\(\int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx\) [166]

Optimal result

Integrand size = 29, antiderivative size = 517 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx=-\frac {2 (d e-c f)^2 (f g-e h)}{(b e-a f)^4 \sqrt {e+f x}}-\frac {(b c-a d)^2 (b g-a h) \sqrt {e+f x}}{3 b^2 (b e-a f)^2 (a+b x)^3}-\frac {(b c-a d) \left (7 a^2 d f h+b^2 (12 d e g-11 c f g+6 c e h)-a b (d f g+18 d e h-5 c f h)\right ) \sqrt {e+f x}}{12 b^2 (b e-a f)^3 (a+b x)^2}+\frac {\left (a^3 d^2 f^2 h+a^2 b d f (d f g-6 d e h+2 c f h)-b^3 \left (8 d^2 e^2 g+c^2 f (19 f g-14 e h)-4 c d e (7 f g-4 e h)\right )+a b^2 \left (5 c^2 f^2 h+2 c d f (5 f g-12 e h)-12 d^2 e (f g-2 e h)\right )\right ) \sqrt {e+f x}}{8 b^2 (b e-a f)^4 (a+b x)}-\frac {\left (a^3 d^2 f^3 h+a^2 b d f^2 (d f g-6 d e h+2 c f h)+a b^2 f \left (5 c^2 f^2 h+2 c d f (5 f g-12 e h)-12 d^2 e (f g-2 e h)\right )-b^3 \left (5 c^2 f^2 (7 f g-6 e h)-12 c d e f (5 f g-4 e h)+8 d^2 e^2 (3 f g-2 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{8 b^{5/2} (b e-a f)^{9/2}} \] Output:

-2*(-c*f+d*e)^2*(-e*h+f*g)/(-a*f+b*e)^4/(f*x+e)^(1/2)-1/3*(-a*d+b*c)^2*(-a 
*h+b*g)*(f*x+e)^(1/2)/b^2/(-a*f+b*e)^2/(b*x+a)^3-1/12*(-a*d+b*c)*(7*a^2*d* 
f*h+b^2*(6*c*e*h-11*c*f*g+12*d*e*g)-a*b*(-5*c*f*h+18*d*e*h+d*f*g))*(f*x+e) 
^(1/2)/b^2/(-a*f+b*e)^3/(b*x+a)^2+1/8*(a^3*d^2*f^2*h+a^2*b*d*f*(2*c*f*h-6* 
d*e*h+d*f*g)-b^3*(8*d^2*e^2*g+c^2*f*(-14*e*h+19*f*g)-4*c*d*e*(-4*e*h+7*f*g 
))+a*b^2*(5*c^2*f^2*h+2*c*d*f*(-12*e*h+5*f*g)-12*d^2*e*(-2*e*h+f*g)))*(f*x 
+e)^(1/2)/b^2/(-a*f+b*e)^4/(b*x+a)-1/8*(a^3*d^2*f^3*h+a^2*b*d*f^2*(2*c*f*h 
-6*d*e*h+d*f*g)+a*b^2*f*(5*c^2*f^2*h+2*c*d*f*(-12*e*h+5*f*g)-12*d^2*e*(-2* 
e*h+f*g))-b^3*(5*c^2*f^2*(-6*e*h+7*f*g)-12*c*d*e*f*(-4*e*h+5*f*g)+8*d^2*e^ 
2*(-2*e*h+3*f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^(5/2) 
/(-a*f+b*e)^(9/2)
 

Mathematica [A] (verified)

