\(\int (d+e x^n) (a+c x^{2 n})^p \, dx\) [49]

Optimal result

Integrand size = 19, antiderivative size = 135 \[ \int \left (d+e x^n\right ) \left (a+c x^{2 n}\right )^p \, dx=d x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2 n},-p,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+\frac {e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2 n},-p,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{1+n} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.81 \[ \int \left (d+e x^n\right ) \left (a+c x^{2 n}\right )^p \, dx=\frac {x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \left (d (1+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2 n},-p,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+e x^n \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2 n},-p,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )\right )}{1+n} \] Input:

Integrate[(d + e*x^n)*(a + c*x^(2*n))^p,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1763, 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.

\(\displaystyle \int \left (d+e x^n\right ) \left (a+c x^{2 n}\right )^p \, dx\)

\(\Big \downarrow \) 1763

\(\displaystyle \int \left (d \left (a+c x^{2 n}\right )^p+e x^n \left (a+c x^{2 n}\right )^p\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[(d + e*x^n)*(a + c*x^(2*n))^p,x]
 

Output:

(d*x*(a + c*x^(2*n))^p*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c 
*x^(2*n))/a)])/(1 + (c*x^(2*n))/a)^p + (e*x^(1 + n)*(a + c*x^(2*n))^p*Hype 
rgeometric2F1[(1 + n)/(2*n), -p, (3 + n^(-1))/2, -((c*x^(2*n))/a)])/((1 + 
n)*(1 + (c*x^(2*n))/a)^p)
 

Defintions of rubi rules used

rule 1763

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> I 
nt[ExpandIntegrand[(d + e*x^n)*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, 
 e, n}, x] && EqQ[n2, 2*n]
 

rule 2009

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

Maple [F]

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

Input:

int((d+e*x^n)*(a+c*x^(2*n))^p,x)
 

Output:

int((d+e*x^n)*(a+c*x^(2*n))^p,x)
 

Fricas [F]

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

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

Output:

integral((e*x^n + d)*(c*x^(2*n) + a)^p, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 138.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07 \[ \int \left (d+e x^n\right ) \left (a+c x^{2 n}\right )^p \, dx=\frac {a^{\frac {1}{2 n}} a^{p - \frac {1}{2 n}} d x \Gamma \left (\frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2 n}, - p \\ 1 + \frac {1}{2 n} \end {matrix}\middle | {\frac {c x^{2 n} e^{i \pi }}{a}} \right )}}{2 n \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {a^{\frac {1}{2} + \frac {1}{2 n}} a^{p - \frac {1}{2} - \frac {1}{2 n}} e x^{n + 1} \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {1}{2} + \frac {1}{2 n} \\ \frac {3}{2} + \frac {1}{2 n} \end {matrix}\middle | {\frac {c x^{2 n} e^{i \pi }}{a}} \right )}}{2 n \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} \] Input:

integrate((d+e*x**n)*(a+c*x**(2*n))**p,x)
 

Output:

a**(1/(2*n))*a**(p - 1/(2*n))*d*x*gamma(1/(2*n))*hyper((1/(2*n), -p), (1 + 
 1/(2*n),), c*x**(2*n)*exp_polar(I*pi)/a)/(2*n*gamma(1 + 1/(2*n))) + a**(1 
/2 + 1/(2*n))*a**(p - 1/2 - 1/(2*n))*e*x**(n + 1)*gamma(1/2 + 1/(2*n))*hyp 
er((-p, 1/2 + 1/(2*n)), (3/2 + 1/(2*n),), c*x**(2*n)*exp_polar(I*pi)/a)/(2 
*n*gamma(3/2 + 1/(2*n)))
 

Maxima [F]

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

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

Output:

integrate((e*x^n + d)*(c*x^(2*n) + a)^p, x)
 

Giac [F]

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

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

Output:

integrate((e*x^n + d)*(c*x^(2*n) + a)^p, x)
 

Mupad [F(-1)]

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

int((a + c*x^(2*n))^p*(d + e*x^n),x)
 

Output:

int((a + c*x^(2*n))^p*(d + e*x^n), x)
 

Reduce [F]

\[ \int \left (d+e x^n\right ) \left (a+c x^{2 n}\right )^p \, dx=\text {too large to display} \] Input:

int((d+e*x^n)*(a+c*x^(2*n))^p,x)
 

Output:

(2*x**n*(x**(2*n)*c + a)**p*e*n*p*x + x**n*(x**(2*n)*c + a)**p*e*x + 2*(x* 
*(2*n)*c + a)**p*d*n*p*x + (x**(2*n)*c + a)**p*d*n*x + (x**(2*n)*c + a)**p 
*d*x + 16*int((x**(2*n)*c + a)**p/(4*x**(2*n)*c*n**2*p**2 + 2*x**(2*n)*c*n 
**2*p + 4*x**(2*n)*c*n*p + x**(2*n)*c*n + x**(2*n)*c + 4*a*n**2*p**2 + 2*a 
*n**2*p + 4*a*n*p + a*n + a),x)*a*d*n**4*p**4 + 16*int((x**(2*n)*c + a)**p 
/(4*x**(2*n)*c*n**2*p**2 + 2*x**(2*n)*c*n**2*p + 4*x**(2*n)*c*n*p + x**(2* 
n)*c*n + x**(2*n)*c + 4*a*n**2*p**2 + 2*a*n**2*p + 4*a*n*p + a*n + a),x)*a 
*d*n**4*p**3 + 4*int((x**(2*n)*c + a)**p/(4*x**(2*n)*c*n**2*p**2 + 2*x**(2 
*n)*c*n**2*p + 4*x**(2*n)*c*n*p + x**(2*n)*c*n + x**(2*n)*c + 4*a*n**2*p** 
2 + 2*a*n**2*p + 4*a*n*p + a*n + a),x)*a*d*n**4*p**2 + 24*int((x**(2*n)*c 
+ a)**p/(4*x**(2*n)*c*n**2*p**2 + 2*x**(2*n)*c*n**2*p + 4*x**(2*n)*c*n*p + 
 x**(2*n)*c*n + x**(2*n)*c + 4*a*n**2*p**2 + 2*a*n**2*p + 4*a*n*p + a*n + 
a),x)*a*d*n**3*p**3 + 16*int((x**(2*n)*c + a)**p/(4*x**(2*n)*c*n**2*p**2 + 
 2*x**(2*n)*c*n**2*p + 4*x**(2*n)*c*n*p + x**(2*n)*c*n + x**(2*n)*c + 4*a* 
n**2*p**2 + 2*a*n**2*p + 4*a*n*p + a*n + a),x)*a*d*n**3*p**2 + 2*int((x**( 
2*n)*c + a)**p/(4*x**(2*n)*c*n**2*p**2 + 2*x**(2*n)*c*n**2*p + 4*x**(2*n)* 
c*n*p + x**(2*n)*c*n + x**(2*n)*c + 4*a*n**2*p**2 + 2*a*n**2*p + 4*a*n*p + 
 a*n + a),x)*a*d*n**3*p + 12*int((x**(2*n)*c + a)**p/(4*x**(2*n)*c*n**2*p* 
*2 + 2*x**(2*n)*c*n**2*p + 4*x**(2*n)*c*n*p + x**(2*n)*c*n + x**(2*n)*c + 
4*a*n**2*p**2 + 2*a*n**2*p + 4*a*n*p + a*n + a),x)*a*d*n**2*p**2 + 4*in...