\(\int (a+b x)^m (c+d x)^{-5-m} (e+f x)^4 \, dx\) [1751]

Optimal result

Integrand size = 26, antiderivative size = 658 \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^4 \, dx=\frac {(d e-c f)^4 (a+b x)^{1+m} (c+d x)^{-4-m}}{d^4 (b c-a d) (4+m)}+\frac {(d e-c f)^3 (3 b d e-4 a d f (4+m)+b c f (13+4 m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{d^4 (b c-a d)^2 (3+m) (4+m)}+\frac {2 (d e-c f)^2 \left (3 a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (7+3 m))+b^2 \left (3 d^2 e^2+2 c d e f (5+2 m)+c^2 f^2 \left (23+17 m+3 m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{d^4 (b c-a d)^3 (2+m) (3+m) (4+m)}-\frac {\left (c (b c-a d)^3 f^4 (2+m) (3+m) (4+m)-(b c-a d)^2 f^3 (4 d e-3 c f) (3+m) (4+m) (b c (1+m)-a d (2+m))-b \left (a^2 d^2 f^2 \left (6 d^2 e^2-8 c d e f+3 c^2 f^2\right ) \left (12+7 m+m^2\right )+b^2 \left (6 d^4 e^4+8 c d^3 e^3 f (1+m)+6 c^2 d^2 e^2 f^2 \left (2+3 m+m^2\right )-8 c^3 d e f^3 \left (3+4 m+m^2\right )+c^4 f^4 \left (10+13 m+3 m^2\right )\right )-2 a b d f \left (4 d^3 e^3 (4+m)+6 c d^2 e^2 f \left (4+5 m+m^2\right )-4 c^2 d e f^2 \left (12+11 m+2 m^2\right )+c^3 f^3 \left (20+17 m+3 m^2\right )\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^4 (b c-a d)^4 (1+m) (2+m) (3+m) (4+m)}+\frac {f^4 (a+b x)^{1+m} (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,1,1-m,\frac {b (c+d x)}{b c-a d}\right )}{d^4 (b c-a d) m} \] Output:

(-c*f+d*e)^4*(b*x+a)^(1+m)*(d*x+c)^(-4-m)/d^4/(-a*d+b*c)/(4+m)+(-c*f+d*e)^ 
3*(3*b*d*e-4*a*d*f*(4+m)+b*c*f*(13+4*m))*(b*x+a)^(1+m)*(d*x+c)^(-3-m)/d^4/ 
(-a*d+b*c)^2/(3+m)/(4+m)+2*(-c*f+d*e)^2*(3*a^2*d^2*f^2*(m^2+7*m+12)-2*a*b* 
d*f*(4+m)*(2*d*e+c*f*(7+3*m))+b^2*(3*d^2*e^2+2*c*d*e*f*(5+2*m)+c^2*f^2*(3* 
m^2+17*m+23)))*(b*x+a)^(1+m)*(d*x+c)^(-2-m)/d^4/(-a*d+b*c)^3/(2+m)/(3+m)/( 
4+m)-(c*(-a*d+b*c)^3*f^4*(2+m)*(3+m)*(4+m)-(-a*d+b*c)^2*f^3*(-3*c*f+4*d*e) 
*(3+m)*(4+m)*(b*c*(1+m)-a*d*(2+m))-b*(a^2*d^2*f^2*(3*c^2*f^2-8*c*d*e*f+6*d 
^2*e^2)*(m^2+7*m+12)+b^2*(6*d^4*e^4+8*c*d^3*e^3*f*(1+m)+6*c^2*d^2*e^2*f^2* 
(m^2+3*m+2)-8*c^3*d*e*f^3*(m^2+4*m+3)+c^4*f^4*(3*m^2+13*m+10))-2*a*b*d*f*( 
4*d^3*e^3*(4+m)+6*c*d^2*e^2*f*(m^2+5*m+4)-4*c^2*d*e*f^2*(2*m^2+11*m+12)+c^ 
3*f^3*(3*m^2+17*m+20))))*(b*x+a)^(1+m)*(d*x+c)^(-1-m)/d^4/(-a*d+b*c)^4/(1+ 
m)/(2+m)/(3+m)/(4+m)+f^4*(b*x+a)^(1+m)*hypergeom([1, 1],[1-m],b*(d*x+c)/(- 
a*d+b*c))/d^4/(-a*d+b*c)/m/((d*x+c)^m)
 

Mathematica [A] (warning: unable to verify)

