\(\int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx\) [97]

Optimal result

Integrand size = 29, antiderivative size = 423 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\frac {2 \left (a^2 d^2 f h+2 a b d (d f g+d e h-3 c f h)+b^2 \left (d^2 e g+6 c^2 f h-3 c d (f g+e h)\right )\right ) \sqrt {e+f x}}{d^5}+\frac {(b c-a d) \left (a d (3 d f g+4 d e h-7 c f h)+b \left (8 d^2 e g+15 c^2 f h-c d (11 f g+12 e h)\right )\right ) \sqrt {e+f x}}{4 d^5 (c+d x)}+\frac {2 b (b d g-3 b c h+2 a d h) (e+f x)^{3/2}}{3 d^4}-\frac {(b c-a d)^2 (d g-c h) (e+f x)^{3/2}}{2 d^4 (c+d x)^2}+\frac {2 b^2 h (e+f x)^{5/2}}{5 d^3 f}-\frac {\left (3 a^2 d^2 f (d f g+4 d e h-5 c f h)+2 a b d \left (35 c^2 f^2 h+4 d^2 e (3 f g+2 e h)-5 c d f (3 f g+8 e h)\right )+b^2 \left (8 d^3 e^2 g-63 c^3 f^2 h-8 c d^2 e (5 f g+3 e h)+7 c^2 d f (5 f g+12 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 d^{11/2} \sqrt {d e-c f}} \] Output:

2*(a^2*d^2*f*h+2*a*b*d*(-3*c*f*h+d*e*h+d*f*g)+b^2*(d^2*e*g+6*c^2*f*h-3*c*d 
*(e*h+f*g)))*(f*x+e)^(1/2)/d^5+1/4*(-a*d+b*c)*(a*d*(-7*c*f*h+4*d*e*h+3*d*f 
*g)+b*(8*d^2*e*g+15*c^2*f*h-c*d*(12*e*h+11*f*g)))*(f*x+e)^(1/2)/d^5/(d*x+c 
)+2/3*b*(2*a*d*h-3*b*c*h+b*d*g)*(f*x+e)^(3/2)/d^4-1/2*(-a*d+b*c)^2*(-c*h+d 
*g)*(f*x+e)^(3/2)/d^4/(d*x+c)^2+2/5*b^2*h*(f*x+e)^(5/2)/d^3/f-1/4*(3*a^2*d 
^2*f*(-5*c*f*h+4*d*e*h+d*f*g)+2*a*b*d*(35*c^2*f^2*h+4*d^2*e*(2*e*h+3*f*g)- 
5*c*d*f*(8*e*h+3*f*g))+b^2*(8*d^3*e^2*g-63*c^3*f^2*h-8*c*d^2*e*(3*e*h+5*f* 
g)+7*c^2*d*f*(12*e*h+5*f*g)))*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/ 
2))/d^(11/2)/(-c*f+d*e)^(1/2)
 

Mathematica [A] (verified)

Time = 1.33 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.18 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\frac {\sqrt {e+f x} \left (10 a b d f \left (-105 c^3 f h+5 c^2 d (9 f g+10 e h-35 f h x)+c d^2 \left (-6 e g+75 f g x+88 e h x-56 f h x^2\right )+4 d^3 x \left (-3 e g+6 f g x+8 e h x+2 f h x^2\right )\right )-15 a^2 d^2 f \left (-15 c^2 f h+c d (3 f g+2 e h-25 f h x)+d^2 (f x (5 g-8 h x)+2 e (g+2 h x))\right )+b^2 \left (945 c^4 f^2 h-105 c^3 d f (6 e h+5 f (g-3 h x))+8 d^4 x^2 \left (3 e^2 h+f^2 x (5 g+3 h x)+e f (20 g+6 h x)\right )-8 c d^3 x \left (-6 e^2 h+f^2 x (35 g+9 h x)+e f (-55 g+48 h x)\right )+c^2 d^2 \left (24 e^2 h+2 e f (125 g-546 h x)+7 f^2 x (-125 g+72 h x)\right )\right )\right )}{60 d^5 f (c+d x)^2}+\frac {\left (3 a^2 d^2 f (d f g+4 d e h-5 c f h)+2 a b d \left (35 c^2 f^2 h+4 d^2 e (3 f g+2 e h)-5 c d f (3 f g+8 e h)\right )+b^2 \left (8 d^3 e^2 g-63 c^3 f^2 h-8 c d^2 e (5 f g+3 e h)+7 c^2 d f (5 f g+12 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{4 d^{11/2} \sqrt {-d e+c f}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 469, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 163