Time = 3.83 (sec) , antiderivative size = 831, normalized size of antiderivative = 1.61 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx=\frac {-3 a^5 d^2 f^2 h (e+f x)-a^4 b d f (e+f x) (6 c f h+d (3 f g-16 e h+8 f h x))-b^5 \left (24 d^2 e^2 x^2 (3 f g x+e (g-2 h x))+12 c d e x \left (-15 f^2 g x^2+2 e^2 (g+2 h x)+e f x (-5 g+12 h x)\right )+c^2 \left (105 f^3 g x^3+5 e f^2 x^2 (7 g-18 h x)+4 e^3 (2 g+3 h x)-2 e^2 f x (7 g+15 h x)\right )\right )+a^3 b^2 \left (3 c^2 f^2 (-16 f g+27 e h+11 f h x)+2 c d f \left (-94 e^2 h+e f (81 g-38 h x)+f^2 x (33 g+8 h x)\right )+d^2 \left (92 e^3 h+f^3 x^2 (8 g+3 h x)+e f^2 x (-38 g+17 h x)+e^2 f (-94 g+58 h x)\right )\right )+a b^4 \left (-12 d^2 e x \left (3 f^2 g x^2+2 e^2 (g-9 h x)+e f x (17 g-6 h x)\right )+c^2 \left (-4 e^3 h+5 f^3 x^2 (-56 g+3 h x)+49 e f^2 x (-2 g+5 h x)+e^2 f (38 g+82 h x)\right )-2 c d \left (-15 f^3 g x^3+4 e^3 (g+6 h x)+e f^2 x^2 (-245 g+36 h x)+2 e^2 f x (-41 g+102 h x)\right )\right )+a^2 b^3 \left (2 c d \left (-8 e^3 h+e^2 f (28 g-250 h x)+e f^2 x (212 g-95 h x)+f^3 x^2 (40 g+3 h x)\right )+c^2 f \left (28 e^2 h+f^2 x (-231 g+40 h x)+e f (-87 g+212 h x)\right )+d^2 \left (3 f^3 g x^3+10 e^2 f x (-25 g+9 h x)-e f^2 x^2 (95 g+18 h x)+e^3 (-8 g+252 h x)\right )\right )}{24 b^2 (b e-a f)^4 (a+b x)^3 \sqrt {e+f x}}+\frac {\left (a^3 d^2 f^3 h+a^2 b d f^2 (d f g-6 d e h+2 c f h)+a b^2 f \left (5 c^2 f^2 h+2 c d f (5 f g-12 e h)+12 d^2 e (-f g+2 e h)\right )+b^3 \left (12 c d e f (5 f g-4 e h)+8 d^2 e^2 (-3 f g+2 e h)+5 c^2 f^2 (-7 f g+6 e h)\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{8 b^{5/2} (-b e+a f)^{9/2}} \] Input:

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

Output:

(-3*a^5*d^2*f^2*h*(e + f*x) - a^4*b*d*f*(e + f*x)*(6*c*f*h + d*(3*f*g - 16 
*e*h + 8*f*h*x)) - b^5*(24*d^2*e^2*x^2*(3*f*g*x + e*(g - 2*h*x)) + 12*c*d* 
e*x*(-15*f^2*g*x^2 + 2*e^2*(g + 2*h*x) + e*f*x*(-5*g + 12*h*x)) + c^2*(105 
*f^3*g*x^3 + 5*e*f^2*x^2*(7*g - 18*h*x) + 4*e^3*(2*g + 3*h*x) - 2*e^2*f*x* 
(7*g + 15*h*x))) + a^3*b^2*(3*c^2*f^2*(-16*f*g + 27*e*h + 11*f*h*x) + 2*c* 
d*f*(-94*e^2*h + e*f*(81*g - 38*h*x) + f^2*x*(33*g + 8*h*x)) + d^2*(92*e^3 
*h + f^3*x^2*(8*g + 3*h*x) + e*f^2*x*(-38*g + 17*h*x) + e^2*f*(-94*g + 58* 
h*x))) + a*b^4*(-12*d^2*e*x*(3*f^2*g*x^2 + 2*e^2*(g - 9*h*x) + e*f*x*(17*g 
 - 6*h*x)) + c^2*(-4*e^3*h + 5*f^3*x^2*(-56*g + 3*h*x) + 49*e*f^2*x*(-2*g 
+ 5*h*x) + e^2*f*(38*g + 82*h*x)) - 2*c*d*(-15*f^3*g*x^3 + 4*e^3*(g + 6*h* 
x) + e*f^2*x^2*(-245*g + 36*h*x) + 2*e^2*f*x*(-41*g + 102*h*x))) + a^2*b^3 
*(2*c*d*(-8*e^3*h + e^2*f*(28*g - 250*h*x) + e*f^2*x*(212*g - 95*h*x) + f^ 
3*x^2*(40*g + 3*h*x)) + c^2*f*(28*e^2*h + f^2*x*(-231*g + 40*h*x) + e*f*(- 
87*g + 212*h*x)) + d^2*(3*f^3*g*x^3 + 10*e^2*f*x*(-25*g + 9*h*x) - e*f^2*x 
^2*(95*g + 18*h*x) + e^3*(-8*g + 252*h*x))))/(24*b^2*(b*e - a*f)^4*(a + b* 
x)^3*Sqrt[e + f*x]) + ((a^3*d^2*f^3*h + a^2*b*d*f^2*(d*f*g - 6*d*e*h + 2*c 
*f*h) + a*b^2*f*(5*c^2*f^2*h + 2*c*d*f*(5*f*g - 12*e*h) + 12*d^2*e*(-(f*g) 
 + 2*e*h)) + b^3*(12*c*d*e*f*(5*f*g - 4*e*h) + 8*d^2*e^2*(-3*f*g + 2*e*h) 
+ 5*c^2*f^2*(-7*f*g + 6*e*h)))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) 
+ a*f]])/(8*b^(5/2)*(-(b*e) + a*f)^(9/2))
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {166, 27, 161, 52, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 161

