3.2.33 \(\int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx\) [133]

3.2.33.1 Optimal result

Integrand size = 31, antiderivative size = 572 \[ \int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\frac {(b c-a d) (d g-c h) (a d f+b (c f (2+m)-d e (3+m))) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^3 f (d e-c f) (3+m)}-\frac {b (d g-c h) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^2 f}-\frac {(b c-a d)^2 h (c+d x)^{-2-m} (e+f x)^{1+m}}{d^3 (d e-c f) (2+m)}-\frac {(d g-c h) \left (b^2 (d e-c f) (2+m) (c f (1+m)-d e (3+m))-2 d f \left (b^2 c e+a^2 d f+a b (c f (1+m)-d e (3+m))\right )\right ) (c+d x)^{-2-m} (e+f x)^{1+m}}{d^3 f (d e-c f)^2 (2+m) (3+m)}-\frac {(b c-a d) h (a d f-b (2 d e (2+m)-c f (3+2 m))) (c+d x)^{-1-m} (e+f x)^{1+m}}{d^3 (d e-c f)^2 (1+m) (2+m)}+\frac {(d g-c h) \left (b^2 (d e-c f) (2+m) (c f (1+m)-d e (3+m))-2 d f \left (b^2 c e+a^2 d f+a b (c f (1+m)-d e (3+m))\right )\right ) (c+d x)^{-1-m} (e+f x)^{1+m}}{d^3 (d e-c f)^3 (1+m) (2+m) (3+m)}-\frac {b^2 h (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d^4 m} \]

output

(-a*d+b*c)*(-c*h+d*g)*(a*d*f+b*(c*f*(2+m)-d*e*(3+m)))*(d*x+c)^(-3-m)*(f*x+ 
e)^(1+m)/d^3/f/(-c*f+d*e)/(3+m)-b*(-c*h+d*g)*(b*x+a)*(d*x+c)^(-3-m)*(f*x+e 
)^(1+m)/d^2/f-(-a*d+b*c)^2*h*(d*x+c)^(-2-m)*(f*x+e)^(1+m)/d^3/(-c*f+d*e)/( 
2+m)-(-c*h+d*g)*(b^2*(-c*f+d*e)*(2+m)*(c*f*(1+m)-d*e*(3+m))-2*d*f*(b^2*c*e 
+a^2*d*f+a*b*(c*f*(1+m)-d*e*(3+m))))*(d*x+c)^(-2-m)*(f*x+e)^(1+m)/d^3/f/(- 
c*f+d*e)^2/(2+m)/(3+m)-(-a*d+b*c)*h*(a*d*f-b*(2*d*e*(2+m)-c*f*(3+2*m)))*(d 
*x+c)^(-1-m)*(f*x+e)^(1+m)/d^3/(-c*f+d*e)^2/(1+m)/(2+m)+(-c*h+d*g)*(b^2*(- 
c*f+d*e)*(2+m)*(c*f*(1+m)-d*e*(3+m))-2*d*f*(b^2*c*e+a^2*d*f+a*b*(c*f*(1+m) 
-d*e*(3+m))))*(d*x+c)^(-1-m)*(f*x+e)^(1+m)/d^3/(-c*f+d*e)^3/(1+m)/(2+m)/(3 
+m)-b^2*h*(f*x+e)^m*hypergeom([-m, -m],[1-m],-f*(d*x+c)/(-c*f+d*e))/d^4/m/ 
((d*x+c)^m)/((d*(f*x+e)/(-c*f+d*e))^m)
 

