\(\int \frac {x^2 (a+b x)^n}{c+d x} \, dx\) [556]

Optimal result

Integrand size = 18, antiderivative size = 108 \[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=-\frac {(b c+a d) (a+b x)^{1+n}}{b^2 d^2 (1+n)}+\frac {(a+b x)^{2+n}}{b^2 d (2+n)}+\frac {c^2 (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 (b c-a d) (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=\frac {(a+b x)^{1+n} \left (-((b c-a d) (a d+b c (2+n)-b d (1+n) x))+b^2 c^2 (2+n) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )\right )}{b^2 d^2 (b c-a d) (1+n) (2+n)} \] Input:

Integrate[(x^2*(a + b*x)^n)/(c + d*x),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {99, 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.

\(\displaystyle \int \frac {x^2 (a+b x)^n}{c+d x} \, dx\)

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

Input:

Int[(x^2*(a + b*x)^n)/(c + d*x),x]
 

Output:

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

Defintions of rubi rules used

rule 99

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, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 2009

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

Maple [F]

\[\int \frac {x^{2} \left (b x +a \right )^{n}}{x d +c}d x\]

Input:

int(x^2*(b*x+a)^n/(d*x+c),x)
 

Output:

int(x^2*(b*x+a)^n/(d*x+c),x)
 

Fricas [F]

\[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{2}}{d x + c} \,d x } \] Input:

integrate(x^2*(b*x+a)^n/(d*x+c),x, algorithm="fricas")
 

Output:

integral((b*x + a)^n*x^2/(d*x + c), x)
 

Sympy [F]

\[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=\int \frac {x^{2} \left (a + b x\right )^{n}}{c + d x}\, dx \] Input:

integrate(x**2*(b*x+a)**n/(d*x+c),x)
 

Output:

Integral(x**2*(a + b*x)**n/(c + d*x), x)
 

Maxima [F]

\[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{2}}{d x + c} \,d x } \] Input:

integrate(x^2*(b*x+a)^n/(d*x+c),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^n*x^2/(d*x + c), x)
 

Giac [F]

\[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{2}}{d x + c} \,d x } \] Input:

integrate(x^2*(b*x+a)^n/(d*x+c),x, algorithm="giac")
 

Output:

integrate((b*x + a)^n*x^2/(d*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 (a+b x)^n}{c+d x} \, dx=\int \frac {x^2\,{\left (a+b\,x\right )}^n}{c+d\,x} \,d x \] Input:

int((x^2*(a + b*x)^n)/(c + d*x),x)
 

Output:

int((x^2*(a + b*x)^n)/(c + d*x), x)
 

Reduce [F]

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

int(x^2*(b*x+a)^n/(d*x+c),x)
 

Output:

( - (a + b*x)**n*a**2*d*n + (a + b*x)**n*a*b*c*n + 2*(a + b*x)**n*a*b*c + 
(a + b*x)**n*a*b*d*n**2*x - (a + b*x)**n*b**2*c*n**2*x - 2*(a + b*x)**n*b* 
*2*c*n*x + (a + b*x)**n*b**2*d*n**2*x**2 + (a + b*x)**n*b**2*d*n*x**2 - in 
t(((a + b*x)**n*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a*b**2*c*d*n**3 - 3 
*int(((a + b*x)**n*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a*b**2*c*d*n**2 
- 2*int(((a + b*x)**n*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a*b**2*c*d*n 
+ int(((a + b*x)**n*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**3*c**2*n**3 
+ 3*int(((a + b*x)**n*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**3*c**2*n** 
2 + 2*int(((a + b*x)**n*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**3*c**2*n 
)/(b**2*d**2*n*(n**2 + 3*n + 2))