\(\int (\frac {d (a+b x)}{-b c+a d})^m (c+d x)^n \, dx\) [800]

Optimal result

Integrand size = 27, antiderivative size = 45 \[ \int \left (\frac {d (a+b x)}{-b c+a d}\right )^m (c+d x)^n \, dx=\frac {(c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (-m,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{d (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.96 \[ \int \left (\frac {d (a+b x)}{-b c+a d}\right )^m (c+d x)^n \, dx=\frac {(a+b x) \left (\frac {d (a+b x)}{-b c+a d}\right )^m (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,\frac {d (a+b x)}{-b c+a d}\right )}{b (1+m)} \] Input:

Integrate[((d*(a + b*x))/(-(b*c) + a*d))^m*(c + d*x)^n,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {204, 79}

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.

\(\displaystyle \int (c+d x)^n \left (\frac {d (a+b x)}{a d-b c}\right )^m \, dx\)

\(\Big \downarrow \) 204

\(\displaystyle \int (c+d x)^n \left (-\frac {b d x}{b c-a d}-\frac {a d}{b c-a d}\right )^mdx\)

\(\Big \downarrow \) 79

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

Input:

Int[((d*(a + b*x))/(-(b*c) + a*d))^m*(c + d*x)^n,x]
 

Output:

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

Defintions of rubi rules used

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 204

Int[(u_)^(m_.)*(v_)^(n_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*ExpandToSum 
[v, x]^n, x] /; FreeQ[{m, n}, x] && LinearQ[{u, v}, x] &&  !LinearMatchQ[{u 
, v}, x]
 

Maple [F]

\[\int \left (\frac {d \left (b x +a \right )}{a d -b c}\right )^{m} \left (x d +c \right )^{n}d x\]

Input:

int((d*(b*x+a)/(a*d-b*c))^m*(d*x+c)^n,x)
 

Output:

int((d*(b*x+a)/(a*d-b*c))^m*(d*x+c)^n,x)
 

Fricas [F]

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

integrate((d*(b*x+a)/(a*d-b*c))^m*(d*x+c)^n,x, algorithm="fricas")
 

Output:

integral((d*x + c)^n*(-(b*d*x + a*d)/(b*c - a*d))^m, x)
 

Sympy [F(-2)]

Exception generated. \[ \int \left (\frac {d (a+b x)}{-b c+a d}\right )^m (c+d x)^n \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((d*(b*x+a)/(a*d-b*c))**m*(d*x+c)**n,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

integrate((d*(b*x+a)/(a*d-b*c))^m*(d*x+c)^n,x, algorithm="maxima")
 

Output:

integrate((d*x + c)^n*(-(b*x + a)*d/(b*c - a*d))^m, x)
 

Giac [F]

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

integrate((d*(b*x+a)/(a*d-b*c))^m*(d*x+c)^n,x, algorithm="giac")
 

Output:

integrate((d*x + c)^n*(-(b*x + a)*d/(b*c - a*d))^m, x)
 

Mupad [F(-1)]

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

int((c + d*x)^n*((d*(a + b*x))/(a*d - b*c))^m,x)
 

Output:

int((c + d*x)^n*((d*(a + b*x))/(a*d - b*c))^m, x)
 

Reduce [F]

\[ \int \left (\frac {d (a+b x)}{-b c+a d}\right )^m (c+d x)^n \, dx=\text {too large to display} \] Input:

int((d*(b*x+a)/(a*d-b*c))^m*(d*x+c)^n,x)
 

Output:

(d**m*((c + d*x)**n*(a + b*x)**m*a*c*m + (c + d*x)**n*(a + b*x)**m*a*c*n + 
 (c + d*x)**n*(a + b*x)**m*a*d*n*x + (c + d*x)**n*(a + b*x)**m*b*c*m*x + i 
nt(((c + d*x)**n*(a + b*x)**m*x)/(a**2*c*d*m*n + a**2*c*d*n**2 + a**2*c*d* 
n + a**2*d**2*m*n*x + a**2*d**2*n**2*x + a**2*d**2*n*x + a*b*c**2*m**2 + a 
*b*c**2*m*n + a*b*c**2*m + a*b*c*d*m**2*x + 2*a*b*c*d*m*n*x + a*b*c*d*m*x 
+ a*b*c*d*n**2*x + a*b*c*d*n*x + a*b*d**2*m*n*x**2 + a*b*d**2*n**2*x**2 + 
a*b*d**2*n*x**2 + b**2*c**2*m**2*x + b**2*c**2*m*n*x + b**2*c**2*m*x + b** 
2*c*d*m**2*x**2 + b**2*c*d*m*n*x**2 + b**2*c*d*m*x**2),x)*a**3*d**3*m**2*n 
**2 + int(((c + d*x)**n*(a + b*x)**m*x)/(a**2*c*d*m*n + a**2*c*d*n**2 + a* 
*2*c*d*n + a**2*d**2*m*n*x + a**2*d**2*n**2*x + a**2*d**2*n*x + a*b*c**2*m 
**2 + a*b*c**2*m*n + a*b*c**2*m + a*b*c*d*m**2*x + 2*a*b*c*d*m*n*x + a*b*c 
*d*m*x + a*b*c*d*n**2*x + a*b*c*d*n*x + a*b*d**2*m*n*x**2 + a*b*d**2*n**2* 
x**2 + a*b*d**2*n*x**2 + b**2*c**2*m**2*x + b**2*c**2*m*n*x + b**2*c**2*m* 
x + b**2*c*d*m**2*x**2 + b**2*c*d*m*n*x**2 + b**2*c*d*m*x**2),x)*a**3*d**3 
*m*n**3 + int(((c + d*x)**n*(a + b*x)**m*x)/(a**2*c*d*m*n + a**2*c*d*n**2 
+ a**2*c*d*n + a**2*d**2*m*n*x + a**2*d**2*n**2*x + a**2*d**2*n*x + a*b*c* 
*2*m**2 + a*b*c**2*m*n + a*b*c**2*m + a*b*c*d*m**2*x + 2*a*b*c*d*m*n*x + a 
*b*c*d*m*x + a*b*c*d*n**2*x + a*b*c*d*n*x + a*b*d**2*m*n*x**2 + a*b*d**2*n 
**2*x**2 + a*b*d**2*n*x**2 + b**2*c**2*m**2*x + b**2*c**2*m*n*x + b**2*c** 
2*m*x + b**2*c*d*m**2*x**2 + b**2*c*d*m*n*x**2 + b**2*c*d*m*x**2),x)*a*...