g)+b^2*(d^2*e*g+c^2*f*h+2*c*d* 
(e*h+f*g)))*(f*x+e)^(1/2)/b^5-1/4*(-a*d+b*c)*(15*a^2*d*f*h+b^2*(4*c*e*h+3* 
c*f*g+8*d*e*g)-a*b*(7*c*f*h+12*d*e*h+11*d*f*g))*(f*x+e)^(1/2)/b^5/(b*x+a)+ 
2/3*d*(-3*a*d*h+2*b*c*h+b*d*g)*(f*x+e)^(3/2)/b^4-1/2*(-a*d+b*c)^2*(-a*h+b* 
g)*(f*x+e)^(3/2)/b^4/(b*x+a)^2+2/5*d^2*h*(f*x+e)^(5/2)/b^3/f+1/4*(63*a^3*d 
^2*f^2*h-7*a^2*b*d*f*(10*c*f*h+12*d*e*h+5*d*f*g)-b^3*(8*d^2*e^2*g+8*c*d*e* 
(2*e*h+3*f*g)+3*c^2*f*(4*e*h+f*g))+a*b^2*(15*c^2*f^2*h+8*d^2*e*(3*e*h+5*f* 
g)+10*c*d*f*(8*e*h+3*f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2) 
)/b^(11/2)/(-a*f+b*e)^(1/2)
 

Mathematica [A] (verified)

Time = 1.42 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.20 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx=\frac {\sqrt {e+f x} \left (945 a^4 d^2 f^2 h-105 a^3 b d f (6 d e h+10 c f h+5 d f (g-3 h x))+b^4 \left (-15 c^2 f (f x (5 g-8 h x)+2 e (g+2 h x))+8 d^2 x^2 \left (3 e^2 h+f^2 x (5 g+3 h x)+e f (20 g+6 h x)\right )+40 c d f x (2 f x (3 g+h x)+e (-3 g+8 h x))\right )-a b^3 \left (15 c^2 f (3 f g+2 e h-25 f h x)+8 d^2 x \left (-6 e^2 h+f^2 x (35 g+9 h x)+e f (-55 g+48 h x)\right )+10 c d f (e (6 g-88 h x)+f x (-75 g+56 h x))\right )+a^2 b^2 \left (225 c^2 f^2 h+50 c d f (9 f g+10 e h-35 f h x)+d^2 \left (24 e^2 h+2 e f (125 g-546 h x)+7 f^2 x (-125 g+72 h x)\right )\right )\right )}{60 b^5 f (a+b x)^2}+\frac {\left (-63 a^3 d^2 f^2 h+7 a^2 b d f (5 d f g+12 d e h+10 c f h)+b^3 \left (8 d^2 e^2 g+8 c d e (3 f g+2 e h)+3 c^2 f (f g+4 e h)\right )-a b^2 \left (15 c^2 f^2 h+8 d^2 e (5 f g+3 e h)+10 c d f (3 f g+8 e h)\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{4 b^{11/2} \sqrt {-b e+a f}} \] Input:

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

Output:

(Sqrt[e + f*x]*(945*a^4*d^2*f^2*h - 105*a^3*b*d*f*(6*d*e*h + 10*c*f*h + 5* 
d*f*(g - 3*h*x)) + b^4*(-15*c^2*f*(f*x*(5*g - 8*h*x) + 2*e*(g + 2*h*x)) + 
8*d^2*x^2*(3*e^2*h + f^2*x*(5*g + 3*h*x) + e*f*(20*g + 6*h*x)) + 40*c*d*f* 
x*(2*f*x*(3*g + h*x) + e*(-3*g + 8*h*x))) - a*b^3*(15*c^2*f*(3*f*g + 2*e*h 
 - 25*f*h*x) + 8*d^2*x*(-6*e^2*h + f^2*x*(35*g + 9*h*x) + e*f*(-55*g + 48* 
h*x)) + 10*c*d*f*(e*(6*g - 88*h*x) + f*x*(-75*g + 56*h*x))) + a^2*b^2*(225 
*c^2*f^2*h + 50*c*d*f*(9*f*g + 10*e*h - 35*f*h*x) + d^2*(24*e^2*h + 2*e*f* 
(125*g - 546*h*x) + 7*f^2*x*(-125*g + 72*h*x)))))/(60*b^5*f*(a + b*x)^2) + 
 ((-63*a^3*d^2*f^2*h + 7*a^2*b*d*f*(5*d*f*g + 12*d*e*h + 10*c*f*h) + b^3*( 
8*d^2*e^2*g + 8*c*d*e*(3*f*g + 2*e*h) + 3*c^2*f*(f*g + 4*e*h)) - a*b^2*(15 
*c^2*f^2*h + 8*d^2*e*(5*f*g + 3*e*h) + 10*c*d*f*(3*f*g + 8*e*h)))*ArcTan[( 
Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/(4*b^(11/2)*Sqrt[-(b*e) + a*f] 
)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {166, 27, 163, 60, 60, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 163

