\(\int x (a+b x^n)^p (c+d x^n) \, dx\) [416]

Optimal result

Integrand size = 18, antiderivative size = 100 \[ \int x \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {d x^2 \left (a+b x^n\right )^{1+p}}{b (2+n+n p)}+\frac {1}{2} \left (c-\frac {2 a d}{b (2+n+n p)}\right ) x^2 \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{n},-p,\frac {2+n}{n},-\frac {b x^n}{a}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.99 \[ \int x \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {x^2 \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c (2+n) \operatorname {Hypergeometric2F1}\left (\frac {2}{n},-p,\frac {2+n}{n},-\frac {b x^n}{a}\right )+2 d x^n \operatorname {Hypergeometric2F1}\left (\frac {2+n}{n},-p,2 \left (1+\frac {1}{n}\right ),-\frac {b x^n}{a}\right )\right )}{2 (2+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {959, 889, 888}

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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(c*x) 
^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0 
] &&  !(ILtQ[p, 0] || GtQ[a, 0])
 

rule 959

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.09 \[ \int x \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {a^{\frac {2}{n}} a^{p - \frac {2}{n}} c x^{2} \Gamma \left (\frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{n}, - p \\ 1 + \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {2}{n}\right )} + \frac {a^{1 + \frac {2}{n}} a^{p - 1 - \frac {2}{n}} d x^{n + 2} \Gamma \left (1 + \frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 + \frac {2}{n} \\ 2 + \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {2}{n}\right )} \] Input:

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

Output:

a**(2/n)*a**(p - 2/n)*c*x**2*gamma(2/n)*hyper((2/n, -p), (1 + 2/n,), b*x** 
n*exp_polar(I*pi)/a)/(n*gamma(1 + 2/n)) + a**(1 + 2/n)*a**(p - 1 - 2/n)*d* 
x**(n + 2)*gamma(1 + 2/n)*hyper((-p, 1 + 2/n), (2 + 2/n,), b*x**n*exp_pola 
r(I*pi)/a)/(n*gamma(2 + 2/n))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(x**n*(x**n*b + a)**p*b*d*n*p*x**2 + 2*x**n*(x**n*b + a)**p*b*d*x**2 + (x* 
*n*b + a)**p*a*d*n*p*x**2 + (x**n*b + a)**p*b*c*n*p*x**2 + (x**n*b + a)**p 
*b*c*n*x**2 + 2*(x**n*b + a)**p*b*c*x**2 - 2*int(((x**n*b + a)**p*x)/(x**n 
*b*n**2*p**2 + x**n*b*n**2*p + 4*x**n*b*n*p + 2*x**n*b*n + 4*x**n*b + a*n* 
*2*p**2 + a*n**2*p + 4*a*n*p + 2*a*n + 4*a),x)*a**2*d*n**3*p**3 - 2*int((( 
x**n*b + a)**p*x)/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 4*x**n*b*n*p + 2*x** 
n*b*n + 4*x**n*b + a*n**2*p**2 + a*n**2*p + 4*a*n*p + 2*a*n + 4*a),x)*a**2 
*d*n**3*p**2 - 8*int(((x**n*b + a)**p*x)/(x**n*b*n**2*p**2 + x**n*b*n**2*p 
 + 4*x**n*b*n*p + 2*x**n*b*n + 4*x**n*b + a*n**2*p**2 + a*n**2*p + 4*a*n*p 
 + 2*a*n + 4*a),x)*a**2*d*n**2*p**2 - 4*int(((x**n*b + a)**p*x)/(x**n*b*n* 
*2*p**2 + x**n*b*n**2*p + 4*x**n*b*n*p + 2*x**n*b*n + 4*x**n*b + a*n**2*p* 
*2 + a*n**2*p + 4*a*n*p + 2*a*n + 4*a),x)*a**2*d*n**2*p - 8*int(((x**n*b + 
 a)**p*x)/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 4*x**n*b*n*p + 2*x**n*b*n + 
4*x**n*b + a*n**2*p**2 + a*n**2*p + 4*a*n*p + 2*a*n + 4*a),x)*a**2*d*n*p + 
 int(((x**n*b + a)**p*x)/(x**n*b*n**2*p**2 + x**n*b*n**2*p + 4*x**n*b*n*p 
+ 2*x**n*b*n + 4*x**n*b + a*n**2*p**2 + a*n**2*p + 4*a*n*p + 2*a*n + 4*a), 
x)*a*b*c*n**4*p**4 + 2*int(((x**n*b + a)**p*x)/(x**n*b*n**2*p**2 + x**n*b* 
n**2*p + 4*x**n*b*n*p + 2*x**n*b*n + 4*x**n*b + a*n**2*p**2 + a*n**2*p + 4 
*a*n*p + 2*a*n + 4*a),x)*a*b*c*n**4*p**3 + int(((x**n*b + a)**p*x)/(x**n*b 
*n**2*p**2 + x**n*b*n**2*p + 4*x**n*b*n*p + 2*x**n*b*n + 4*x**n*b + a*n...