\(\int (a+b x)^m (c+d x)^{-4-m} (e+f x)^3 \, dx\) [1744]

Optimal result

Integrand size = 26, antiderivative size = 328 \[ \int (a+b x)^m (c+d x)^{-4-m} (e+f x)^3 \, dx=\frac {(d e-c f)^3 (a+b x)^{1+m} (c+d x)^{-3-m}}{d^3 (b c-a d) (3+m)}+\frac {(d e-c f)^2 (2 b d e-3 a d f (3+m)+b c f (7+3 m)) (a+b x)^{1+m} (c+d x)^{-2-m}}{d^3 (b c-a d)^2 (2+m) (3+m)}+\frac {(d e-c f) \left (3 a^2 d^2 f^2 \left (6+5 m+m^2\right )-3 a b d f (3+m) (d e+c f (3+2 m))+b^2 \left (2 d^2 e^2+c d e f (5+3 m)+c^2 f^2 \left (11+12 m+3 m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^3 (b c-a d)^3 (1+m) (2+m) (3+m)}+\frac {f^3 (a+b x)^{1+m} (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,1,1-m,\frac {b (c+d x)}{b c-a d}\right )}{d^3 (b c-a d) m} \] Output:

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

Mathematica [A] (verified)

Time = 12.48 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.85 \[ \int (a+b x)^m (c+d x)^{-4-m} (e+f x)^3 \, dx=\frac {(a+b x)^m (c+d x)^{-m} \left (\frac {6 f^2 (a+b x)^3 (e+f x)}{(b c-a d)^3 (c+d x)}-\frac {3 f (a+b x)^2 (-a d (1+m)+b c (3+m)+2 b d x) (e+f x)^2}{(b c-a d)^3 (c+d x)^2}+\frac {(a+b x) (e+f x)^3 \left (a^2 d^2 \left (2+3 m+m^2\right )-2 a b d (1+m) (c (3+m)+d x)+b^2 \left (c^2 \left (6+5 m+m^2\right )+2 c d (3+m) x+2 d^2 x^2\right )\right )}{(b c-a d)^3 (c+d x)^3}-\frac {6 f^3 \left (\frac {d (a+b x)}{-b c+a d}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-3-m,-m,1-m,\frac {b (c+d x)}{b c-a d}\right )}{d^4 m}\right )}{(1+m) (2+m) (3+m)} \] Input:

Integrate[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x)^3,x]
 

Output:

((a + b*x)^m*((6*f^2*(a + b*x)^3*(e + f*x))/((b*c - a*d)^3*(c + d*x)) - (3 
*f*(a + b*x)^2*(-(a*d*(1 + m)) + b*c*(3 + m) + 2*b*d*x)*(e + f*x)^2)/((b*c 
 - a*d)^3*(c + d*x)^2) + ((a + b*x)*(e + f*x)^3*(a^2*d^2*(2 + 3*m + m^2) - 
 2*a*b*d*(1 + m)*(c*(3 + m) + d*x) + b^2*(c^2*(6 + 5*m + m^2) + 2*c*d*(3 + 
 m)*x + 2*d^2*x^2)))/((b*c - a*d)^3*(c + d*x)^3) - (6*f^3*Hypergeometric2F 
1[-3 - m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(d^4*m*((d*(a + b*x))/(-( 
b*c) + a*d))^m)))/((1 + m)*(2 + m)*(3 + m)*(c + d*x)^m)
 

Rubi [A] (warning: unable to verify)

Time = 0.54 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.24, 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.

\(\displaystyle \int (e+f x)^3 (a+b x)^m (c+d x)^{-m-4} \, dx\)

\(\Big \downarrow \) 137

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x)^3,x]
 

Output:

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

\[\int \left (b x +a \right )^{m} \left (x d +c \right )^{-4-m} \left (f x +e \right )^{3}d x\]

Input:

int((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)^3,x)
 

Output:

int((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)^3,x)
 

Fricas [F]

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

integrate((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)^3,x, algorithm="fricas")
 

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int (a+b x)^m (c+d x)^{-4-m} (e+f x)^3 \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((b*x+a)**m*(d*x+c)**(-4-m)*(f*x+e)**3,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

integrate((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)^3,x, algorithm="maxima")
 

Output:

integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 4), x)
 

Giac [F]

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

integrate((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)^3,x, algorithm="giac")
 

Output:

integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 4), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)^3,x)
 

Output:

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