3.2.33.2 Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.74 \[ \int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\frac {(c+d x)^{-3-m} (e+f x)^m \left (-d (d g-c h) (e+f x) \left (-\left ((b c-a d) (d e-c f)^2 (1+m) (2+m) (a d f+b c f (2+m)-b d e (3+m))\right )+b d (d e-c f)^3 (1+m) (2+m) (3+m) (a+b x)+\left (b^2 (d e-c f) (2+m) (c f (1+m)-d e (3+m))+2 d f \left (-a^2 d f-b (b c e+a c f (1+m)-a d e (3+m))\right )\right ) (c+d x) (-c f (2+m)+d (e+e m-f x))\right )-(d e-c f) h (3+m) (c+d x) \left (d (b c-a d)^2 f (d e-c f) (1+m) (e+f x)-(c+d x) \left (d \left (a^2 d^2 f^2+2 a b d f (c f (1+m)-d e (2+m))+b^2 \left (-c^2 f^2 (1+m)+d^2 e^2 (2+m)\right )\right ) (e+f x)-b^2 (d e-c f)^3 (2+m) \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-1-m,-1-m,-m,\frac {f (c+d x)}{-d e+c f}\right )\right )\right )\right )}{d^4 f (d e-c f)^3 (1+m) (2+m) (3+m)} \]

input

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

output

((c + d*x)^(-3 - m)*(e + f*x)^m*(-(d*(d*g - c*h)*(e + f*x)*(-((b*c - a*d)* 
(d*e - c*f)^2*(1 + m)*(2 + m)*(a*d*f + b*c*f*(2 + m) - b*d*e*(3 + m))) + b 
*d*(d*e - c*f)^3*(1 + m)*(2 + m)*(3 + m)*(a + b*x) + (b^2*(d*e - c*f)*(2 + 
 m)*(c*f*(1 + m) - d*e*(3 + m)) + 2*d*f*(-(a^2*d*f) - b*(b*c*e + a*c*f*(1 
+ m) - a*d*e*(3 + m))))*(c + d*x)*(-(c*f*(2 + m)) + d*(e + e*m - f*x)))) - 
 (d*e - c*f)*h*(3 + m)*(c + d*x)*(d*(b*c - a*d)^2*f*(d*e - c*f)*(1 + m)*(e 
 + f*x) - (c + d*x)*(d*(a^2*d^2*f^2 + 2*a*b*d*f*(c*f*(1 + m) - d*e*(2 + m) 
) + b^2*(-(c^2*f^2*(1 + m)) + d^2*e^2*(2 + m)))*(e + f*x) - (b^2*(d*e - c* 
f)^3*(2 + m)*Hypergeometric2F1[-1 - m, -1 - m, -m, (f*(c + d*x))/(-(d*e) + 
 c*f)])/((d*(e + f*x))/(d*e - c*f))^m))))/(d^4*f*(d*e - c*f)^3*(1 + m)*(2 
+ m)*(3 + m))
 

3.2.33.3 Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 507, normalized size of antiderivative = 0.89, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {177, 100, 25, 88, 80, 79, 101, 25, 88, 55, 48}

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.

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

\(\Big \downarrow \) 177

\(\displaystyle \frac {(d g-c h) \int (a+b x)^2 (c+d x)^{-m-4} (e+f x)^mdx}{d}+\frac {h \int (a+b x)^2 (c+d x)^{-m-3} (e+f x)^mdx}{d}\)

\(\Big \downarrow \) 100

\(\displaystyle \frac {h \left (\frac {\int -(c+d x)^{-m-2} (e+f x)^m \left (-c (c f (m+1)-d e (m+2)) b^2-d (d e-c f) (m+2) x b^2+2 a d (c f (m+1)-d e (m+2)) b+a^2 d^2 f\right )dx}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}+\frac {(d g-c h) \int (a+b x)^2 (c+d x)^{-m-4} (e+f x)^mdx}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {h \left (-\frac {\int (c+d x)^{-m-2} (e+f x)^m \left (-c (c f (m+1)-d e (m+2)) b^2-d (d e-c f) (m+2) x b^2+2 a d (c f (m+1)-d e (m+2)) b+a^2 d^2 f\right )dx}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}+\frac {(d g-c h) \int (a+b x)^2 (c+d x)^{-m-4} (e+f x)^mdx}{d}\)

