Integrand size = 31, antiderivative size = 815 \[ \int (a+b x)^3 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\frac {(b c-a d)^2 (a d f+b (c f (2+m)-d e (3+m))) (c f h (4+m)-d (f g+e h (3+m))) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^4 f^2 (d e-c f) (3+m)}-\frac {b (b c-a d) (c f h (4+m)-d (f g+e h (3+m))) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^3 f^2}+\frac {h (a+b x)^3 (c+d x)^{-3-m} (e+f x)^{1+m}}{d f}-\frac {(b c-a d)^2 (3 a d f h-b (c f h (4+m)-d (f g+e h m))) (c+d x)^{-2-m} (e+f x)^{1+m}}{d^4 f (d e-c f) (2+m)}+\frac {(b c-a d) (c f h (4+m)-d (f g+e h (3+m))) \left (2 a^2 d^2 f^2+2 a b d f (c f (1+m)-d e (3+m))+b^2 \left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f \left (3+4 m+m^2\right )+d^2 e^2 \left (6+5 m+m^2\right )\right )\right ) (c+d x)^{-2-m} (e+f x)^{1+m}}{d^4 f^2 (d e-c f)^2 (2+m) (3+m)}-\frac {(b c-a d) (a d f-b (2 d e (2+m)-c f (3+2 m))) (3 a d f h-b (c f h (4+m)-d (f g+e h m))) (c+d x)^{-1-m} (e+f x)^{1+m}}{d^4 f (d e-c f)^2 (1+m) (2+m)}-\frac {(b c-a d) (c f h (4+m)-d (f g+e h (3+m))) \left (2 a^2 d^2 f^2+2 a b d f (c f (1+m)-d e (3+m))+b^2 \left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f \left (3+4 m+m^2\right )+d^2 e^2 \left (6+5 m+m^2\right )\right )\right ) (c+d x)^{-1-m} (e+f x)^{1+m}}{d^4 f (d e-c f)^3 (1+m) (2+m) (3+m)}-\frac {b^2 (3 a d f h-b (c f h (4+m)-d (f g+e h m))) (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d^5 f m} \]
(-a*d+b*c)^2*(a*d*f+b*(c*f*(2+m)-d*e*(3+m)))*(c*f*h*(4+m)-d*(f*g+e*h*(3+m) ))*(d*x+c)^(-3-m)*(f*x+e)^(1+m)/d^4/f^2/(-c*f+d*e)/(3+m)-b*(-a*d+b*c)*(c*f *h*(4+m)-d*(f*g+e*h*(3+m)))*(b*x+a)*(d*x+c)^(-3-m)*(f*x+e)^(1+m)/d^3/f^2+h *(b*x+a)^3*(d*x+c)^(-3-m)*(f*x+e)^(1+m)/d/f-(-a*d+b*c)^2*(3*a*d*f*h-b*(c*f *h*(4+m)-d*(e*h*m+f*g)))*(d*x+c)^(-2-m)*(f*x+e)^(1+m)/d^4/f/(-c*f+d*e)/(2+ m)+(-a*d+b*c)*(c*f*h*(4+m)-d*(f*g+e*h*(3+m)))*(2*a^2*d^2*f^2+2*a*b*d*f*(c* f*(1+m)-d*e*(3+m))+b^2*(c^2*f^2*(m^2+3*m+2)-2*c*d*e*f*(m^2+4*m+3)+d^2*e^2* (m^2+5*m+6)))*(d*x+c)^(-2-m)*(f*x+e)^(1+m)/d^4/f^2/(-c*f+d*e)^2/(2+m)/(3+m )-(-a*d+b*c)*(a*d*f-b*(2*d*e*(2+m)-c*f*(3+2*m)))*(3*a*d*f*h-b*(c*f*h*(4+m) -d*(e*h*m+f*g)))*(d*x+c)^(-1-m)*(f*x+e)^(1+m)/d^4/f/(-c*f+d*e)^2/(1+m)/(2+ m)-(-a*d+b*c)*(c*f*h*(4+m)-d*(f*g+e*h*(3+m)))*(2*a^2*d^2*f^2+2*a*b*d*f*(c* f*(1+m)-d*e*(3+m))+b^2*(c^2*f^2*(m^2+3*m+2)-2*c*d*e*f*(m^2+4*m+3)+d^2*e^2* (m^2+5*m+6)))*(d*x+c)^(-1-m)*(f*x+e)^(1+m)/d^4/f/(-c*f+d*e)^3/(1+m)/(2+m)/ (3+m)-b^2*(3*a*d*f*h-b*(c*f*h*(4+m)-d*(e*h*m+f*g)))*(f*x+e)^m*hypergeom([- m, -m],[1-m],-f*(d*x+c)/(-c*f+d*e))/d^5/f/m/((d*x+c)^m)/((d*(f*x+e)/(-c*f+ d*e))^m)
\[ \int (a+b x)^3 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\int (a+b x)^3 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx \]
Time = 0.90 (sec) , antiderivative size = 598, normalized size of antiderivative = 0.73, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {170, 25, 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.