\(\displaystyle \frac {\frac {(e+f x)^{5/2} \left (63 a^3 d^2 f^2 h-a^2 b d f (70 c f h+66 d e h+35 d f g)+a b^2 \left (25 c^2 f^2 h+30 c d f (2 e h+f g)+2 d^2 e (4 e h+15 f g)\right )+2 b d^2 x (b e-a f) (-9 a f h+4 b e h+5 b f g)-5 b^3 c f (4 c e h+c f g+4 d e g)\right )}{5 b^2 f (a+b x) (b e-a f)}-\frac {\left (63 a^3 d^2 f^2 h-7 a^2 b d f (10 c f h+12 d e h+5 d f g)+a b^2 \left (15 c^2 f^2 h+10 c d f (8 e h+3 f g)+8 d^2 e (3 e h+5 f g)\right )-b^3 \left (3 c^2 f (4 e h+f g)+8 c d e (2 e h+3 f g)+8 d^2 e^2 g\right )\right ) \int \frac {(e+f x)^{3/2}}{a+b x}dx}{2 b^2 (b e-a f)}}{4 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{2 b (a+b x)^2 (b e-a f)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {(e+f x)^{5/2} \left (63 a^3 d^2 f^2 h-a^2 b d f (70 c f h+66 d e h+35 d f g)+a b^2 \left (25 c^2 f^2 h+30 c d f (2 e h+f g)+2 d^2 e (4 e h+15 f g)\right )+2 b d^2 x (b e-a f) (-9 a f h+4 b e h+5 b f g)-5 b^3 c f (4 c e h+c f g+4 d e g)\right )}{5 b^2 f (a+b x) (b e-a f)}-\frac {\left (63 a^3 d^2 f^2 h-7 a^2 b d f (10 c f h+12 d e h+5 d f g)+a b^2 \left (15 c^2 f^2 h+10 c d f (8 e h+3 f g)+8 d^2 e (3 e h+5 f g)\right )-b^3 \left (3 c^2 f (4 e h+f g)+8 c d e (2 e h+3 f g)+8 d^2 e^2 g\right )\right ) \left (\frac {(b e-a f) \int \frac {\sqrt {e+f x}}{a+b x}dx}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right )}{2 b^2 (b e-a f)}}{4 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{2 b (a+b x)^2 (b e-a f)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {(e+f x)^{5/2} \left (63 a^3 d^2 f^2 h-a^2 b d f (70 c f h+66 d e h+35 d f g)+a b^2 \left (25 c^2 f^2 h+30 c d f (2 e h+f g)+2 d^2 e (4 e h+15 f g)\right )+2 b d^2 x (b e-a f) (-9 a f h+4 b e h+5 b f g)-5 b^3 c f (4 c e h+c f g+4 d e g)\right )}{5 b^2 f (a+b x) (b e-a f)}-\frac {\left (63 a^3 d^2 f^2 h-7 a^2 b d f (10 c f h+12 d e h+5 d f g)+a b^2 \left (15 c^2 f^2 h+10 c d f (8 e h+3 f g)+8 d^2 e (3 e h+5 f g)\right )-b^3 \left (3 c^2 f (4 e h+f g)+8 c d e (2 e h+3 f g)+8 d^2 e^2 g\right )\right ) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b}+\frac {2 \sqrt {e+f x}}{b}\right )}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right )}{2 b^2 (b e-a f)}}{4 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{2 b (a+b x)^2 (b e-a f)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {(e+f x)^{5/2} \left (63 a^3 d^2 f^2 h-a^2 b d f (70 c f h+66 d e h+35 d f g)+a b^2 \left (25 c^2 f^2 h+30 c d f (2 e h+f g)+2 d^2 e (4 e h+15 f g)\right )+2 b d^2 x (b e-a f) (-9 a f h+4 b e h+5 b f g)-5 b^3 c f (4 c e h+c f g+4 d e g)\right )}{5 b^2 f (a+b x) (b e-a f)}-\frac {\left (63 a^3 d^2 f^2 h-7 a^2 b d f (10 c f h+12 d e h+5 d f g)+a b^2 \left (15 c^2 f^2 h+10 c d f (8 e h+3 f g)+8 d^2 e (3 e h+5 f g)\right )-b^3 \left (3 c^2 f (4 e h+f g)+8 c d e (2 e h+3 f g)+8 d^2 e^2 g\right )\right ) \left (\frac {(b e-a f) \left (\frac {2 (b e-a f) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b f}+\frac {2 \sqrt {e+f x}}{b}\right )}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right )}{2 b^2 (b e-a f)}}{4 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{2 b (a+b x)^2 (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {(e+f x)^{5/2} \left (63 a^3 d^2 f^2 h-a^2 b d f (70 c f h+66 d e h+35 d f g)+a b^2 \left (25 c^2 f^2 h+30 c d f (2 e h+f g)+2 d^2 e (4 e h+15 f g)\right )+2 b d^2 x (b e-a f) (-9 a f h+4 b e h+5 b f g)-5 b^3 c f (4 c e h+c f g+4 d e g)\right )}{5 b^2 f (a+b x) (b e-a f)}-\frac {\left (\frac {(b e-a f) \left (\frac {2 \sqrt {e+f x}}{b}-\frac {2 \sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right ) \left (63 a^3 d^2 f^2 h-7 a^2 b d f (10 c f h+12 d e h+5 d f g)+a b^2 \left (15 c^2 f^2 h+10 c d f (8 e h+3 f g)+8 d^2 e (3 e h+5 f g)\right )-b^3 \left (3 c^2 f (4 e h+f g)+8 c d e (2 e h+3 f g)+8 d^2 e^2 g\right )\right )}{2 b^2 (b e-a f)}}{4 b (b e-a f)}-\frac {(c+d x)^2 (e+f x)^{5/2} (b g-a h)}{2 b (a+b x)^2 (b e-a f)}\)

