\(\int \frac {(c+d x)^m (e+f x)}{(a+b x)^3} \, dx\) [247]

Optimal result

Integrand size = 20, antiderivative size = 118 \[ \int \frac {(c+d x)^m (e+f x)}{(a+b x)^3} \, dx=-\frac {f (c+d x)^{1+m}}{b d (1-m) (a+b x)^2}+\frac {d (2 b c f-b d e (1-m)-a d f (1+m)) (c+d x)^{1+m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^3 (1-m) (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.90 \[ \int \frac {(c+d x)^m (e+f x)}{(a+b x)^3} \, dx=\frac {(c+d x)^{1+m} \left (\frac {-b e+a f}{(a+b x)^2}+\frac {d (2 b c f+b d e (-1+m)-a d f (1+m)) \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d)^2 (1+m)}\right )}{2 b (b c-a d)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {87, 78}

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.

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 78

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 78

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

rule 87

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)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((c + d*x)**m*(e + f*x)/(a + b*x)**3, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^m (e+f x)}{(a+b x)^3} \, dx=\text {too large to display} \] Input:

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

Output:

( - (c + d*x)**m*a*c*f + (c + d*x)**m*a*d*f*m*x + (c + d*x)**m*b*c*e*m - ( 
c + d*x)**m*b*c*e - 2*(c + d*x)**m*b*c*f*x - int(((c + d*x)**m*x)/(a**4*c* 
d*m**2 - a**4*c*d*m + a**4*d**2*m**2*x - a**4*d**2*m*x - 2*a**3*b*c**2*m + 
 2*a**3*b*c**2 + 3*a**3*b*c*d*m**2*x - 5*a**3*b*c*d*m*x + 2*a**3*b*c*d*x + 
 3*a**3*b*d**2*m**2*x**2 - 3*a**3*b*d**2*m*x**2 - 6*a**2*b**2*c**2*m*x + 6 
*a**2*b**2*c**2*x + 3*a**2*b**2*c*d*m**2*x**2 - 9*a**2*b**2*c*d*m*x**2 + 6 
*a**2*b**2*c*d*x**2 + 3*a**2*b**2*d**2*m**2*x**3 - 3*a**2*b**2*d**2*m*x**3 
 - 6*a*b**3*c**2*m*x**2 + 6*a*b**3*c**2*x**2 + a*b**3*c*d*m**2*x**3 - 7*a* 
b**3*c*d*m*x**3 + 6*a*b**3*c*d*x**3 + a*b**3*d**2*m**2*x**4 - a*b**3*d**2* 
m*x**4 - 2*b**4*c**2*m*x**3 + 2*b**4*c**2*x**3 - 2*b**4*c*d*m*x**4 + 2*b** 
4*c*d*x**4),x)*a**5*d**3*f*m**4 + int(((c + d*x)**m*x)/(a**4*c*d*m**2 - a* 
*4*c*d*m + a**4*d**2*m**2*x - a**4*d**2*m*x - 2*a**3*b*c**2*m + 2*a**3*b*c 
**2 + 3*a**3*b*c*d*m**2*x - 5*a**3*b*c*d*m*x + 2*a**3*b*c*d*x + 3*a**3*b*d 
**2*m**2*x**2 - 3*a**3*b*d**2*m*x**2 - 6*a**2*b**2*c**2*m*x + 6*a**2*b**2* 
c**2*x + 3*a**2*b**2*c*d*m**2*x**2 - 9*a**2*b**2*c*d*m*x**2 + 6*a**2*b**2* 
c*d*x**2 + 3*a**2*b**2*d**2*m**2*x**3 - 3*a**2*b**2*d**2*m*x**3 - 6*a*b**3 
*c**2*m*x**2 + 6*a*b**3*c**2*x**2 + a*b**3*c*d*m**2*x**3 - 7*a*b**3*c*d*m* 
x**3 + 6*a*b**3*c*d*x**3 + a*b**3*d**2*m**2*x**4 - a*b**3*d**2*m*x**4 - 2* 
b**4*c**2*m*x**3 + 2*b**4*c**2*x**3 - 2*b**4*c*d*m*x**4 + 2*b**4*c*d*x**4) 
,x)*a**5*d**3*f*m**2 + int(((c + d*x)**m*x)/(a**4*c*d*m**2 - a**4*c*d*m...