\(\displaystyle \int (a+b x)^3 (g+h x) (c+d x)^{-m-4} (e+f x)^m \, dx\) |
\(\Big \downarrow \) 170 |
\(\displaystyle \frac {\int -(a+b x)^2 (c+d x)^{-m-4} (e+f x)^m (3 b c e h-a (d f g-c f h (m+1)+d e h (m+3))-(b d f g+3 a d f h+b d e h m-b c f h (m+4)) x)dx}{d f}+\frac {h (a+b x)^3 (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {h (a+b x)^3 (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}-\frac {\int (a+b x)^2 (c+d x)^{-m-4} (e+f x)^m (3 b c e h+a c f (m+1) h-a d (f g+e h (m+3))-(3 a d f h-b c f (m+4) h+b d (f g+e h m)) x)dx}{d f}\) |
\(\Big \downarrow \) 177 |
\(\displaystyle \frac {h (a+b x)^3 (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}-\frac {\frac {(b c-a d) (-c f h (m+4)+d e h (m+3)+d f g) \int (a+b x)^2 (c+d x)^{-m-4} (e+f x)^mdx}{d}-\frac {(3 a d f h-b c f h (m+4)+b d (e h m+f g)) \int (a+b x)^2 (c+d x)^{-m-3} (e+f x)^mdx}{d}}{d f}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {h (a+b x)^3 (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}-\frac {\frac {(b c-a d) (-c f h (m+4)+d e h (m+3)+d f g) \int (a+b x)^2 (c+d x)^{-m-4} (e+f x)^mdx}{d}-\frac {(3 a d f h-b c f h (m+4)+b d (e h m+f g)) \left (\frac {\int -(c+d x)^{-m-2} (e+f x)^m \left (-c (c f (m+1)-d e (m+2)) b^2-d (d e-c f) (m+2) x b^2+2 a d (c f (m+1)-d e (m+2)) b+a^2 d^2 f\right )dx}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}}{d f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {h (a+b x)^3 (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}-\frac {\frac {(b c-a d) (-c f h (m+4)+d e h (m+3)+d f g) \int (a+b x)^2 (c+d x)^{-m-4} (e+f x)^mdx}{d}-\frac {(3 a d f h-b c f h (m+4)+b d (e h m+f g)) \left (-\frac {\int (c+d x)^{-m-2} (e+f x)^m \left (-c (c f (m+1)-d e (m+2)) b^2-d (d e-c f) (m+2) x b^2+2 a d (c f (m+1)-d e (m+2)) b+a^2 d^2 f\right )dx}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}}{d f}\) |
\(\Big \downarrow \) 88 |
\(\displaystyle \frac {h (a+b x)^3 (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}-\frac {\frac {(b c-a d) (-c f h (m+4)+d e h (m+3)+d f g) \int (a+b x)^2 (c+d x)^{-m-4} (e+f x)^mdx}{d}-\frac {(3 a d f h-b c f h (m+4)+b d (e h m+f g)) \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}-b^2 (m+2) (d e-c f) \int (c+d x)^{-m-1} (e+f x)^mdx}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}}{d f}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {h (a+b x)^3 (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}-\frac {\frac {(b c-a d) (-c f h (m+4)+d e h (m+3)+d f g) \int (a+b x)^2 (c+d x)^{-m-4} (e+f x)^mdx}{d}-\frac {(3 a d f h-b c f h (m+4)+b d (e h m+f g)) \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}-b^2 (m+2) (d e-c f) (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \int (c+d x)^{-m-1} \left (\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}\right )^mdx}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}}{d f}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {h (a+b x)^3 (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}-\frac {\frac {(b c-a d) (-c f h (m+4)+d e h (m+3)+d f g) \int (a+b x)^2 (c+d x)^{-m-4} (e+f x)^mdx}{d}-\frac {(3 a d f h-b c f h (m+4)+b d (e h m+f g)) \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}+\frac {b^2 (m+2) (d e-c f) (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d m}}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}}{d f}\) |