Input:

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

Output:

-1/2*((b*g - a*h)*(c + d*x)^2*(e + f*x)^(5/2))/(b*(b*e - a*f)*(a + b*x)^2) 
 + (((e + f*x)^(5/2)*(63*a^3*d^2*f^2*h - 5*b^3*c*f*(4*d*e*g + c*f*g + 4*c* 
e*h) - a^2*b*d*f*(35*d*f*g + 66*d*e*h + 70*c*f*h) + a*b^2*(25*c^2*f^2*h + 
30*c*d*f*(f*g + 2*e*h) + 2*d^2*e*(15*f*g + 4*e*h)) + 2*b*d^2*(b*e - a*f)*( 
5*b*f*g + 4*b*e*h - 9*a*f*h)*x))/(5*b^2*f*(b*e - a*f)*(a + b*x)) - ((63*a^ 
3*d^2*f^2*h - 7*a^2*b*d*f*(5*d*f*g + 12*d*e*h + 10*c*f*h) - b^3*(8*d^2*e^2 
*g + 8*c*d*e*(3*f*g + 2*e*h) + 3*c^2*f*(f*g + 4*e*h)) + a*b^2*(15*c^2*f^2* 
h + 8*d^2*e*(5*f*g + 3*e*h) + 10*c*d*f*(3*f*g + 8*e*h)))*((2*(e + f*x)^(3/ 
2))/(3*b) + ((b*e - a*f)*((2*Sqrt[e + f*x])/b - (2*Sqrt[b*e - a*f]*ArcTanh 
[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/b^(3/2)))/b))/(2*b^2*(b*e - a*f 
)))/(4*b*(b*e - a*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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 163

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n 
+ 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)* 
(m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + 
d*x)^(n + 1), 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*(b*c - a*d)*(m + 1)*(m + n + 3))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, - 
1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && 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 = 0.61 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.21