\(\displaystyle -\frac {\frac {3 \left (a^3 d^2 f^3 h+a^2 b d f^2 (2 c f h-6 d e h+d f g)+a b^2 f \left (5 c^2 f^2 h+2 c d f (5 f g-12 e h)-12 d^2 e (f g-2 e h)\right )-\left (b^3 \left (5 c^2 f^2 (7 f g-6 e h)-12 c d e f (5 f g-4 e h)+8 d^2 e^2 (3 f g-2 e h)\right )\right )\right ) \int \frac {1}{(a+b x)^2 \sqrt {e+f x}}dx}{4 b f (b e-a f)^2}-\frac {3 a^3 d^2 e f^2 h+a^2 b f \left (-4 c^2 f^2 h+6 c d e f h+d^2 e (3 f g-14 e h)\right )+x \left (3 a^3 d^2 f^3 h+a^2 b d f^2 (-2 c f h-10 d e h+3 d f g)-a b^2 f \left (5 c^2 f^2 h+2 c d f (5 f g-12 e h)-4 d^2 e (f g-e h)\right )+b^3 \left (5 c^2 f^2 (7 f g-6 e h)-12 c d e f (5 f g-4 e h)+4 d^2 e^2 (7 f g-6 e h)\right )\right )-a b^2 \left (-c^2 f^2 (28 f g-25 e h)+2 c d e f (33 f g-32 e h)-8 d^2 e^2 (4 f g-3 e h)\right )-b^3 c e f (6 c e h-7 c f g+4 d e g)}{2 b f (a+b x)^2 \sqrt {e+f x} (b e-a f)^2}}{6 b (b e-a f)}-\frac {(c+d x)^2 (b g-a h)}{3 b (a+b x)^3 \sqrt {e+f x} (b e-a f)}\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {\frac {3 \left (a^3 d^2 f^3 h+a^2 b d f^2 (2 c f h-6 d e h+d f g)+a b^2 f \left (5 c^2 f^2 h+2 c d f (5 f g-12 e h)-12 d^2 e (f g-2 e h)\right )-\left (b^3 \left (5 c^2 f^2 (7 f g-6 e h)-12 c d e f (5 f g-4 e h)+8 d^2 e^2 (3 f g-2 e h)\right )\right )\right ) \left (-\frac {f \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{2 (b e-a f)}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 b f (b e-a f)^2}-\frac {3 a^3 d^2 e f^2 h+a^2 b f \left (-4 c^2 f^2 h+6 c d e f h+d^2 e (3 f g-14 e h)\right )+x \left (3 a^3 d^2 f^3 h+a^2 b d f^2 (-2 c f h-10 d e h+3 d f g)-a b^2 f \left (5 c^2 f^2 h+2 c d f (5 f g-12 e h)-4 d^2 e (f g-e h)\right )+b^3 \left (5 c^2 f^2 (7 f g-6 e h)-12 c d e f (5 f g-4 e h)+4 d^2 e^2 (7 f g-6 e h)\right )\right )-a b^2 \left (-c^2 f^2 (28 f g-25 e h)+2 c d e f (33 f g-32 e h)-8 d^2 e^2 (4 f g-3 e h)\right )-b^3 c e f (6 c e h-7 c f g+4 d e g)}{2 b f (a+b x)^2 \sqrt {e+f x} (b e-a f)^2}}{6 b (b e-a f)}-\frac {(c+d x)^2 (b g-a h)}{3 b (a+b x)^3 \sqrt {e+f x} (b e-a f)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\frac {3 \left (a^3 d^2 f^3 h+a^2 b d f^2 (2 c f h-6 d e h+d f g)+a b^2 f \left (5 c^2 f^2 h+2 c d f (5 f g-12 e h)-12 d^2 e (f g-2 e h)\right )-\left (b^3 \left (5 c^2 f^2 (7 f g-6 e h)-12 c d e f (5 f g-4 e h)+8 d^2 e^2 (3 f g-2 e h)\right )\right )\right ) \left (-\frac {\int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b e-a f}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 b f (b e-a f)^2}-\frac {3 a^3 d^2 e f^2 h+a^2 b f \left (-4 c^2 f^2 h+6 c d e f h+d^2 e (3 f g-14 e h)\right )+x \left (3 a^3 d^2 f^3 h+a^2 b d f^2 (-2 c f h-10 d e h+3 d f g)-a b^2 f \left (5 c^2 f^2 h+2 c d f (5 f g-12 e h)-4 d^2 e (f g-e h)\right )+b^3 \left (5 c^2 f^2 (7 f g-6 e h)-12 c d e f (5 f g-4 e h)+4 d^2 e^2 (7 f g-6 e h)\right )\right )-a b^2 \left (-c^2 f^2 (28 f g-25 e h)+2 c d e f (33 f g-32 e h)-8 d^2 e^2 (4 f g-3 e h)\right )-b^3 c e f (6 c e h-7 c f g+4 d e g)}{2 b f (a+b x)^2 \sqrt {e+f x} (b e-a f)^2}}{6 b (b e-a f)}-\frac {(c+d x)^2 (b g-a h)}{3 b (a+b x)^3 \sqrt {e+f x} (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {3 \left (\frac {f \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b e-a f)^{3/2}}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right ) \left (a^3 d^2 f^3 h+a^2 b d f^2 (2 c f h-6 d e h+d f g)+a b^2 f \left (5 c^2 f^2 h+2 c d f (5 f g-12 e h)-12 d^2 e (f g-2 e h)\right )-\left (b^3 \left (5 c^2 f^2 (7 f g-6 e h)-12 c d e f (5 f g-4 e h)+8 d^2 e^2 (3 f g-2 e h)\right )\right )\right )}{4 b f (b e-a f)^2}-\frac {3 a^3 d^2 e f^2 h+a^2 b f \left (-4 c^2 f^2 h+6 c d e f h+d^2 e (3 f g-14 e h)\right )+x \left (3 a^3 d^2 f^3 h+a^2 b d f^2 (-2 c f h-10 d e h+3 d f g)-a b^2 f \left (5 c^2 f^2 h+2 c d f (5 f g-12 e h)-4 d^2 e (f g-e h)\right )+b^3 \left (5 c^2 f^2 (7 f g-6 e h)-12 c d e f (5 f g-4 e h)+4 d^2 e^2 (7 f g-6 e h)\right )\right )-a b^2 \left (-c^2 f^2 (28 f g-25 e h)+2 c d e f (33 f g-32 e h)-8 d^2 e^2 (4 f g-3 e h)\right )-b^3 c e f (6 c e h-7 c f g+4 d e g)}{2 b f (a+b x)^2 \sqrt {e+f x} (b e-a f)^2}}{6 b (b e-a f)}-\frac {(c+d x)^2 (b g-a h)}{3 b (a+b x)^3 \sqrt {e+f x} (b e-a f)}\)

