\(\int x (a+b x)^n (c+d x)^p \, dx\) [576]

Optimal result

Integrand size = 16, antiderivative size = 119 \[ \int x (a+b x)^n (c+d x)^p \, dx=\frac {(a+b x)^{1+n} (c+d x)^{1+p}}{b d (2+n+p)}-\frac {(b c (1+n)+a d (1+p)) (a+b x)^{1+n} (c+d x)^{1+p} \operatorname {Hypergeometric2F1}\left (1,2+n+p,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{b d (b c-a d) (1+n) (2+n+p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.88 \[ \int x (a+b x)^n (c+d x)^p \, dx=\frac {(a+b x)^{1+n} (c+d x)^p \left (b (c+d x)-\frac {(b c (1+n)+a d (1+p)) \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1+n,-p,2+n,\frac {d (a+b x)}{-b c+a d}\right )}{1+n}\right )}{b^2 d (2+n+p)} \] Input:

Integrate[x*(a + b*x)^n*(c + d*x)^p,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {90, 80, 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.

\(\displaystyle \int x (a+b x)^n (c+d x)^p \, dx\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {(a+b x)^{n+1} (c+d x)^{p+1}}{b d (n+p+2)}-\frac {(a d (p+1)+b c (n+1)) \int (a+b x)^n (c+d x)^pdx}{b d (n+p+2)}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {(a+b x)^{n+1} (c+d x)^{p+1}}{b d (n+p+2)}-\frac {(c+d x)^p (a d (p+1)+b c (n+1)) \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \int (a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^pdx}{b d (n+p+2)}\)

\(\Big \downarrow \) 79

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

Input:

Int[x*(a + b*x)^n*(c + d*x)^p,x]
 

Output:

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

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 80

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])
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

Maple [F]

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

Input:

int(x*(b*x+a)^n*(d*x+c)^p,x)
 

Output:

int(x*(b*x+a)^n*(d*x+c)^p,x)
 

Fricas [F]

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

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

Output:

integral((b*x + a)^n*(d*x + c)^p*x, x)
 

Sympy [F(-2)]

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

integrate(x*(b*x+a)**n*(d*x+c)**p,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

integrate((b*x + a)^n*(d*x + c)^p*x, x)
 

Giac [F]

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

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

Output:

integrate((b*x + a)^n*(d*x + c)^p*x, x)
 

Mupad [F(-1)]

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

int(x*(a + b*x)^n*(c + d*x)^p,x)
 

Output:

int(x*(a + b*x)^n*(c + d*x)^p, x)
 

Reduce [F]

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

int(x*(b*x+a)^n*(d*x+c)^p,x)
 

Output:

( - (c + d*x)**p*(a + b*x)**n*a**2*c*d*n + (c + d*x)**p*(a + b*x)**n*a**2* 
d**2*n*p*x - (c + d*x)**p*(a + b*x)**n*a*b*c**2*p + (c + d*x)**p*(a + b*x) 
**n*a*b*c*d*n**2*x + (c + d*x)**p*(a + b*x)**n*a*b*c*d*p**2*x + (c + d*x)* 
*p*(a + b*x)**n*a*b*d**2*n*p*x**2 + (c + d*x)**p*(a + b*x)**n*a*b*d**2*p** 
2*x**2 + (c + d*x)**p*(a + b*x)**n*a*b*d**2*p*x**2 + (c + d*x)**p*(a + b*x 
)**n*b**2*c**2*n*p*x + (c + d*x)**p*(a + b*x)**n*b**2*c*d*n**2*x**2 + (c + 
 d*x)**p*(a + b*x)**n*b**2*c*d*n*p*x**2 + (c + d*x)**p*(a + b*x)**n*b**2*c 
*d*n*x**2 - int(((c + d*x)**p*(a + b*x)**n*x)/(a**2*c*d*n**2*p + 2*a**2*c* 
d*n*p**2 + 3*a**2*c*d*n*p + a**2*c*d*p**3 + 3*a**2*c*d*p**2 + 2*a**2*c*d*p 
 + a**2*d**2*n**2*p*x + 2*a**2*d**2*n*p**2*x + 3*a**2*d**2*n*p*x + a**2*d* 
*2*p**3*x + 3*a**2*d**2*p**2*x + 2*a**2*d**2*p*x + a*b*c**2*n**3 + 2*a*b*c 
**2*n**2*p + 3*a*b*c**2*n**2 + a*b*c**2*n*p**2 + 3*a*b*c**2*n*p + 2*a*b*c* 
*2*n + a*b*c*d*n**3*x + 3*a*b*c*d*n**2*p*x + 3*a*b*c*d*n**2*x + 3*a*b*c*d* 
n*p**2*x + 6*a*b*c*d*n*p*x + 2*a*b*c*d*n*x + a*b*c*d*p**3*x + 3*a*b*c*d*p* 
*2*x + 2*a*b*c*d*p*x + a*b*d**2*n**2*p*x**2 + 2*a*b*d**2*n*p**2*x**2 + 3*a 
*b*d**2*n*p*x**2 + a*b*d**2*p**3*x**2 + 3*a*b*d**2*p**2*x**2 + 2*a*b*d**2* 
p*x**2 + b**2*c**2*n**3*x + 2*b**2*c**2*n**2*p*x + 3*b**2*c**2*n**2*x + b* 
*2*c**2*n*p**2*x + 3*b**2*c**2*n*p*x + 2*b**2*c**2*n*x + b**2*c*d*n**3*x** 
2 + 2*b**2*c*d*n**2*p*x**2 + 3*b**2*c*d*n**2*x**2 + b**2*c*d*n*p**2*x**2 + 
 3*b**2*c*d*n*p*x**2 + 2*b**2*c*d*n*x**2),x)*a**4*d**4*n**3*p**3 - int(...