method	result	size
pseudoelliptic	\(-\frac {63 \left (\left (\left (-\frac {g \,f^{2} c^{2}}{21}-\frac {4 c e \left (c h +2 d g \right ) f}{21}-\frac {16 \left (c h +\frac {d g}{2}\right ) d \,e^{2}}{63}\right ) b^{3}+\frac {5 \left (\left (h \,c^{2}+2 c d g \right ) f^{2}+\frac {16 \left (c h +\frac {d g}{2}\right ) d e f}{3}+\frac {8 d^{2} e^{2} h}{5}\right ) a \,b^{2}}{21}-\frac {10 a^{2} \left (\left (c h +\frac {d g}{2}\right ) f +\frac {6 d e h}{5}\right ) d f b}{9}+a^{3} d^{2} f^{2} h \right ) \left (b x +a \right )^{2} f \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )-\left (\left (-\frac {5 x \left (-\frac {8 x^{2} \left (\frac {3 h x}{5}+g \right ) d^{2}}{15}-\frac {16 x c \left (\frac {h x}{3}+g \right ) d}{5}+c^{2} \left (-\frac {8 h x}{5}+g \right )\right ) f^{2}}{63}-\frac {2 \left (\left (-\frac {8}{5} h \,x^{3}-\frac {16}{3} g \,x^{2}\right ) d^{2}+4 \left (-\frac {8 h x}{3}+g \right ) x c d +c^{2} \left (2 h x +g \right )\right ) e f}{63}+\frac {8 d^{2} e^{2} h \,x^{2}}{315}\right ) b^{4}-\frac {2 a \left (\left (\frac {28 x^{2} \left (\frac {9 h x}{35}+g \right ) d^{2}}{3}-25 \left (-\frac {56 h x}{75}+g \right ) x c d +\frac {3 c^{2} \left (-\frac {25 h x}{3}+g \right )}{2}\right ) f^{2}+\left (\left (\frac {64}{5} h \,x^{2}-\frac {44}{3} g x \right ) d^{2}+2 c \left (-\frac {44 h x}{3}+g \right ) d +h \,c^{2}\right ) e f -\frac {8 d^{2} e^{2} h x}{5}\right ) b^{3}}{63}+\frac {5 a^{2} \left (\left (\left (\frac {56}{25} h \,x^{2}-\frac {35}{9} g x \right ) d^{2}+2 c \left (-\frac {35 h x}{9}+g \right ) d +h \,c^{2}\right ) f^{2}+\frac {20 d \left (\left (-\frac {273 h x}{125}+\frac {g}{2}\right ) d +c h \right ) e f}{9}+\frac {8 d^{2} e^{2} h}{75}\right ) b^{2}}{21}-\frac {10 a^{3} \left (\left (\left (-\frac {3 h x}{2}+\frac {g}{2}\right ) d +c h \right ) f +\frac {3 d e h}{5}\right ) d f b}{9}+a^{4} d^{2} f^{2} h \right ) \sqrt {\left (a f -b e \right ) b}\, \sqrt {f x +e}\right )}{4 \sqrt {\left (a f -b e \right ) b}\, \left (b x +a \right )^{2} b^{5} f}\)	\(510\)
risch	\(\frac {2 \left (3 x^{2} h \,b^{2} d^{2} f^{2}-15 a b \,d^{2} f^{2} h x +10 b^{2} c d \,f^{2} h x +6 b^{2} d^{2} e f h x +5 b^{2} d^{2} f^{2} g x +90 a^{2} d^{2} f^{2} h -90 a b c d \,f^{2} h -60 a b \,d^{2} e f h -45 a b \,d^{2} f^{2} g +15 c^{2} b^{2} f^{2} h +40 b^{2} c d e f h +30 b^{2} c d \,f^{2} g +3 b^{2} d^{2} e^{2} h +20 b^{2} d^{2} e f g \right ) \sqrt {f x +e}}{15 f \,b^{5}}-\frac {\frac {2 \left (-\frac {17}{8} a^{3} b \,d^{2} f^{2} h +\frac {13}{4} a^{2} b^{2} c d \,f^{2} h +\frac {3}{2} a^{2} b^{2} d^{2} e f h +\frac {13}{8} a^{2} b^{2} d^{2} f^{2} g -\frac {9}{8} a \,b^{3} c^{2} f^{2} h -2 a \,b^{3} c d e f h -\frac {9}{4} a \,b^{3} c d \,f^{2} g -a \,b^{3} d^{2} e f g +\frac {1}{2} b^{4} c^{2} e f h +\frac {5}{8} b^{4} c^{2} f^{2} g +b^{4} c d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {f \left (15 a^{4} d^{2} f^{2} h -22 a^{3} b c d \,f^{2} h -27 a^{3} b \,d^{2} e f h -11 a^{3} b \,d^{2} f^{2} g +7 a^{2} b^{2} c^{2} f^{2} h +38 a^{2} b^{2} c d e f h +14 a^{2} b^{2} c d \,f^{2} g +12 a^{2} b^{2} d^{2} e^{2} h +19 a^{2} b^{2} d^{2} e f g -11 a \,b^{3} c^{2} e f h -3 a \,b^{3} c^{2} f^{2} g -16 a \,b^{3} c d \,e^{2} h -22 a \,b^{3} c d e f g -8 a \,b^{3} d^{2} e^{2} g +4 b^{4} c^{2} e^{2} h +3 b^{4} c^{2} e f g +8 b^{4} c d \,e^{2} g \right ) \sqrt {f x +e}}{4}}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (63 a^{3} d^{2} f^{2} h -70 a^{2} b c d \,f^{2} h -84 a^{2} b \,d^{2} e f h -35 a^{2} b \,d^{2} f^{2} g +15 a \,b^{2} c^{2} f^{2} h +80 a \,b^{2} c d e f h +30 a \,b^{2} c d \,f^{2} g +24 a \,b^{2} d^{2} e^{2} h +40 a \,b^{2} d^{2} e f g -12 b^{3} c^{2} e f h -3 b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h -24 b^{3} c d e f g -8 b^{3} d^{2} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{4 \sqrt {\left (a f -b e \right ) b}}}{b^{5}}\)	\(781\)
