\(\int (a+b x^n)^p (a+b (3+2 p) x^n) \, dx\) [428]

Optimal result

Integrand size = 22, antiderivative size = 91 \[ \int \left (a+b x^n\right )^p \left (a+b (3+2 p) x^n\right ) \, dx=\frac {(3+2 p) x \left (a+b x^n\right )^{1+p}}{1+n+n p}-\frac {a (2-n) (1+p) x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {b x^n}{a}\right )}{1+n+n p} \] Output:

(3+2*p)*x*(a+b*x^n)^(p+1)/(n*p+n+1)-a*(2-n)*(p+1)*x*(a+b*x^n)^p*hypergeom( 
[-p, 1/n],[1+1/n],-b*x^n/a)/(n*p+n+1)/((1+b*x^n/a)^p)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97 \[ \int \left (a+b x^n\right )^p \left (a+b (3+2 p) x^n\right ) \, dx=\frac {x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left ((3+2 p) \left (a+b x^n\right ) \left (1+\frac {b x^n}{a}\right )^p+a (-2+n) (1+p) \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {b x^n}{a}\right )\right )}{1+n+n p} \] Input:

Integrate[(a + b*x^n)^p*(a + b*(3 + 2*p)*x^n),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.

\(\displaystyle \int \left (a+b x^n\right )^p \left (a+b (2 p+3) x^n\right ) \, dx\)

\(\Big \downarrow \) 913

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

\(\Big \downarrow \) 779

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

\(\Big \downarrow \) 778

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

Input:

Int[(a + b*x^n)^p*(a + b*(3 + 2*p)*x^n),x]
 

Output:

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

\[\int \left (a +b \,x^{n}\right )^{p} \left (a +b \left (3+2 p \right ) x^{n}\right )d x\]

Input:

int((a+b*x^n)^p*(a+b*(3+2*p)*x^n),x)
 

Output:

int((a+b*x^n)^p*(a+b*(3+2*p)*x^n),x)
 

Fricas [F]

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

integrate((a+b*x^n)^p*(a+b*(3+2*p)*x^n),x, algorithm="fricas")
 

Output:

integral(((2*b*p + 3*b)*x^n + a)*(b*x^n + a)^p, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.70 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.89 \[ \int \left (a+b x^n\right )^p \left (a+b (3+2 p) x^n\right ) \, dx=\frac {a a^{\frac {1}{n}} a^{p - \frac {1}{n}} x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{n}, - p \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} + \frac {2 a^{1 + \frac {1}{n}} a^{p - 1 - \frac {1}{n}} b p x^{n + 1} \Gamma \left (1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 + \frac {1}{n} \\ 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {1}{n}\right )} + \frac {3 a^{1 + \frac {1}{n}} a^{p - 1 - \frac {1}{n}} b x^{n + 1} \Gamma \left (1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 + \frac {1}{n} \\ 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {1}{n}\right )} \] Input:

integrate((a+b*x**n)**p*(a+b*(3+2*p)*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)) + 2*a**(1 + 1/n)*a**(p - 1 - 1/n)*b*p 
*x**(n + 1)*gamma(1 + 1/n)*hyper((-p, 1 + 1/n), (2 + 1/n,), b*x**n*exp_pol 
ar(I*pi)/a)/(n*gamma(2 + 1/n)) + 3*a**(1 + 1/n)*a**(p - 1 - 1/n)*b*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]

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

integrate((a+b*x^n)^p*(a+b*(3+2*p)*x^n),x, algorithm="maxima")
 

Output:

integrate((b*(2*p + 3)*x^n + a)*(b*x^n + a)^p, x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (a+b x^n\right )^p \left (a+b (3+2 p) x^n\right ) \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*x^n)^p*(a+b*(3+2*p)*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,[0,0,2,2,1,2,2]%%%}+%%%{2,[0,0,2,2,1,1,2]%%%}+%%%{1,[0,0 
,2,2,1,0,
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^n\right )^p \left (a+b (3+2 p) x^n\right ) \, dx=\int {\left (a+b\,x^n\right )}^p\,\left (a+b\,x^n\,\left (2\,p+3\right )\right ) \,d x \] Input:

int((a + b*x^n)^p*(a + b*x^n*(2*p + 3)),x)
 

Output:

int((a + b*x^n)^p*(a + b*x^n*(2*p + 3)), x)
 

Reduce [F]

\[ \int \left (a+b x^n\right )^p \left (a+b (3+2 p) x^n\right ) \, dx =\text {Too large to display} \] Input:

int((a+b*x^n)^p*(a+b*(3+2*p)*x^n),x)
 

Output:

(2*x**n*(x**n*b + a)**p*b*n*p**2*x + 3*x**n*(x**n*b + a)**p*b*n*p*x + 2*x* 
*n*(x**n*b + a)**p*b*p*x + 3*x**n*(x**n*b + a)**p*b*x + 2*(x**n*b + a)**p* 
a*n*p**2*x + 4*(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**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**3*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**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 - 4*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**3...