Input:

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

Output:

-1/3*((b*g - a*h)*(c + d*x)^2)/(b*(b*e - a*f)*(a + b*x)^3*Sqrt[e + f*x]) - 
 (-1/2*(3*a^3*d^2*e*f^2*h - b^3*c*e*f*(4*d*e*g - 7*c*f*g + 6*c*e*h) + a^2* 
b*f*(6*c*d*e*f*h - 4*c^2*f^2*h + d^2*e*(3*f*g - 14*e*h)) - a*b^2*(2*c*d*e* 
f*(33*f*g - 32*e*h) - c^2*f^2*(28*f*g - 25*e*h) - 8*d^2*e^2*(4*f*g - 3*e*h 
)) + (3*a^3*d^2*f^3*h + a^2*b*d*f^2*(3*d*f*g - 10*d*e*h - 2*c*f*h) + b^3*( 
4*d^2*e^2*(7*f*g - 6*e*h) + 5*c^2*f^2*(7*f*g - 6*e*h) - 12*c*d*e*f*(5*f*g 
- 4*e*h)) - a*b^2*f*(5*c^2*f^2*h + 2*c*d*f*(5*f*g - 12*e*h) - 4*d^2*e*(f*g 
 - e*h)))*x)/(b*f*(b*e - a*f)^2*(a + b*x)^2*Sqrt[e + f*x]) + (3*(a^3*d^2*f 
^3*h + a^2*b*d*f^2*(d*f*g - 6*d*e*h + 2*c*f*h) + a*b^2*f*(5*c^2*f^2*h + 2* 
c*d*f*(5*f*g - 12*e*h) - 12*d^2*e*(f*g - 2*e*h)) - b^3*(5*c^2*f^2*(7*f*g - 
 6*e*h) - 12*c*d*e*f*(5*f*g - 4*e*h) + 8*d^2*e^2*(3*f*g - 2*e*h)))*(-(Sqrt 
[e + f*x]/((b*e - a*f)*(a + b*x))) + (f*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sq 
rt[b*e - a*f]])/(Sqrt[b]*(b*e - a*f)^(3/2))))/(4*b*f*(b*e - a*f)^2))/(6*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 52