derivativedivides	\(\frac {\frac {2 \left (\frac {d^{2} h \left (f x +e \right )^{\frac {5}{2}} b^{2}}{5}-a b \,d^{2} f h \left (f x +e \right )^{\frac {3}{2}}+\frac {2 b^{2} c d f h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{2} d^{2} f g \left (f x +e \right )^{\frac {3}{2}}}{3}+6 a^{2} d^{2} f^{2} h \sqrt {f x +e}-6 a b c d \,f^{2} h \sqrt {f x +e}-3 a b \,d^{2} e f h \sqrt {f x +e}-3 a b \,d^{2} f^{2} g \sqrt {f x +e}+b^{2} c^{2} f^{2} h \sqrt {f x +e}+2 b^{2} c d e f h \sqrt {f x +e}+2 b^{2} c d \,f^{2} g \sqrt {f x +e}+b^{2} d^{2} e f g \sqrt {f x +e}\right )}{b^{5}}-\frac {2 f \left (\frac {\left (-\frac {17}{8} a^{3} b \,d^{2} f^{2} h +\frac {13}{4} a^{2} b^{2} c d \,f^{2} h +\frac {3}{2} a^{2} b^{2} d^{2} e f h +\frac {13}{8} a^{2} b^{2} d^{2} f^{2} g -\frac {9}{8} a \,b^{3} c^{2} f^{2} h -2 a \,b^{3} c d e f h -\frac {9}{4} a \,b^{3} c d \,f^{2} g -a \,b^{3} d^{2} e f g +\frac {1}{2} b^{4} c^{2} e f h +\frac {5}{8} b^{4} c^{2} f^{2} g +b^{4} c d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {f \left (15 a^{4} d^{2} f^{2} h -22 a^{3} b c d \,f^{2} h -27 a^{3} b \,d^{2} e f h -11 a^{3} b \,d^{2} f^{2} g +7 a^{2} b^{2} c^{2} f^{2} h +38 a^{2} b^{2} c d e f h +14 a^{2} b^{2} c d \,f^{2} g +12 a^{2} b^{2} d^{2} e^{2} h +19 a^{2} b^{2} d^{2} e f g -11 a \,b^{3} c^{2} e f h -3 a \,b^{3} c^{2} f^{2} g -16 a \,b^{3} c d \,e^{2} h -22 a \,b^{3} c d e f g -8 a \,b^{3} d^{2} e^{2} g +4 b^{4} c^{2} e^{2} h +3 b^{4} c^{2} e f g +8 b^{4} c d \,e^{2} g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (63 a^{3} d^{2} f^{2} h -70 a^{2} b c d \,f^{2} h -84 a^{2} b \,d^{2} e f h -35 a^{2} b \,d^{2} f^{2} g +15 a \,b^{2} c^{2} f^{2} h +80 a \,b^{2} c d e f h +30 a \,b^{2} c d \,f^{2} g +24 a \,b^{2} d^{2} e^{2} h +40 a \,b^{2} d^{2} e f g -12 b^{3} c^{2} e f h -3 b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h -24 b^{3} c d e f g -8 b^{3} d^{2} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 \sqrt {\left (a f -b e \right ) b}}\right )}{b^{5}}}{f}\)	\(818\)
default	\(\frac {\frac {2 \left (\frac {d^{2} h \left (f x +e \right )^{\frac {5}{2}} b^{2}}{5}-a b \,d^{2} f h \left (f x +e \right )^{\frac {3}{2}}+\frac {2 b^{2} c d f h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{2} d^{2} f g \left (f x +e \right )^{\frac {3}{2}}}{3}+6 a^{2} d^{2} f^{2} h \sqrt {f x +e}-6 a b c d \,f^{2} h \sqrt {f x +e}-3 a b \,d^{2} e f h \sqrt {f x +e}-3 a b \,d^{2} f^{2} g \sqrt {f x +e}+b^{2} c^{2} f^{2} h \sqrt {f x +e}+2 b^{2} c d e f h \sqrt {f x +e}+2 b^{2} c d \,f^{2} g \sqrt {f x +e}+b^{2} d^{2} e f g \sqrt {f x +e}\right )}{b^{5}}-\frac {2 f \left (\frac {\left (-\frac {17}{8} a^{3} b \,d^{2} f^{2} h +\frac {13}{4} a^{2} b^{2} c d \,f^{2} h +\frac {3}{2} a^{2} b^{2} d^{2} e f h +\frac {13}{8} a^{2} b^{2} d^{2} f^{2} g -\frac {9}{8} a \,b^{3} c^{2} f^{2} h -2 a \,b^{3} c d e f h -\frac {9}{4} a \,b^{3} c d \,f^{2} g -a \,b^{3} d^{2} e f g +\frac {1}{2} b^{4} c^{2} e f h +\frac {5}{8} b^{4} c^{2} f^{2} g +b^{4} c d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {f \left (15 a^{4} d^{2} f^{2} h -22 a^{3} b c d \,f^{2} h -27 a^{3} b \,d^{2} e f h -11 a^{3} b \,d^{2} f^{2} g +7 a^{2} b^{2} c^{2} f^{2} h +38 a^{2} b^{2} c d e f h +14 a^{2} b^{2} c d \,f^{2} g +12 a^{2} b^{2} d^{2} e^{2} h +19 a^{2} b^{2} d^{2} e f g -11 a \,b^{3} c^{2} e f h -3 a \,b^{3} c^{2} f^{2} g -16 a \,b^{3} c d \,e^{2} h -22 a \,b^{3} c d e f g -8 a \,b^{3} d^{2} e^{2} g +4 b^{4} c^{2} e^{2} h +3 b^{4} c^{2} e f g +8 b^{4} c d \,e^{2} g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (63 a^{3} d^{2} f^{2} h -70 a^{2} b c d \,f^{2} h -84 a^{2} b \,d^{2} e f h -35 a^{2} b \,d^{2} f^{2} g +15 a \,b^{2} c^{2} f^{2} h +80 a \,b^{2} c d e f h +30 a \,b^{2} c d \,f^{2} g +24 a \,b^{2} d^{2} e^{2} h +40 a \,b^{2} d^{2} e f g -12 b^{3} c^{2} e f h -3 b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h -24 b^{3} c d e f g -8 b^{3} d^{2} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 \sqrt {\left (a f -b e \right ) b}}\right )}{b^{5}}}{f}\)	\(818\)