\(\Big \downarrow \) 101 |
\(\displaystyle \frac {h (a+b x)^3 (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}-\frac {\frac {(b c-a d) (-c f h (m+4)+d e h (m+3)+d f g) \left (-\frac {\int -(c+d x)^{-m-4} (e+f x)^m \left (d f a^2+b (b c e-a d (m+3) e+a c f (m+1))-b^2 (d e-c f) (m+2) x\right )dx}{d f}-\frac {b (a+b x) (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}\right )}{d}-\frac {(3 a d f h-b c f h (m+4)+b d (e h m+f g)) \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}+\frac {b^2 (m+2) (d e-c f) (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d m}}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}}{d f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {h (a+b x)^3 (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}-\frac {\frac {(b c-a d) (-c f h (m+4)+d e h (m+3)+d f g) \left (\frac {\int (c+d x)^{-m-4} (e+f x)^m \left (d f a^2+b (b c e-a d (m+3) e+a c f (m+1))-b^2 (d e-c f) (m+2) x\right )dx}{d f}-\frac {b (a+b x) (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}\right )}{d}-\frac {(3 a d f h-b c f h (m+4)+b d (e h m+f g)) \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}+\frac {b^2 (m+2) (d e-c f) (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d m}}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}}{d f}\) |
\(\Big \downarrow \) 88 |
\(\displaystyle \frac {h (a+b x)^3 (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}-\frac {\frac {(b c-a d) (-c f h (m+4)+d e h (m+3)+d f g) \left (\frac {\frac {\left (\frac {b^2 (m+2) (c f (m+1)-d e (m+3))}{d}-\frac {2 f \left (a^2 d f+b (a c f (m+1)-a d e (m+3)+b c e)\right )}{d e-c f}\right ) \int (c+d x)^{-m-3} (e+f x)^mdx}{m+3}+\frac {(b c-a d) (c+d x)^{-m-3} (e+f x)^{m+1} (a d f+b c f (m+2)-b d e (m+3))}{d (m+3) (d e-c f)}}{d f}-\frac {b (a+b x) (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}\right )}{d}-\frac {(3 a d f h-b c f h (m+4)+b d (e h m+f g)) \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}+\frac {b^2 (m+2) (d e-c f) (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d m}}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}}{d f}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {h (a+b x)^3 (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}-\frac {\frac {(b c-a d) (-c f h (m+4)+d e h (m+3)+d f g) \left (\frac {\frac {\left (\frac {b^2 (m+2) (c f (m+1)-d e (m+3))}{d}-\frac {2 f \left (a^2 d f+b (a c f (m+1)-a d e (m+3)+b c e)\right )}{d e-c f}\right ) \left (-\frac {f \int (c+d x)^{-m-2} (e+f x)^mdx}{(m+2) (d e-c f)}-\frac {(c+d x)^{-m-2} (e+f x)^{m+1}}{(m+2) (d e-c f)}\right )}{m+3}+\frac {(b c-a d) (c+d x)^{-m-3} (e+f x)^{m+1} (a d f+b c f (m+2)-b d e (m+3))}{d (m+3) (d e-c f)}}{d f}-\frac {b (a+b x) (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}\right )}{d}-\frac {(3 a d f h-b c f h (m+4)+b d (e h m+f g)) \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}+\frac {b^2 (m+2) (d e-c f) (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d m}}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}}{d