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*((d*g - c*h)*(a + b*x)^2*(e + f*x)^(5/2))/(d*(d*e - c*f)*(c + d*x)^2) 
 + (-1/5*((e + f*x)^(5/2)*(5*a^2*d^2*f*(d*f*g + 4*d*e*h - 5*c*f*h) + 10*a* 
b*d*f*(2*d^2*e*g + 7*c^2*f*h - 3*c*d*(f*g + 2*e*h)) - b^2*c*(63*c^2*f^2*h 
+ 2*d^2*e*(15*f*g + 4*e*h) - c*d*f*(35*f*g + 66*e*h)) - 2*b^2*d*(d*e - c*f 
)*(5*d*f*g + 4*d*e*h - 9*c*f*h)*x))/(d^2*f*(d*e - c*f)*(c + d*x)) + ((3*a^ 
2*d^2*f*(d*f*g + 4*d*e*h - 5*c*f*h) + 2*a*b*d*(35*c^2*f^2*h + 4*d^2*e*(3*f 
*g + 2*e*h) - 5*c*d*f*(3*f*g + 8*e*h)) + b^2*(8*d^3*e^2*g - 63*c^3*f^2*h - 
 8*c*d^2*e*(5*f*g + 3*e*h) + 7*c^2*d*f*(5*f*g + 12*e*h)))*((2*(e + f*x)^(3 
/2))/(3*d) + ((d*e - c*f)*((2*Sqrt[e + f*x])/d - (2*Sqrt[d*e - c*f]*ArcTan 
h[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(3/2)))/d))/(2*d^2*(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 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 = 0.63 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.21

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

Input:

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

Output:

15/4/((c*f-d*e)*d)^(1/2)*(-(d*x+c)^2*f*((-1/5*a^2*f^2*g-4/5*a*e*(a*h+2*b*g 
)*f-16/15*b*e^2*(a*h+1/2*b*g))*d^3+((a^2*h+2*a*b*g)*f^2+16/3*(a*h+1/2*b*g) 
*b*e*f+8/5*b^2*e^2*h)*c*d^2-14/3*c^2*((a*h+1/2*b*g)*f+6/5*e*h*b)*b*f*d+21/ 
5*b^2*c^3*f^2*h)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))+((c*f-d*e)*d) 
^(1/2)*(f*x+e)^(1/2)*((-1/3*x*(-8/15*x^2*(3/5*h*x+g)*b^2-16/5*a*x*(1/3*h*x 
+g)*b+a^2*(-8/5*h*x+g))*f^2-2/15*((-8/5*h*x^3-16/3*g*x^2)*b^2+4*a*x*(-8/3* 
h*x+g)*b+a^2*(2*h*x+g))*e*f+8/75*b^2*e^2*h*x^2)*d^4-2/15*c*((28/3*x^2*(9/3 
5*h*x+g)*b^2-25*(-56/75*h*x+g)*a*x*b+3/2*a^2*(-25/3*h*x+g))*f^2+e*((64/5*h 
*x^2-44/3*g*x)*b^2+2*a*(-44/3*h*x+g)*b+a^2*h)*f-8/5*b^2*e^2*h*x)*d^3+(((56 
/25*h*x^2-35/9*g*x)*b^2+2*a*(-35/9*h*x+g)*b+a^2*h)*f^2+20/9*((-273/125*h*x 
+1/2*g)*b+a*h)*b*e*f+8/75*b^2*e^2*h)*c^2*d^2-14/3*(((-3/2*h*x+1/2*g)*b+a*h 
)*f+3/5*e*h*b)*c^3*b*f*d+21/5*b^2*c^4*f^2*h))/d^5/(d*x+c)^2/f
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1426 vs. \(2 (393) = 786\).