Input:

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

Output:

-63/4*(((-1/21*g*f^2*c^2-4/21*c*e*(c*h+2*d*g)*f-16/63*(c*h+1/2*d*g)*d*e^2) 
*b^3+5/21*((c^2*h+2*c*d*g)*f^2+16/3*(c*h+1/2*d*g)*d*e*f+8/5*d^2*e^2*h)*a*b 
^2-10/9*a^2*((c*h+1/2*d*g)*f+6/5*d*e*h)*d*f*b+a^3*d^2*f^2*h)*(b*x+a)^2*f*a 
rctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))-((-5/63*x*(-8/15*x^2*(3/5*h*x+g 
)*d^2-16/5*x*c*(1/3*h*x+g)*d+c^2*(-8/5*h*x+g))*f^2-2/63*((-8/5*h*x^3-16/3* 
g*x^2)*d^2+4*(-8/3*h*x+g)*x*c*d+c^2*(2*h*x+g))*e*f+8/315*d^2*e^2*h*x^2)*b^ 
4-2/63*a*((28/3*x^2*(9/35*h*x+g)*d^2-25*(-56/75*h*x+g)*x*c*d+3/2*c^2*(-25/ 
3*h*x+g))*f^2+((64/5*h*x^2-44/3*g*x)*d^2+2*c*(-44/3*h*x+g)*d+h*c^2)*e*f-8/ 
5*d^2*e^2*h*x)*b^3+5/21*a^2*(((56/25*h*x^2-35/9*g*x)*d^2+2*c*(-35/9*h*x+g) 
*d+h*c^2)*f^2+20/9*d*((-273/125*h*x+1/2*g)*d+c*h)*e*f+8/75*d^2*e^2*h)*b^2- 
10/9*a^3*(((-3/2*h*x+1/2*g)*d+c*h)*f+3/5*d*e*h)*d*f*b+a^4*d^2*f^2*h)*((a*f 
-b*e)*b)^(1/2)*(f*x+e)^(1/2))/((a*f-b*e)*b)^(1/2)/(b*x+a)^2/b^5/f
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1426 vs. \(2 (392) = 784\).

Time = 0.23 (sec) , antiderivative size = 2866, normalized size of antiderivative = 6.79 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/120*(15*sqrt(b^2*e - a*b*f)*(((8*b^5*d^2*e^2*f + 8*(3*b^5*c*d - 5*a*b^ 
4*d^2)*e*f^2 + (3*b^5*c^2 - 30*a*b^4*c*d + 35*a^2*b^3*d^2)*f^3)*g + (8*(2* 
b^5*c*d - 3*a*b^4*d^2)*e^2*f + 4*(3*b^5*c^2 - 20*a*b^4*c*d + 21*a^2*b^3*d^ 
2)*e*f^2 - (15*a*b^4*c^2 - 70*a^2*b^3*c*d + 63*a^3*b^2*d^2)*f^3)*h)*x^2 + 
(8*a^2*b^3*d^2*e^2*f + 8*(3*a^2*b^3*c*d - 5*a^3*b^2*d^2)*e*f^2 + (3*a^2*b^ 
3*c^2 - 30*a^3*b^2*c*d + 35*a^4*b*d^2)*f^3)*g + (8*(2*a^2*b^3*c*d - 3*a^3* 
b^2*d^2)*e^2*f + 4*(3*a^2*b^3*c^2 - 20*a^3*b^2*c*d + 21*a^4*b*d^2)*e*f^2 - 
 (15*a^3*b^2*c^2 - 70*a^4*b*c*d + 63*a^5*d^2)*f^3)*h + 2*((8*a*b^4*d^2*e^2 
*f + 8*(3*a*b^4*c*d - 5*a^2*b^3*d^2)*e*f^2 + (3*a*b^4*c^2 - 30*a^2*b^3*c*d 
 + 35*a^3*b^2*d^2)*f^3)*g + (8*(2*a*b^4*c*d - 3*a^2*b^3*d^2)*e^2*f + 4*(3* 
a*b^4*c^2 - 20*a^2*b^3*c*d + 21*a^3*b^2*d^2)*e*f^2 - (15*a^2*b^3*c^2 - 70* 
a^3*b^2*c*d + 63*a^4*b*d^2)*f^3)*h)*x)*log((b*f*x + 2*b*e - a*f + 2*sqrt(b 
^2*e - a*b*f)*sqrt(f*x + e))/(b*x + a)) - 2*(24*(b^6*d^2*e*f^2 - a*b^5*d^2 
*f^3)*h*x^4 + 8*(5*(b^6*d^2*e*f^2 - a*b^5*d^2*f^3)*g + (6*b^6*d^2*e^2*f + 
5*(2*b^6*c*d - 3*a*b^5*d^2)*e*f^2 - (10*a*b^5*c*d - 9*a^2*b^4*d^2)*f^3)*h) 
*x^3 + 8*(5*(4*b^6*d^2*e^2*f + (6*b^6*c*d - 11*a*b^5*d^2)*e*f^2 - (6*a*b^5 
*c*d - 7*a^2*b^4*d^2)*f^3)*g + (3*b^6*d^2*e^3 + (40*b^6*c*d - 51*a*b^5*d^2 
)*e^2*f + (15*b^6*c^2 - 110*a*b^5*c*d + 111*a^2*b^4*d^2)*e*f^2 - (15*a*b^5 
*c^2 - 70*a^2*b^4*c*d + 63*a^3*b^3*d^2)*f^3)*h)*x^2 - 5*(2*(3*b^6*c^2 + 6* 
a*b^5*c*d - 25*a^2*b^4*d^2)*e^2*f + (3*a*b^5*c^2 - 102*a^2*b^4*c*d + 15...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((d*x+c)^2*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^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(a*f-b*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. 1040 vs. \(2 (392) = 784\).