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

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 161

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_)) 
*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 
 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m + n + 2)) 
 + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1 
) - c*d*(f*g + e*h)*(m + 1) + d^2*e*g*(m + n + 2)))*x)/(b*d*(b*c - a*d)^2*( 
m + 1)*(n + 1)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f* 
h*(2 + 3*n + 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*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
 d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b*c - a*d)^2*(m + 1)*( 
n + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h}, x] && LtQ[m, -1] && LtQ[n, -1]
 

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 = 0.77 (sec) , antiderivative size = 862, normalized size of antiderivative = 1.67

method	result	size
pseudoelliptic	\(\frac {\left (b x +a \right )^{3} \sqrt {f x +e}\, \left (\left (-35 c^{2} g \,f^{3}+30 c e \left (c h +2 d g \right ) f^{2}-48 \left (c h +\frac {d g}{2}\right ) d \,e^{2} f +16 d^{2} e^{3} h \right ) b^{3}+5 a \left (\left (h \,c^{2}+2 c d g \right ) f^{2}-\frac {24 \left (c h +\frac {d g}{2}\right ) d e f}{5}+\frac {24 d^{2} e^{2} h}{5}\right ) f \,b^{2}+2 a^{2} d \left (\left (c h +\frac {d g}{2}\right ) f -3 d e h \right ) f^{2} b +a^{3} d^{2} f^{3} h \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )-\sqrt {\left (a f -b e \right ) b}\, \left (\left (35 c^{2} f^{3} g \,x^{3}+\frac {35 x^{2} c \left (\frac {18 \left (-c h -2 d g \right ) x}{7}+c g \right ) e \,f^{2}}{3}-\frac {14 x \left (\frac {36 \left (-2 c d h -d^{2} g \right ) x^{2}}{7}+\frac {15 c \left (c h +2 d g \right ) x}{7}+c^{2} g \right ) e^{2} f}{3}+\frac {8 \left (-6 d^{2} h \,x^{3}+3 \left (2 c d h +d^{2} g \right ) x^{2}+3 \left (\frac {1}{2} h \,c^{2}+c d g \right ) x +c^{2} g \right ) e^{3}}{3}\right ) b^{5}+\frac {4 a \left (5 \left (-\frac {3 c \left (c h +2 d g \right ) x^{3}}{4}+14 c^{2} g \,x^{2}\right ) f^{3}+\frac {49 \left (\frac {18 \left (2 c d h +d^{2} g \right ) x^{2}}{49}+5 \left (-c d g -\frac {1}{2} h \,c^{2}\right ) x +c^{2} g \right ) x e \,f^{2}}{2}-\frac {19 \left (\frac {36 d^{2} h \,x^{3}}{19}-\frac {204 \left (c h +\frac {d g}{2}\right ) d \,x^{2}}{19}+\frac {41 c \left (c h +2 d g \right ) x}{19}+c^{2} g \right ) e^{2} f}{2}+\left (-54 d^{2} h \,x^{2}+6 \left (2 c d h +d^{2} g \right ) x +h \,c^{2}+2 c d g \right ) e^{3}\right ) b^{4}}{3}-\frac {28 a^{2} \left (-\frac {33 x \left (\frac {\left (-2 c d h -d^{2} g \right ) x^{2}}{77}-\frac {40 c \left (c h +2 d g \right ) x}{231}+c^{2} g \right ) f^{3}}{4}-\frac {87 \left (\frac {6 d^{2} h \,x^{3}}{29}+\frac {95 \left (2 c d h +d^{2} g \right ) x^{2}}{87}-\frac {212 c \left (c h +2 d g \right ) x}{87}+c^{2} g \right ) e \,f^{2}}{28}+\left (\frac {45 d^{2} h \,x^{2}}{14}-\frac {125 \left (c h +\frac {d g}{2}\right ) d x}{7}+h \,c^{2}+2 c d g \right ) e^{2} f -\frac {4 \left (-\frac {63}{4} d h x +c h +\frac {1}{2} d g \right ) d \,e^{3}}{7}\right ) b^{3}}{3}-27 a^{3} \left (\frac {\left (d^{2} h \,x^{3}+\frac {8 \left (2 c d h +d^{2} g \right ) x^{2}}{3}+11 c \left (c h +2 d g \right ) x -16 c^{2} g \right ) f^{3}}{27}+\left (\frac {17 d^{2} h \,x^{2}}{81}+\frac {38 \left (-2 c d h -d^{2} g \right ) x}{81}+h \,c^{2}+2 c d g \right ) e \,f^{2}-\frac {188 d \left (-\frac {29}{94} d h x +c h +\frac {1}{2} d g \right ) e^{2} f}{81}+\frac {92 d^{2} e^{3} h}{81}\right ) b^{2}+2 a^{4} d \left (\left (\frac {4}{3} d h x +c h +\frac {1}{2} d g \right ) f -\frac {8 d e h}{3}\right ) \left (f x +e \right ) f b +a^{5} d^{2} f^{2} h \left (f x +e \right )\right )}{8 \sqrt {f x +e}\, \sqrt {\left (a f -b e \right ) b}\, \left (b x +a \right )^{3} \left (a f -b e \right )^{4} b^{2}}\)	\(862\)
derivativedivides	\(\text {Expression too large to display}\)	\(1192\)
default	\(\text {Expression too large to display}\)	\(1192\)