Time = 15.01 (sec) , antiderivative size = 523, normalized size of antiderivative = 0.79 \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^4 \, dx=(a+b x)^m (c+d x)^{-m} \left (-\frac {24 f^3 (a+b x)^4 (e+f x)}{(b c-a d)^4 (1+m) (2+m) (3+m) (4+m) (c+d x)}+\frac {12 f^2 (a+b x)^3 (-a d (1+m)+b c (4+m)+3 b d x) (e+f x)^2}{(b c-a d)^4 (1+m) (2+m) (3+m) (4+m) (c+d x)^2}+\frac {(e+f x)^4 \left (\frac {6 b^4}{(b c-a d)^4 (1+m) (2+m) (3+m) (4+m)}-\frac {1}{(4+m) (c+d x)^4}+\frac {b m}{(b c-a d) (3+m) (4+m) (c+d x)^3}+\frac {3 b^2 m}{(b c-a d)^2 (2+m) \left (12+7 m+m^2\right ) (c+d x)^2}+\frac {6 b^3 m}{(b c-a d)^3 (1+m) \left (24+26 m+9 m^2+m^3\right ) (c+d x)}\right )}{d}-\frac {4 f (a+b x)^2 (e+f x)^3 \left (a^2 d^2 \left (2+3 m+m^2\right )-2 a b d (1+m) (c (4+m)+2 d x)+b^2 \left (c^2 \left (12+7 m+m^2\right )+4 c d (4+m) x+6 d^2 x^2\right )\right )}{(b c-a d)^4 (1+m) (2+m) (3+m) (4+m) (c+d x)^3}-\frac {24 f^4 \left (\frac {d (a+b x)}{-b c+a d}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-4-m,-m,1-m,\frac {b (c+d x)}{b c-a d}\right )}{d^5 m (1+m) (2+m) (3+m) (4+m)}\right ) \] Input:

Integrate[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x)^4,x]
 

Output:

((a + b*x)^m*((-24*f^3*(a + b*x)^4*(e + f*x))/((b*c - a*d)^4*(1 + m)*(2 + 
m)*(3 + m)*(4 + m)*(c + d*x)) + (12*f^2*(a + b*x)^3*(-(a*d*(1 + m)) + b*c* 
(4 + m) + 3*b*d*x)*(e + f*x)^2)/((b*c - a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 
+ m)*(c + d*x)^2) + ((e + f*x)^4*((6*b^4)/((b*c - a*d)^4*(1 + m)*(2 + m)*( 
3 + m)*(4 + m)) - 1/((4 + m)*(c + d*x)^4) + (b*m)/((b*c - a*d)*(3 + m)*(4 
+ m)*(c + d*x)^3) + (3*b^2*m)/((b*c - a*d)^2*(2 + m)*(12 + 7*m + m^2)*(c + 
 d*x)^2) + (6*b^3*m)/((b*c - a*d)^3*(1 + m)*(24 + 26*m + 9*m^2 + m^3)*(c + 
 d*x))))/d - (4*f*(a + b*x)^2*(e + f*x)^3*(a^2*d^2*(2 + 3*m + m^2) - 2*a*b 
*d*(1 + m)*(c*(4 + m) + 2*d*x) + b^2*(c^2*(12 + 7*m + m^2) + 4*c*d*(4 + m) 
*x + 6*d^2*x^2)))/((b*c - a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(c + d*x) 
^3) - (24*f^4*Hypergeometric2F1[-4 - m, -m, 1 - m, (b*(c + d*x))/(b*c - a* 
d)])/(d^5*m*(1 + m)*(2 + m)*(3 + m)*(4 + m)*((d*(a + b*x))/(-(b*c) + a*d)) 
^m)))/(c + d*x)^m
 

Rubi [A] (warning: unable to verify)

Time = 0.84 (sec) , antiderivative size = 650, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {137, 2009}

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.

\(\displaystyle \int (e+f x)^4 (a+b x)^m (c+d x)^{-m-5} \, dx\)