Time = 0.17 (sec) , antiderivative size = 1040, normalized size of antiderivative = 2.46 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*(8*b^3*d^2*e^2*g + 24*b^3*c*d*e*f*g - 40*a*b^2*d^2*e*f*g + 3*b^3*c^2*f 
^2*g - 30*a*b^2*c*d*f^2*g + 35*a^2*b*d^2*f^2*g + 16*b^3*c*d*e^2*h - 24*a*b 
^2*d^2*e^2*h + 12*b^3*c^2*e*f*h - 80*a*b^2*c*d*e*f*h + 84*a^2*b*d^2*e*f*h 
- 15*a*b^2*c^2*f^2*h + 70*a^2*b*c*d*f^2*h - 63*a^3*d^2*f^2*h)*arctan(sqrt( 
f*x + e)*b/sqrt(-b^2*e + a*b*f))/(sqrt(-b^2*e + a*b*f)*b^5) - 1/4*(8*(f*x 
+ e)^(3/2)*b^4*c*d*e*f*g - 8*(f*x + e)^(3/2)*a*b^3*d^2*e*f*g - 8*sqrt(f*x 
+ e)*b^4*c*d*e^2*f*g + 8*sqrt(f*x + e)*a*b^3*d^2*e^2*f*g + 5*(f*x + e)^(3/ 
2)*b^4*c^2*f^2*g - 18*(f*x + e)^(3/2)*a*b^3*c*d*f^2*g + 13*(f*x + e)^(3/2) 
*a^2*b^2*d^2*f^2*g - 3*sqrt(f*x + e)*b^4*c^2*e*f^2*g + 22*sqrt(f*x + e)*a* 
b^3*c*d*e*f^2*g - 19*sqrt(f*x + e)*a^2*b^2*d^2*e*f^2*g + 3*sqrt(f*x + e)*a 
*b^3*c^2*f^3*g - 14*sqrt(f*x + e)*a^2*b^2*c*d*f^3*g + 11*sqrt(f*x + e)*a^3 
*b*d^2*f^3*g + 4*(f*x + e)^(3/2)*b^4*c^2*e*f*h - 16*(f*x + e)^(3/2)*a*b^3* 
c*d*e*f*h + 12*(f*x + e)^(3/2)*a^2*b^2*d^2*e*f*h - 4*sqrt(f*x + e)*b^4*c^2 
*e^2*f*h + 16*sqrt(f*x + e)*a*b^3*c*d*e^2*f*h - 12*sqrt(f*x + e)*a^2*b^2*d 
^2*e^2*f*h - 9*(f*x + e)^(3/2)*a*b^3*c^2*f^2*h + 26*(f*x + e)^(3/2)*a^2*b^ 
2*c*d*f^2*h - 17*(f*x + e)^(3/2)*a^3*b*d^2*f^2*h + 11*sqrt(f*x + e)*a*b^3* 
c^2*e*f^2*h - 38*sqrt(f*x + e)*a^2*b^2*c*d*e*f^2*h + 27*sqrt(f*x + e)*a^3* 
b*d^2*e*f^2*h - 7*sqrt(f*x + e)*a^2*b^2*c^2*f^3*h + 22*sqrt(f*x + e)*a^3*b 
*c*d*f^3*h - 15*sqrt(f*x + e)*a^4*d^2*f^3*h)/(((f*x + e)*b - b*e + a*f)^2* 
b^5) + 2/15*(5*(f*x + e)^(3/2)*b^12*d^2*f^5*g + 15*sqrt(f*x + e)*b^12*d...
 