Input:

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

Output:

1/8*((b*x+a)^3*(f*x+e)^(1/2)*((-35*c^2*g*f^3+30*c*e*(c*h+2*d*g)*f^2-48*(c* 
h+1/2*d*g)*d*e^2*f+16*d^2*e^3*h)*b^3+5*a*((c^2*h+2*c*d*g)*f^2-24/5*(c*h+1/ 
2*d*g)*d*e*f+24/5*d^2*e^2*h)*f*b^2+2*a^2*d*((c*h+1/2*d*g)*f-3*d*e*h)*f^2*b 
+a^3*d^2*f^3*h)*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))-((a*f-b*e)*b)^ 
(1/2)*((35*c^2*f^3*g*x^3+35/3*x^2*c*(18/7*(-c*h-2*d*g)*x+c*g)*e*f^2-14/3*x 
*(36/7*(-2*c*d*h-d^2*g)*x^2+15/7*c*(c*h+2*d*g)*x+c^2*g)*e^2*f+8/3*(-6*d^2* 
h*x^3+3*(2*c*d*h+d^2*g)*x^2+3*(1/2*h*c^2+c*d*g)*x+c^2*g)*e^3)*b^5+4/3*a*(5 
*(-3/4*c*(c*h+2*d*g)*x^3+14*c^2*g*x^2)*f^3+49/2*(18/49*(2*c*d*h+d^2*g)*x^2 
+5*(-c*d*g-1/2*h*c^2)*x+c^2*g)*x*e*f^2-19/2*(36/19*d^2*h*x^3-204/19*(c*h+1 
/2*d*g)*d*x^2+41/19*c*(c*h+2*d*g)*x+c^2*g)*e^2*f+(-54*d^2*h*x^2+6*(2*c*d*h 
+d^2*g)*x+h*c^2+2*c*d*g)*e^3)*b^4-28/3*a^2*(-33/4*x*(1/77*(-2*c*d*h-d^2*g) 
*x^2-40/231*c*(c*h+2*d*g)*x+c^2*g)*f^3-87/28*(6/29*d^2*h*x^3+95/87*(2*c*d* 
h+d^2*g)*x^2-212/87*c*(c*h+2*d*g)*x+c^2*g)*e*f^2+(45/14*d^2*h*x^2-125/7*(c 
*h+1/2*d*g)*d*x+h*c^2+2*c*d*g)*e^2*f-4/7*(-63/4*d*h*x+c*h+1/2*d*g)*d*e^3)* 
b^3-27*a^3*(1/27*(d^2*h*x^3+8/3*(2*c*d*h+d^2*g)*x^2+11*c*(c*h+2*d*g)*x-16* 
c^2*g)*f^3+(17/81*d^2*h*x^2+38/81*(-2*c*d*h-d^2*g)*x+h*c^2+2*c*d*g)*e*f^2- 
188/81*d*(-29/94*d*h*x+c*h+1/2*d*g)*e^2*f+92/81*d^2*e^3*h)*b^2+2*a^4*d*((4 
/3*d*h*x+c*h+1/2*d*g)*f-8/3*d*e*h)*(f*x+e)*f*b+a^5*d^2*f^2*h*(f*x+e)))/(f* 
x+e)^(1/2)/((a*f-b*e)*b)^(1/2)/(b*x+a)^3/(a*f-b*e)^4/b^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2940 vs. \(2 (491) = 982\).