f}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {h (a+b x)^3 (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}-\frac {\frac {(b c-a d) (-c f h (m+4)+d e h (m+3)+d f g) \left (\frac {\frac {\left (\frac {f (c+d x)^{-m-1} (e+f x)^{m+1}}{(m+1) (m+2) (d e-c f)^2}-\frac {(c+d x)^{-m-2} (e+f x)^{m+1}}{(m+2) (d e-c f)}\right ) \left (\frac {b^2 (m+2) (c f (m+1)-d e (m+3))}{d}-\frac {2 f \left (a^2 d f+b (a c f (m+1)-a d e (m+3)+b c e)\right )}{d e-c f}\right )}{m+3}+\frac {(b c-a d) (c+d x)^{-m-3} (e+f x)^{m+1} (a d f+b c f (m+2)-b d e (m+3))}{d (m+3) (d e-c f)}}{d f}-\frac {b (a+b x) (c+d x)^{-m-3} (e+f x)^{m+1}}{d f}\right )}{d}-\frac {(3 a d f h-b c f h (m+4)+b d (e h m+f g)) \left (-\frac {\frac {(b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{(m+1) (d e-c f)}+\frac {b^2 (m+2) (d e-c f) (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {f (c+d x)}{d e-c f}\right )}{d m}}{d^2 (m+2) (d e-c f)}-\frac {(b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^2 (m+2) (d e-c f)}\right )}{d}}{d f}\) |
(h*(a + b*x)^3*(c + d*x)^(-3 - m)*(e + f*x)^(1 + m))/(d*f) - (((b*c - a*d) *(d*f*g + d*e*h*(3 + m) - c*f*h*(4 + m))*(-((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 - ((3*a*d*f*h - b*c*f*h*(4 + m) + b*d*(f*g + e*h*m))*(-(((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*Hypergeometric2 F1[-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)/(d*f)
3.2.32.3.1 Defintions of rubi rules used
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]
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])
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]))
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])
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]
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])))
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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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]))
\[\int \left (b x +a \right )^{3} \left (d x +c \right )^{-4-m} \left (f x +e \right )^{m} \left (h x +g \right )d x\]
\[ \int (a+b x)^3 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\int { {\left (b x + a\right )}^{3} {\left (h x + g\right )} {\left (d x + c\right )}^{-m - 4} {\left (f x + e\right )}^{m} \,d x } \]
integral((b^3*h*x^4 + a^3*g + (b^3*g + 3*a*b^2*h)*x^3 + 3*(a*b^2*g + a^2*b *h)*x^2 + (3*a^2*b*g + a^3*h)*x)*(d*x + c)^(-m - 4)*(f*x + e)^m, x)
Timed out. \[ \int (a+b x)^3 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\text {Timed out} \]
\[ \int (a+b x)^3 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\int { {\left (b x + a\right )}^{3} {\left (h x + g\right )} {\left (d x + c\right )}^{-m - 4} {\left (f x + e\right )}^{m} \,d x } \]
\[ \int (a+b x)^3 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\int { {\left (b x + a\right )}^{3} {\left (h x + g\right )} {\left (d x + c\right )}^{-m - 4} {\left (f x + e\right )}^{m} \,d x } \]
Timed out. \[ \int (a+b x)^3 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx=\int \frac {{\left (e+f\,x\right )}^m\,\left (g+h\,x\right )\,{\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{m+4}} \,d x \]