\(\Big \downarrow \) 88

\(\displaystyle \frac {h \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}-b^2 (m+2) (d e-c f) \int (c+d x)^{-m-1} (e+f x)^mdx}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}+\frac {(d g-c h) \int (a+b x)^2 (c+d x)^{-m-4} (e+f x)^mdx}{d}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {h \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}-b^2 (m+2) (d e-c f) (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \int (c+d x)^{-m-1} \left (\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}\right )^mdx}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}+\frac {(d g-c h) \int (a+b x)^2 (c+d x)^{-m-4} (e+f x)^mdx}{d}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {(d g-c h) \int (a+b x)^2 (c+d x)^{-m-4} (e+f x)^mdx}{d}+\frac {h \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}+\frac {b^2 (m+2) (d e-c f) (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d m}}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 101

\(\displaystyle \frac {(d g-c h) \left (-\frac {\int -(c+d x)^{-m-4} (e+f x)^m \left (d f a^2+b (b c e-a d (m+3) e+a c f (m+1))-b^2 (d e-c f) (m+2) x\right )dx}{d f}-\frac {b (a+b x) (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}\right )}{d}+\frac {h \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}+\frac {b^2 (m+2) (d e-c f) (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d m}}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(d g-c h) \left (\frac {\int (c+d x)^{-m-4} (e+f x)^m \left (d f a^2+b (b c e-a d (m+3) e+a c f (m+1))-b^2 (d e-c f) (m+2) x\right )dx}{d f}-\frac {b (a+b x) (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}\right )}{d}+\frac {h \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}+\frac {b^2 (m+2) (d e-c f) (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d m}}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 88

\(\displaystyle \frac {(d g-c h) \left (\frac {\frac {\left (\frac {b^2 (m+2) (c f (m+1)-d e (m+3))}{d}-\frac {2 f \left (a^2 d f+b (a c f (m+1)-a d e (m+3)+b c e)\right )}{d e-c f}\right ) \int (c+d x)^{-m-3} (e+f x)^mdx}{m+3}+\frac {(b c-a d) (c+d x)^{-m-3} (e+f x)^{m+1} (a d f+b c f (m+2)-b d e (m+3))}{d (m+3) (d e-c f)}}{d f}-\frac {b (a+b x) (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}\right )}{d}+\frac {h \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}+\frac {b^2 (m+2) (d e-c f) (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d m}}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(d g-c h) \left (\frac {\frac {\left (\frac {b^2 (m+2) (c f (m+1)-d e (m+3))}{d}-\frac {2 f \left (a^2 d f+b (a c f (m+1)-a d e (m+3)+b c e)\right )}{d e-c f}\right ) \left (-\frac {f \int (c+d x)^{-m-2} (e+f x)^mdx}{(m+2) (d e-c f)}-\frac {(c+d x)^{-m-2} (e+f x)^{m+1}}{(m+2) (d e-c f)}\right )}{m+3}+\frac {(b c-a d) (c+d x)^{-m-3} (e+f x)^{m+1} (a d f+b c f (m+2)-b d e (m+3))}{d (m+3) (d e-c f)}}{d f}-\frac {b (a+b x) (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}\right )}{d}+\frac {h \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}+\frac {b^2 (m+2) (d e-c f) (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d m}}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {(d g-c h) \left (\frac {\frac {\left (\frac {f (c+d x)^{-m-1} (e+f x)^{m+1}}{(m+1) (m+2) (d e-c f)^2}-\frac {(c+d x)^{-m-2} (e+f x)^{m+1}}{(m+2) (d e-c f)}\right ) \left (\frac {b^2 (m+2) (c f (m+1)-d e (m+3))}{d}-\frac {2 f \left (a^2 d f+b (a c f (m+1)-a d e (m+3)+b c e)\right )}{d e-c f}\right )}{m+3}+\frac {(b c-a d) (c+d x)^{-m-3} (e+f x)^{m+1} (a d f+b c f (m+2)-b d e (m+3))}{d (m+3) (d e-c f)}}{d f}-\frac {b (a+b x) (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}\right )}{d}+\frac {h \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}+\frac {b^2 (m+2) (d e-c f) (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d m}}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}\)