Mupad [B] (verification not implemented)

Time = 2.66 (sec) , antiderivative size = 856, normalized size of antiderivative = 2.03 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(e + f*x)^(3/2)*((2*d^2*f*g - 6*d^2*e*h + 4*c*d*f*h)/(3*b^3*f) - (2*d^2*h* 
(a*f - b*e))/(b^4*f)) - (e + f*x)^(1/2)*((3*((2*d^2*f*g - 6*d^2*e*h + 4*c* 
d*f*h)/(b^3*f) - (6*d^2*h*(a*f - b*e))/(b^4*f))*(a*f - b*e))/b - (2*(c*f - 
 d*e)*(c*f*h - 3*d*e*h + 2*d*f*g))/(b^3*f) + (6*d^2*h*(a*f - b*e)^2)/(b^5* 
f)) - ((e + f*x)^(3/2)*((5*b^4*c^2*f^2*g)/4 + b^4*c^2*e*f*h - (9*a*b^3*c^2 
*f^2*h)/4 - (17*a^3*b*d^2*f^2*h)/4 + (13*a^2*b^2*d^2*f^2*g)/4 + 2*b^4*c*d* 
e*f*g - (9*a*b^3*c*d*f^2*g)/2 - 2*a*b^3*d^2*e*f*g + (13*a^2*b^2*c*d*f^2*h) 
/2 + 3*a^2*b^2*d^2*e*f*h - 4*a*b^3*c*d*e*f*h) - (e + f*x)^(1/2)*((15*a^4*d 
^2*f^3*h)/4 - (3*a*b^3*c^2*f^3*g)/4 - (11*a^3*b*d^2*f^3*g)/4 + (3*b^4*c^2* 
e*f^2*g)/4 + b^4*c^2*e^2*f*h + (7*a^2*b^2*c^2*f^3*h)/4 + (19*a^2*b^2*d^2*e 
*f^2*g)/4 + 3*a^2*b^2*d^2*e^2*f*h - (11*a^3*b*c*d*f^3*h)/2 + 2*b^4*c*d*e^2 
*f*g + (7*a^2*b^2*c*d*f^3*g)/2 - (11*a*b^3*c^2*e*f^2*h)/4 - 2*a*b^3*d^2*e^ 
2*f*g - (27*a^3*b*d^2*e*f^2*h)/4 + (19*a^2*b^2*c*d*e*f^2*h)/2 - (11*a*b^3* 
c*d*e*f^2*g)/2 - 4*a*b^3*c*d*e^2*f*h))/(b^7*(e + f*x)^2 - (e + f*x)*(2*b^7 
*e - 2*a*b^6*f) + b^7*e^2 + a^2*b^5*f^2 - 2*a*b^6*e*f) + (atan((b^(1/2)*(e 
 + f*x)^(1/2))/(a*f - b*e)^(1/2))*(3*b^3*c^2*f^2*g + 8*b^3*d^2*e^2*g - 63* 
a^3*d^2*f^2*h + 16*b^3*c*d*e^2*h + 12*b^3*c^2*e*f*h - 15*a*b^2*c^2*f^2*h - 
 24*a*b^2*d^2*e^2*h + 35*a^2*b*d^2*f^2*g + 24*b^3*c*d*e*f*g - 30*a*b^2*c*d 
*f^2*g + 70*a^2*b*c*d*f^2*h - 40*a*b^2*d^2*e*f*g + 84*a^2*b*d^2*e*f*h - 80 
*a*b^2*c*d*e*f*h))/(4*b^(11/2)*(a*f - b*e)^(1/2)) + (2*d^2*h*(e + f*x)^...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 3464, normalized size of antiderivative = 8.21 \[ \int \frac {(c+d x)^2 (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 945*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - 
 b*e)))*a**5*d**2*f**3*h + 1050*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x 
)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*c*d*f**3*h + 1260*sqrt(b)*sqrt(a*f 
- b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*d**2*e*f** 
2*h + 525*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f 
 - b*e)))*a**4*b*d**2*f**3*g - 1890*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + 
 f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*d**2*f**3*h*x - 225*sqrt(b)*sqr 
t(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c 
**2*f**3*h - 1200*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)* 
sqrt(a*f - b*e)))*a**3*b**2*c*d*e*f**2*h - 450*sqrt(b)*sqrt(a*f - b*e)*ata 
n((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c*d*f**3*g + 2100 
*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e))) 
*a**3*b**2*c*d*f**3*h*x - 360*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)* 
b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*d**2*e**2*f*h - 600*sqrt(b)*sqrt(a 
*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*d**2 
*e*f**2*g + 2520*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*s 
qrt(a*f - b*e)))*a**3*b**2*d**2*e*f**2*h*x + 1050*sqrt(b)*sqrt(a*f - b*e)* 
atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*d**2*f**3*g*x 
- 945*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b 
*e)))*a**3*b**2*d**2*f**3*h*x**2 + 180*sqrt(b)*sqrt(a*f - b*e)*atan((sq...