\(\Big \downarrow \) 137

\(\displaystyle \int \left (\frac {4 f^3 (a+b x)^m (d e-c f) (c+d x)^{-m-2}}{d^4}+\frac {6 f^2 (a+b x)^m (d e-c f)^2 (c+d x)^{-m-3}}{d^4}+\frac {(a+b x)^m (d e-c f)^4 (c+d x)^{-m-5}}{d^4}+\frac {4 f (a+b x)^m (d e-c f)^3 (c+d x)^{-m-4}}{d^4}+\frac {f^4 (a+b x)^m (c+d x)^{-m-1}}{d^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6 b^3 (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac {6 b^2 (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-2}}{d^4 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac {8 b^2 f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (m+3) (b c-a d)^3}-\frac {f^4 (a+b x)^m (c+d x)^{-m} \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,\frac {b (c+d x)}{b c-a d}\right )}{d^5 m}+\frac {4 f^3 (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d^4 (m+1) (b c-a d)}+\frac {6 f^2 (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^4 (m+2) (b c-a d)}+\frac {6 b f^2 (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-1}}{d^4 (m+1) (m+2) (b c-a d)^2}+\frac {(a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-4}}{d^4 (m+4) (b c-a d)}+\frac {4 f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-3}}{d^4 (m+3) (b c-a d)}+\frac {3 b (a+b x)^{m+1} (d e-c f)^4 (c+d x)^{-m-3}}{d^4 (m+3) (m+4) (b c-a d)^2}+\frac {8 b f (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-2}}{d^4 (m+2) (m+3) (b c-a d)^2}\)

Input:

Int[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x)^4,x]
 

Output:

((d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/(d^4*(b*c - a*d)*(4 + 
 m)) + (4*f*(d*e - c*f)^3*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/(d^4*(b*c 
- a*d)*(3 + m)) + (3*b*(d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m)) 
/(d^4*(b*c - a*d)^2*(3 + m)*(4 + m)) + (6*f^2*(d*e - c*f)^2*(a + b*x)^(1 + 
 m)*(c + d*x)^(-2 - m))/(d^4*(b*c - a*d)*(2 + m)) + (8*b*f*(d*e - c*f)^3*( 
a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d^4*(b*c - a*d)^2*(2 + m)*(3 + m)) + 
 (6*b^2*(d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d^4*(b*c - a* 
d)^3*(2 + m)*(3 + m)*(4 + m)) + (4*f^3*(d*e - c*f)*(a + b*x)^(1 + m)*(c + 
d*x)^(-1 - m))/(d^4*(b*c - a*d)*(1 + m)) + (6*b*f^2*(d*e - c*f)^2*(a + b*x 
)^(1 + m)*(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)^2*(1 + m)*(2 + m)) + (8*b^2 
*f*(d*e - c*f)^3*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^4*(b*c - a*d)^3* 
(1 + m)*(2 + m)*(3 + m)) + (6*b^3*(d*e - c*f)^4*(a + b*x)^(1 + m)*(c + d*x 
)^(-1 - m))/(d^4*(b*c - a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)) - (f^4*(a 
+ b*x)^m*Hypergeometric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(d^5 
*m*(-((d*(a + b*x))/(b*c - a*d)))^m*(c + d*x)^m)
 

Defintions of rubi rules used

rule 137

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (IGtQ[m, 0] || (ILtQ[m, 0] && 
ILtQ[n, 0]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \left (b x +a \right )^{m} \left (x d +c \right )^{-5-m} \left (f x +e \right )^{4}d x\]

Input:

int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x)
 

Output:

int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x)
 

Fricas [F]

\[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^4 \, dx=\int { {\left (f x + e\right )}^{4} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x, algorithm="fricas")
 

Output:

integral((f^4*x^4 + 4*e*f^3*x^3 + 6*e^2*f^2*x^2 + 4*e^3*f*x + e^4)*(b*x + 
a)^m*(d*x + c)^(-m - 5), x)
 

Sympy [F(-2)]

Exception generated. \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^4 \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((b*x+a)**m*(d*x+c)**(-5-m)*(f*x+e)**4,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^4 \, dx=\int { {\left (f x + e\right )}^{4} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x, algorithm="maxima")
 

Output:

integrate((f*x + e)^4*(b*x + a)^m*(d*x + c)^(-m - 5), x)
 

Giac [F]

\[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^4 \, dx=\int { {\left (f x + e\right )}^{4} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x, algorithm="giac")
 

Output:

integrate((f*x + e)^4*(b*x + a)^m*(d*x + c)^(-m - 5), x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^4 \, dx=\int \frac {{\left (e+f\,x\right )}^4\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+5}} \,d x \] Input:

int(((e + f*x)^4*(a + b*x)^m)/(c + d*x)^(m + 5),x)
 

Output:

int(((e + f*x)^4*(a + b*x)^m)/(c + d*x)^(m + 5), x)
 