input

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

output

((d*g - c*h)*(-((b*(a + b*x)*(c + d*x)^(-3 - m)*(e + f*x)^(1 + m))/(d*f)) 
+ (((b*c - a*d)*(a*d*f + b*c*f*(2 + m) - b*d*e*(3 + m))*(c + d*x)^(-3 - m) 
*(e + f*x)^(1 + m))/(d*(d*e - c*f)*(3 + m)) + (((b^2*(2 + m)*(c*f*(1 + m) 
- d*e*(3 + m)))/d - (2*f*(a^2*d*f + b*(b*c*e + a*c*f*(1 + m) - a*d*e*(3 + 
m))))/(d*e - c*f))*(-(((c + d*x)^(-2 - m)*(e + f*x)^(1 + m))/((d*e - c*f)* 
(2 + m))) + (f*(c + d*x)^(-1 - m)*(e + f*x)^(1 + m))/((d*e - c*f)^2*(1 + m 
)*(2 + m))))/(3 + m))/(d*f)))/d + (h*(-(((b*c - a*d)^2*(c + d*x)^(-2 - m)* 
(e + f*x)^(1 + m))/(d^2*(d*e - c*f)*(2 + m))) - (((b*c - a*d)*(a*d*f - 2*b 
*d*e*(2 + m) + b*c*f*(3 + 2*m))*(c + d*x)^(-1 - m)*(e + f*x)^(1 + m))/((d* 
e - c*f)*(1 + m)) + (b^2*(d*e - c*f)*(2 + m)*(e + f*x)^m*Hypergeometric2F1 
[-m, -m, 1 - m, -((f*(c + d*x))/(d*e - c*f))])/(d*m*(c + d*x)^m*((d*(e + f 
*x))/(d*e - c*f))^m))/(d^2*(d*e - c*f)*(2 + m))))/d
 

3.2.33.3.1 Defintions of rubi rules used

rule 25

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

rule 48

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] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

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*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 1]
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 177

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h/b   Int[(a + b*x)^(m + 1)*(c + d 
*x)^n*(e + f*x)^p, x], x] + Simp[(b*g - a*h)/b   Int[(a + b*x)^m*(c + d*x)^ 
n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (Su 
mSimplerQ[m, 1] || ( !SumSimplerQ[n, 1] &&  !SumSimplerQ[p, 1]))
 

3.2.33.4 Maple [F]

\[\int \left (b x +a \right )^{2} \left (d x +c \right )^{-4-m} \left (f x +e \right )^{m} \left (h x +g \right )d x\]

input

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

output

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

3.2.33.5 Fricas [F]

\[ \int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\int { {\left (b x + a\right )}^{2} {\left (h x + g\right )} {\left (d x + c\right )}^{-m - 4} {\left (f x + e\right )}^{m} \,d x } \]

input

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

output

integral((b^2*h*x^3 + a^2*g + (b^2*g + 2*a*b*h)*x^2 + (2*a*b*g + a^2*h)*x) 
*(d*x + c)^(-m - 4)*(f*x + e)^m, x)
 

3.2.33.6 Sympy [F(-2)]

Exception generated. \[ \int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\text {Exception raised: HeuristicGCDFailed} \]

input

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

output

Exception raised: HeuristicGCDFailed >> no luck
 

3.2.33.7 Maxima [F]

\[ \int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\int { {\left (b x + a\right )}^{2} {\left (h x + g\right )} {\left (d x + c\right )}^{-m - 4} {\left (f x + e\right )}^{m} \,d x } \]

input

integrate((b*x+a)^2*(d*x+c)^(-4-m)*(f*x+e)^m*(h*x+g),x, algorithm="maxima" 
)
 

output

integrate((b*x + a)^2*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m, x)
 

3.2.33.8 Giac [F]

\[ \int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\int { {\left (b x + a\right )}^{2} {\left (h x + g\right )} {\left (d x + c\right )}^{-m - 4} {\left (f x + e\right )}^{m} \,d x } \]

input

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

output

integrate((b*x + a)^2*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m, x)
 

3.2.33.9 Mupad [F(-1)]

Timed out. \[ \int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\int \frac {{\left (e+f\,x\right )}^m\,\left (g+h\,x\right )\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^{m+4}} \,d x \]

input

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

output

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