Time = 0.72 (sec) , antiderivative size = 5894, normalized size of antiderivative = 11.40 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((d*x+c)^2*(h*x+g)/(b*x+a)^4/(f*x+e)^(3/2),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(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. 1761 vs. \(2 (491) = 982\).

Time = 0.17 (sec) , antiderivative size = 1761, normalized size of antiderivative = 3.41 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*(24*b^3*d^2*e^2*f*g - 60*b^3*c*d*e*f^2*g + 12*a*b^2*d^2*e*f^2*g + 35* 
b^3*c^2*f^3*g - 10*a*b^2*c*d*f^3*g - a^2*b*d^2*f^3*g - 16*b^3*d^2*e^3*h + 
48*b^3*c*d*e^2*f*h - 24*a*b^2*d^2*e^2*f*h - 30*b^3*c^2*e*f^2*h + 24*a*b^2* 
c*d*e*f^2*h + 6*a^2*b*d^2*e*f^2*h - 5*a*b^2*c^2*f^3*h - 2*a^2*b*c*d*f^3*h 
- a^3*d^2*f^3*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^6*e^4 - 
4*a*b^5*e^3*f + 6*a^2*b^4*e^2*f^2 - 4*a^3*b^3*e*f^3 + a^4*b^2*f^4)*sqrt(-b 
^2*e + a*b*f)) - 2*(d^2*e^2*f*g - 2*c*d*e*f^2*g + c^2*f^3*g - d^2*e^3*h + 
2*c*d*e^2*f*h - c^2*e*f^2*h)/((b^4*e^4 - 4*a*b^3*e^3*f + 6*a^2*b^2*e^2*f^2 
 - 4*a^3*b*e*f^3 + a^4*f^4)*sqrt(f*x + e)) - 1/24*(24*(f*x + e)^(5/2)*b^5* 
d^2*e^2*f*g - 48*(f*x + e)^(3/2)*b^5*d^2*e^3*f*g + 24*sqrt(f*x + e)*b^5*d^ 
2*e^4*f*g - 84*(f*x + e)^(5/2)*b^5*c*d*e*f^2*g + 36*(f*x + e)^(5/2)*a*b^4* 
d^2*e*f^2*g + 192*(f*x + e)^(3/2)*b^5*c*d*e^2*f^2*g - 48*(f*x + e)^(3/2)*a 
*b^4*d^2*e^2*f^2*g - 108*sqrt(f*x + e)*b^5*c*d*e^3*f^2*g + 12*sqrt(f*x + e 
)*a*b^4*d^2*e^3*f^2*g + 57*(f*x + e)^(5/2)*b^5*c^2*f^3*g - 30*(f*x + e)^(5 
/2)*a*b^4*c*d*f^3*g - 3*(f*x + e)^(5/2)*a^2*b^3*d^2*f^3*g - 136*(f*x + e)^ 
(3/2)*b^5*c^2*e*f^3*g - 112*(f*x + e)^(3/2)*a*b^4*c*d*e*f^3*g + 104*(f*x + 
 e)^(3/2)*a^2*b^3*d^2*e*f^3*g + 87*sqrt(f*x + e)*b^5*c^2*e^2*f^3*g + 150*s 
qrt(f*x + e)*a*b^4*c*d*e^2*f^3*g - 93*sqrt(f*x + e)*a^2*b^3*d^2*e^2*f^3*g 
+ 136*(f*x + e)^(3/2)*a*b^4*c^2*f^4*g - 80*(f*x + e)^(3/2)*a^2*b^3*c*d*f^4 
*g - 8*(f*x + e)^(3/2)*a^3*b^2*d^2*f^4*g - 174*sqrt(f*x + e)*a*b^4*c^2*...
 