Reduce [F]

\[ \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^4 \, dx=\left (\int \frac {\left (b x +a \right )^{m}}{\left (d x +c \right )^{m} c^{5}+5 \left (d x +c \right )^{m} c^{4} d x +10 \left (d x +c \right )^{m} c^{3} d^{2} x^{2}+10 \left (d x +c \right )^{m} c^{2} d^{3} x^{3}+5 \left (d x +c \right )^{m} c \,d^{4} x^{4}+\left (d x +c \right )^{m} d^{5} x^{5}}d x \right ) e^{4}+\left (\int \frac {\left (b x +a \right )^{m} x^{4}}{\left (d x +c \right )^{m} c^{5}+5 \left (d x +c \right )^{m} c^{4} d x +10 \left (d x +c \right )^{m} c^{3} d^{2} x^{2}+10 \left (d x +c \right )^{m} c^{2} d^{3} x^{3}+5 \left (d x +c \right )^{m} c \,d^{4} x^{4}+\left (d x +c \right )^{m} d^{5} x^{5}}d x \right ) f^{4}+4 \left (\int \frac {\left (b x +a \right )^{m} x^{3}}{\left (d x +c \right )^{m} c^{5}+5 \left (d x +c \right )^{m} c^{4} d x +10 \left (d x +c \right )^{m} c^{3} d^{2} x^{2}+10 \left (d x +c \right )^{m} c^{2} d^{3} x^{3}+5 \left (d x +c \right )^{m} c \,d^{4} x^{4}+\left (d x +c \right )^{m} d^{5} x^{5}}d x \right ) e \,f^{3}+6 \left (\int \frac {\left (b x +a \right )^{m} x^{2}}{\left (d x +c \right )^{m} c^{5}+5 \left (d x +c \right )^{m} c^{4} d x +10 \left (d x +c \right )^{m} c^{3} d^{2} x^{2}+10 \left (d x +c \right )^{m} c^{2} d^{3} x^{3}+5 \left (d x +c \right )^{m} c \,d^{4} x^{4}+\left (d x +c \right )^{m} d^{5} x^{5}}d x \right ) e^{2} f^{2}+4 \left (\int \frac {\left (b x +a \right )^{m} x}{\left (d x +c \right )^{m} c^{5}+5 \left (d x +c \right )^{m} c^{4} d x +10 \left (d x +c \right )^{m} c^{3} d^{2} x^{2}+10 \left (d x +c \right )^{m} c^{2} d^{3} x^{3}+5 \left (d x +c \right )^{m} c \,d^{4} x^{4}+\left (d x +c \right )^{m} d^{5} x^{5}}d x \right ) e^{3} f \] Input:

int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^4,x)
 

Output:

int((a + b*x)**m/((c + d*x)**m*c**5 + 5*(c + d*x)**m*c**4*d*x + 10*(c + d* 
x)**m*c**3*d**2*x**2 + 10*(c + d*x)**m*c**2*d**3*x**3 + 5*(c + d*x)**m*c*d 
**4*x**4 + (c + d*x)**m*d**5*x**5),x)*e**4 + int(((a + b*x)**m*x**4)/((c + 
 d*x)**m*c**5 + 5*(c + d*x)**m*c**4*d*x + 10*(c + d*x)**m*c**3*d**2*x**2 + 
 10*(c + d*x)**m*c**2*d**3*x**3 + 5*(c + d*x)**m*c*d**4*x**4 + (c + d*x)** 
m*d**5*x**5),x)*f**4 + 4*int(((a + b*x)**m*x**3)/((c + d*x)**m*c**5 + 5*(c 
 + d*x)**m*c**4*d*x + 10*(c + d*x)**m*c**3*d**2*x**2 + 10*(c + d*x)**m*c** 
2*d**3*x**3 + 5*(c + d*x)**m*c*d**4*x**4 + (c + d*x)**m*d**5*x**5),x)*e*f* 
*3 + 6*int(((a + b*x)**m*x**2)/((c + d*x)**m*c**5 + 5*(c + d*x)**m*c**4*d* 
x + 10*(c + d*x)**m*c**3*d**2*x**2 + 10*(c + d*x)**m*c**2*d**3*x**3 + 5*(c 
 + d*x)**m*c*d**4*x**4 + (c + d*x)**m*d**5*x**5),x)*e**2*f**2 + 4*int(((a 
+ b*x)**m*x)/((c + d*x)**m*c**5 + 5*(c + d*x)**m*c**4*d*x + 10*(c + d*x)** 
m*c**3*d**2*x**2 + 10*(c + d*x)**m*c**2*d**3*x**3 + 5*(c + d*x)**m*c*d**4* 
x**4 + (c + d*x)**m*d**5*x**5),x)*e**3*f