Time = 0.26 (sec) , antiderivative size = 2866, normalized size of antiderivative = 6.78 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/120*(15*sqrt(d^2*e - c*d*f)*(((8*b^2*d^5*e^2*f - 8*(5*b^2*c*d^4 - 3*a* 
b*d^5)*e*f^2 + (35*b^2*c^2*d^3 - 30*a*b*c*d^4 + 3*a^2*d^5)*f^3)*g - (8*(3* 
b^2*c*d^4 - 2*a*b*d^5)*e^2*f - 4*(21*b^2*c^2*d^3 - 20*a*b*c*d^4 + 3*a^2*d^ 
5)*e*f^2 + (63*b^2*c^3*d^2 - 70*a*b*c^2*d^3 + 15*a^2*c*d^4)*f^3)*h)*x^2 + 
(8*b^2*c^2*d^3*e^2*f - 8*(5*b^2*c^3*d^2 - 3*a*b*c^2*d^3)*e*f^2 + (35*b^2*c 
^4*d - 30*a*b*c^3*d^2 + 3*a^2*c^2*d^3)*f^3)*g - (8*(3*b^2*c^3*d^2 - 2*a*b* 
c^2*d^3)*e^2*f - 4*(21*b^2*c^4*d - 20*a*b*c^3*d^2 + 3*a^2*c^2*d^3)*e*f^2 + 
 (63*b^2*c^5 - 70*a*b*c^4*d + 15*a^2*c^3*d^2)*f^3)*h + 2*((8*b^2*c*d^4*e^2 
*f - 8*(5*b^2*c^2*d^3 - 3*a*b*c*d^4)*e*f^2 + (35*b^2*c^3*d^2 - 30*a*b*c^2* 
d^3 + 3*a^2*c*d^4)*f^3)*g - (8*(3*b^2*c^2*d^3 - 2*a*b*c*d^4)*e^2*f - 4*(21 
*b^2*c^3*d^2 - 20*a*b*c^2*d^3 + 3*a^2*c*d^4)*e*f^2 + (63*b^2*c^4*d - 70*a* 
b*c^3*d^2 + 15*a^2*c^2*d^3)*f^3)*h)*x)*log((d*f*x + 2*d*e - c*f + 2*sqrt(d 
^2*e - c*d*f)*sqrt(f*x + e))/(d*x + c)) - 2*(24*(b^2*d^6*e*f^2 - b^2*c*d^5 
*f^3)*h*x^4 + 8*(5*(b^2*d^6*e*f^2 - b^2*c*d^5*f^3)*g + (6*b^2*d^6*e^2*f - 
5*(3*b^2*c*d^5 - 2*a*b*d^6)*e*f^2 + (9*b^2*c^2*d^4 - 10*a*b*c*d^5)*f^3)*h) 
*x^3 + 8*(5*(4*b^2*d^6*e^2*f - (11*b^2*c*d^5 - 6*a*b*d^6)*e*f^2 + (7*b^2*c 
^2*d^4 - 6*a*b*c*d^5)*f^3)*g + (3*b^2*d^6*e^3 - (51*b^2*c*d^5 - 40*a*b*d^6 
)*e^2*f + (111*b^2*c^2*d^4 - 110*a*b*c*d^5 + 15*a^2*d^6)*e*f^2 - (63*b^2*c 
^3*d^3 - 70*a*b*c^2*d^4 + 15*a^2*c*d^5)*f^3)*h)*x^2 + 5*(2*(25*b^2*c^2*d^4 
 - 6*a*b*c*d^5 - 3*a^2*d^6)*e^2*f - (155*b^2*c^3*d^3 - 102*a*b*c^2*d^4 ...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((b*x+a)^2*(f*x+e)^(3/2)*(h*x+g)/(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. 1040 vs. \(2 (393) = 786\).

Time = 0.16 (sec) , antiderivative size = 1040, normalized size of antiderivative = 2.46 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 856, normalized size of antiderivative = 2.02 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 3464, normalized size of antiderivative = 8.19 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 225*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - 
 d*e)))*a**2*c**3*d**2*f**3*h + 180*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + 
 f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*c**2*d**3*e*f**2*h + 45*sqrt(d)*s 
qrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*c**2 
*d**3*f**3*g - 450*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d) 
*sqrt(c*f - d*e)))*a**2*c**2*d**3*f**3*h*x + 360*sqrt(d)*sqrt(c*f - d*e)*a 
tan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*c*d**4*e*f**2*h*x + 
90*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e) 
))*a**2*c*d**4*f**3*g*x - 225*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)* 
d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*c*d**4*f**3*h*x**2 + 180*sqrt(d)*sqrt(c 
*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*d**5*e*f* 
*2*h*x**2 + 45*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqr 
t(c*f - d*e)))*a**2*d**5*f**3*g*x**2 + 1050*sqrt(d)*sqrt(c*f - d*e)*atan(( 
sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b*c**4*d*f**3*h - 1200*sqrt( 
d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b*c 
**3*d**2*e*f**2*h - 450*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sq 
rt(d)*sqrt(c*f - d*e)))*a*b*c**3*d**2*f**3*g + 2100*sqrt(d)*sqrt(c*f - d*e 
)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b*c**3*d**2*f**3*h*x 
 + 240*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - 
d*e)))*a*b*c**2*d**3*e**2*f*h + 360*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(...