Mupad [B] (verification not implemented)

Time = 3.16 (sec) , antiderivative size = 1085, normalized size of antiderivative = 2.10 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(atan(((e + f*x)^(1/2)*(b^6*e^4 + a^4*b^2*f^4 - 4*a^3*b^3*e*f^3 + 6*a^2*b^ 
4*e^2*f^2 - 4*a*b^5*e^3*f))/(b^(3/2)*(a*f - b*e)^(9/2)))*(a^3*d^2*f^3*h - 
35*b^3*c^2*f^3*g + 16*b^3*d^2*e^3*h + 5*a*b^2*c^2*f^3*h + a^2*b*d^2*f^3*g 
+ 30*b^3*c^2*e*f^2*h - 24*b^3*d^2*e^2*f*g + 10*a*b^2*c*d*f^3*g + 2*a^2*b*c 
*d*f^3*h + 60*b^3*c*d*e*f^2*g - 48*b^3*c*d*e^2*f*h - 12*a*b^2*d^2*e*f^2*g 
+ 24*a*b^2*d^2*e^2*f*h - 6*a^2*b*d^2*e*f^2*h - 24*a*b^2*c*d*e*f^2*h))/(8*b 
^(5/2)*(a*f - b*e)^(9/2)) - ((2*(c^2*f^3*g - d^2*e^3*h - c^2*e*f^2*h + d^2 
*e^2*f*g - 2*c*d*e*f^2*g + 2*c*d*e^2*f*h))/(a*f - b*e) - ((e + f*x)^3*(a^3 
*d^2*f^3*h - 35*b^3*c^2*f^3*g + 16*b^3*d^2*e^3*h + 5*a*b^2*c^2*f^3*h + a^2 
*b*d^2*f^3*g + 30*b^3*c^2*e*f^2*h - 24*b^3*d^2*e^2*f*g + 10*a*b^2*c*d*f^3* 
g + 2*a^2*b*c*d*f^3*h + 60*b^3*c*d*e*f^2*g - 48*b^3*c*d*e^2*f*h - 12*a*b^2 
*d^2*e*f^2*g + 24*a*b^2*d^2*e^2*f*h - 6*a^2*b*d^2*e*f^2*h - 24*a*b^2*c*d*e 
*f^2*h))/(8*(a*f - b*e)^4) - ((e + f*x)^2*(18*b^3*d^2*e^3*h - a^3*d^2*f^3* 
h - 35*b^3*c^2*f^3*g + 5*a*b^2*c^2*f^3*h + a^2*b*d^2*f^3*g + 30*b^3*c^2*e* 
f^2*h - 24*b^3*d^2*e^2*f*g + 10*a*b^2*c*d*f^3*g + 2*a^2*b*c*d*f^3*h + 60*b 
^3*c*d*e*f^2*g - 48*b^3*c*d*e^2*f*h - 12*a*b^2*d^2*e*f^2*g + 18*a*b^2*d^2* 
e^2*f*h - 24*a*b^2*c*d*e*f^2*h))/(3*b*(a*f - b*e)^3) + ((e + f*x)*(77*b^3* 
c^2*f^3*g + a^3*d^2*f^3*h - 48*b^3*d^2*e^3*h - 11*a*b^2*c^2*f^3*h + a^2*b* 
d^2*f^3*g - 66*b^3*c^2*e*f^2*h + 56*b^3*d^2*e^2*f*g - 22*a*b^2*c*d*f^3*g + 
 2*a^2*b*c*d*f^3*h - 132*b^3*c*d*e*f^2*g + 112*b^3*c*d*e^2*f*h + 20*a*b...
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 5440, normalized size of antiderivative = 10.52 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*s 
qrt(a*f - b*e)))*a**6*d**2*f**3*h + 6*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e 
)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*b*c*d*f**3*h - 18 
*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqr 
t(a*f - b*e)))*a**5*b*d**2*e*f**2*h + 3*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b 
*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*b*d**2*f**3*g + 
 9*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*s 
qrt(a*f - b*e)))*a**5*b*d**2*f**3*h*x + 15*sqrt(b)*sqrt(e + f*x)*sqrt(a*f 
- b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*c**2*f* 
*3*h - 72*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sq 
rt(b)*sqrt(a*f - b*e)))*a**4*b**2*c*d*e*f**2*h + 30*sqrt(b)*sqrt(e + f*x)* 
sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b** 
2*c*d*f**3*g + 18*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x 
)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*c*d*f**3*h*x + 72*sqrt(b)*sqrt(e 
 + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))* 
a**4*b**2*d**2*e**2*f*h - 36*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((s 
qrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*d**2*e*f**2*g - 54*sq 
rt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a 
*f - b*e)))*a**4*b**2*d**2*e*f**2*h*x + 9*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - 
 b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*d**2*...