\(\int x^2 (a+b x)^n (c+d x)^{-n} \, dx\) [581]

Optimal result

Integrand size = 20, antiderivative size = 189 \[ \int x^2 (a+b x)^n (c+d x)^{-n} \, dx=-\frac {(a d (4-n)+b c (2+n)) (a+b x)^{1+n} (c+d x)^{1-n}}{6 b^2 d^2}+\frac {(a+b x)^{2+n} (c+d x)^{1-n}}{3 b^2 d}+\frac {\left (2 a b c d \left (1-n^2\right )+a^2 d^2 \left (2-3 n+n^2\right )+b^2 c^2 \left (2+3 n+n^2\right )\right ) (a+b x)^{1+n} (c+d x)^{1-n} \operatorname {Hypergeometric2F1}\left (1,2,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{6 b^2 d^2 (b c-a d) (1+n)} \] Output:

-1/6*(a*d*(4-n)+b*c*(2+n))*(b*x+a)^(1+n)*(d*x+c)^(1-n)/b^2/d^2+1/3*(b*x+a) 
^(2+n)*(d*x+c)^(1-n)/b^2/d+1/6*(2*a*b*c*d*(-n^2+1)+a^2*d^2*(n^2-3*n+2)+b^2 
*c^2*(n^2+3*n+2))*(b*x+a)^(1+n)*(d*x+c)^(1-n)*hypergeom([1, 2],[2+n],-d*(b 
*x+a)/(-a*d+b*c))/b^2/d^2/(-a*d+b*c)/(1+n)
 

Mathematica [A] (warning: unable to verify)

Time = 0.12 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.81 \[ \int x^2 (a+b x)^n (c+d x)^{-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{-n} \left (b (a d (-2+n)-b c (2+n)) (c+d x)+2 b^2 d x (c+d x)+\frac {\left (-2 a b c d \left (-1+n^2\right )+a^2 d^2 \left (2-3 n+n^2\right )+b^2 c^2 \left (2+3 n+n^2\right )\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )}{1+n}\right )}{6 b^3 d^2} \] Input:

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

Output:

((a + b*x)^(1 + n)*(b*(a*d*(-2 + n) - b*c*(2 + n))*(c + d*x) + 2*b^2*d*x*( 
c + d*x) + ((-2*a*b*c*d*(-1 + n^2) + a^2*d^2*(2 - 3*n + n^2) + b^2*c^2*(2 
+ 3*n + n^2))*((b*(c + d*x))/(b*c - a*d))^n*Hypergeometric2F1[n, 1 + n, 2 
+ n, (d*(a + b*x))/(-(b*c) + a*d)])/(1 + n)))/(6*b^3*d^2*(c + d*x)^n)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {101, 25, 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.

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

\(\Big \downarrow \) 101

\(\displaystyle \frac {\int -(a+b x)^n (c+d x)^{-n} (a c+(a d (2-n)+b c (n+2)) x)dx}{3 b d}+\frac {x (a+b x)^{n+1} (c+d x)^{1-n}}{3 b d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x (a+b x)^{n+1} (c+d x)^{1-n}}{3 b d}-\frac {\int (a+b x)^n (c+d x)^{-n} (a c+(a d (2-n)+b c (n+2)) x)dx}{3 b d}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {x (a+b x)^{n+1} (c+d x)^{1-n}}{3 b d}-\frac {\frac {(2 a b c d-(a d (1-n)+b c (n+1)) (a d (2-n)+b c (n+2))) \int (a+b x)^n (c+d x)^{-n}dx}{2 b d}+\frac {(a+b x)^{n+1} (c+d x)^{1-n} (a d (2-n)+b c (n+2))}{2 b d}}{3 b d}\)

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int x^2 (a+b x)^n (c+d x)^{-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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