\(\int (a+b x^n)^p (A+B x^n) \, dx\) [68]

Optimal result

Integrand size = 17, antiderivative size = 89 \[ \int \left (a+b x^n\right )^p \left (A+B x^n\right ) \, dx=\frac {B x \left (a+b x^n\right )^{1+p}}{b (1+n+n p)}+\left (A-\frac {a B}{b+b n+b n p}\right ) x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {b x^n}{a}\right ) \] Output:

B*x*(a+b*x^n)^(p+1)/b/(n*p+n+1)+(A-a*B/(b*n*p+b*n+b))*x*(a+b*x^n)^p*hyperg 
eom([-p, 1/n],[1+1/n],-b*x^n/a)/((1+b*x^n/a)^p)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^n\right )^p \left (A+B x^n\right ) \, dx=\frac {x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (B \left (a+b x^n\right ) \left (1+\frac {b x^n}{a}\right )^p+(-a B+A b (1+n+n p)) \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {b x^n}{a}\right )\right )}{b (1+n+n p)} \] Input:

Integrate[(a + b*x^n)^p*(A + B*x^n),x]
 

Output:

(x*(a + b*x^n)^p*(B*(a + b*x^n)*(1 + (b*x^n)/a)^p + (-(a*B) + A*b*(1 + n + 
 n*p))*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*x^n)/a)]))/(b*(1 + n 
 + n*p)*(1 + (b*x^n)/a)^p)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {913, 779, 778}

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.

\(\displaystyle \int \left (A+B x^n\right ) \left (a+b x^n\right )^p \, dx\)

\(\Big \downarrow \) 913

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

\(\Big \downarrow \) 779

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

\(\Big \downarrow \) 778

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

Input:

Int[(a + b*x^n)^p*(A + B*x^n),x]
 

Output:

(B*x*(a + b*x^n)^(1 + p))/(b*(1 + n + n*p)) + ((A - (a*B)/(b + b*n + b*n*p 
))*x*(a + b*x^n)^p*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*x^n)/a)] 
)/(1 + (b*x^n)/a)^p
 

Defintions of rubi rules used

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 779

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x 
^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(1 + b*(x^n/a))^p, x], x 
] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Si 
mplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 913

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si 
mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( 
p + 1) + 1))/(b*(n*(p + 1) + 1))   Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b 
, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
 

Maple [F]

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

Input:

int((a+b*x^n)^p*(A+B*x^n),x)
 

Output:

int((a+b*x^n)^p*(A+B*x^n),x)
 

Fricas [F]

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

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

Output:

integral((B*x^n + A)*(b*x^n + a)^p, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.20 \[ \int \left (a+b x^n\right )^p \left (A+B x^n\right ) \, dx=\frac {A a^{\frac {1}{n}} a^{p - \frac {1}{n}} x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{n}, - p \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} + \frac {B a^{1 + \frac {1}{n}} a^{p - 1 - \frac {1}{n}} x^{n + 1} \Gamma \left (1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 + \frac {1}{n} \\ 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {1}{n}\right )} \] Input:

integrate((a+b*x**n)**p*(A+B*x**n),x)
 

Output:

A*a**(1/n)*a**(p - 1/n)*x*gamma(1/n)*hyper((1/n, -p), (1 + 1/n,), b*x**n*e 
xp_polar(I*pi)/a)/(n*gamma(1 + 1/n)) + B*a**(1 + 1/n)*a**(p - 1 - 1/n)*x** 
(n + 1)*gamma(1 + 1/n)*hyper((-p, 1 + 1/n), (2 + 1/n,), b*x**n*exp_polar(I 
*pi)/a)/(n*gamma(2 + 1/n))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

integrate((B*x^n + A)*(b*x^n + a)^p, x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (a+b x^n\right )^p \left (A+B x^n\right ) \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{1,[0,0,2,2,1,2,1,0,1]%%%}+%%%{2,[0,0,2,2,1,1,1,0,1]%%%}+%% 
%{1,[0,0,
 

Mupad [F(-1)]

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

int((A + B*x^n)*(a + b*x^n)^p,x)
 

Output:

int((A + B*x^n)*(a + b*x^n)^p, x)
 

Reduce [F]

\[ \int \left (a+b x^n\right )^p \left (A+B x^n\right ) \, dx=\frac {x^{n} \left (x^{n} b +a \right )^{p} b n p x +x^{n} \left (x^{n} b +a \right )^{p} b x +2 \left (x^{n} b +a \right )^{p} a n p x +\left (x^{n} b +a \right )^{p} a n x +\left (x^{n} b +a \right )^{p} a x +\left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n} b \,n^{2} p^{2}+x^{n} b \,n^{2} p +2 x^{n} b n p +x^{n} b n +x^{n} b +a \,n^{2} p^{2}+a \,n^{2} p +2 a n p +a n +a}d x \right ) a^{2} n^{4} p^{4}+2 \left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n} b \,n^{2} p^{2}+x^{n} b \,n^{2} p +2 x^{n} b n p +x^{n} b n +x^{n} b +a \,n^{2} p^{2}+a \,n^{2} p +2 a n p +a n +a}d x \right ) a^{2} n^{4} p^{3}+\left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n} b \,n^{2} p^{2}+x^{n} b \,n^{2} p +2 x^{n} b n p +x^{n} b n +x^{n} b +a \,n^{2} p^{2}+a \,n^{2} p +2 a n p +a n +a}d x \right ) a^{2} n^{4} p^{2}+2 \left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n} b \,n^{2} p^{2}+x^{n} b \,n^{2} p +2 x^{n} b n p +x^{n} b n +x^{n} b +a \,n^{2} p^{2}+a \,n^{2} p +2 a n p +a n +a}d x \right ) a^{2} n^{3} p^{3}+3 \left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n} b \,n^{2} p^{2}+x^{n} b \,n^{2} p +2 x^{n} b n p +x^{n} b n +x^{n} b +a \,n^{2} p^{2}+a \,n^{2} p +2 a n p +a n +a}d x \right ) a^{2} n^{3} p^{2}+\left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n} b \,n^{2} p^{2}+x^{n} b \,n^{2} p +2 x^{n} b n p +x^{n} b n +x^{n} b +a \,n^{2} p^{2}+a \,n^{2} p +2 a n p +a n +a}d x \right ) a^{2} n^{3} p +\left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n} b \,n^{2} p^{2}+x^{n} b \,n^{2} p +2 x^{n} b n p +x^{n} b n +x^{n} b +a \,n^{2} p^{2}+a \,n^{2} p +2 a n p +a n +a}d x \right ) a^{2} n^{2} p^{2}+\left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n} b \,n^{2} p^{2}+x^{n} b \,n^{2} p +2 x^{n} b n p +x^{n} b n +x^{n} b +a \,n^{2} p^{2}+a \,n^{2} p +2 a n p +a n +a}d x \right ) a^{2} n^{2} p}{n^{2} p^{2}+n^{2} p +2 n p +n +1} \] Input:

int((a+b*x^n)^p*(A+B*x^n),x)
 

Output:

(x**n*(x**n*b + a)**p*b*n*p*x + x**n*(x**n*b + a)**p*b*x + 2*(x**n*b + a)* 
*p*a*n*p*x + (x**n*b + a)**p*a*n*x + (x**n*b + a)**p*a*x + int((x**n*b + a 
)**p/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 2*x**n*b*n*p + x**n*b*n + x**n*b 
+ a*n**2*p**2 + a*n**2*p + 2*a*n*p + a*n + a),x)*a**2*n**4*p**4 + 2*int((x 
**n*b + a)**p/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 2*x**n*b*n*p + x**n*b*n 
+ x**n*b + a*n**2*p**2 + a*n**2*p + 2*a*n*p + a*n + a),x)*a**2*n**4*p**3 + 
 int((x**n*b + a)**p/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 2*x**n*b*n*p + x* 
*n*b*n + x**n*b + a*n**2*p**2 + a*n**2*p + 2*a*n*p + a*n + a),x)*a**2*n**4 
*p**2 + 2*int((x**n*b + a)**p/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 2*x**n*b 
*n*p + x**n*b*n + x**n*b + a*n**2*p**2 + a*n**2*p + 2*a*n*p + a*n + a),x)* 
a**2*n**3*p**3 + 3*int((x**n*b + a)**p/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 
 2*x**n*b*n*p + x**n*b*n + x**n*b + a*n**2*p**2 + a*n**2*p + 2*a*n*p + a*n 
 + a),x)*a**2*n**3*p**2 + int((x**n*b + a)**p/(x**n*b*n**2*p**2 + x**n*b*n 
**2*p + 2*x**n*b*n*p + x**n*b*n + x**n*b + a*n**2*p**2 + a*n**2*p + 2*a*n* 
p + a*n + a),x)*a**2*n**3*p + int((x**n*b + a)**p/(x**n*b*n**2*p**2 + x**n 
*b*n**2*p + 2*x**n*b*n*p + x**n*b*n + x**n*b + a*n**2*p**2 + a*n**2*p + 2* 
a*n*p + a*n + a),x)*a**2*n**2*p**2 + int((x**n*b + a)**p/(x**n*b*n**2*p**2 
 + x**n*b*n**2*p + 2*x**n*b*n*p + x**n*b*n + x**n*b + a*n**2*p**2 + a*n**2 
*p + 2*a*n*p + a*n + a),x)*a**2*n**2*p)/(n**2*p**2 + n**2*p + 2*n*p + n + 
1)