\(\int (d+e x^n) (a+b x^n+c x^{2 n})^p \, dx\) [91]

Optimal result

Integrand size = 24, antiderivative size = 288 \[ \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^p \, dx=\frac {e x^{1+n} \left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \operatorname {AppellF1}\left (1+\frac {1}{n},-p,-p,2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{1+n}+d x \left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \operatorname {AppellF1}\left (\frac {1}{n},-p,-p,1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right ) \] Output:

e*x^(1+n)*(a+b*x^n+c*x^(2*n))^p*AppellF1(1+1/n,-p,-p,2+1/n,-2*c*x^n/(b-(-4 
*a*c+b^2)^(1/2)),-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))/(1+n)/((1+2*c*x^n/(b-(-4 
*a*c+b^2)^(1/2)))^p)/((1+2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))^p)+d*x*(a+b*x^n+c 
*x^(2*n))^p*AppellF1(1/n,-p,-p,1+1/n,-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)),-2*c* 
x^n/(b+(-4*a*c+b^2)^(1/2)))/((1+2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))^p)/((1+2*c 
*x^n/(b+(-4*a*c+b^2)^(1/2)))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 0.67 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.84 \[ \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^p \, dx=\frac {x \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+x^n \left (b+c x^n\right )\right )^p \left (e x^n \operatorname {AppellF1}\left (1+\frac {1}{n},-p,-p,2+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )+d (1+n) \operatorname {AppellF1}\left (\frac {1}{n},-p,-p,1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )}{1+n} \] Input:

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

Output:

(x*(a + x^n*(b + c*x^n))^p*(e*x^n*AppellF1[1 + n^(-1), -p, -p, 2 + n^(-1), 
 (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + 
 d*(1 + n)*AppellF1[n^(-1), -p, -p, 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 
 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/((1 + n)*((b - Sqrt[b^2 - 
4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c 
*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1762, 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.

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

\(\Big \downarrow \) 1762

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

\(\Big \downarrow \) 2009

\(\displaystyle d x \left (\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \operatorname {AppellF1}\left (\frac {1}{n},-p,-p,1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )+\frac {e x^{n+1} \left (\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \operatorname {AppellF1}\left (1+\frac {1}{n},-p,-p,2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{n+1}\)

Input:

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

Output:

(e*x^(1 + n)*(a + b*x^n + c*x^(2*n))^p*AppellF1[1 + n^(-1), -p, -p, 2 + n^ 
(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c 
])])/((1 + n)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b 
+ Sqrt[b^2 - 4*a*c]))^p) + (d*x*(a + b*x^n + c*x^(2*n))^p*AppellF1[n^(-1), 
 -p, -p, 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + S 
qrt[b^2 - 4*a*c])])/((1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x 
^n)/(b + Sqrt[b^2 - 4*a*c]))^p)
 

Defintions of rubi rules used

rule 1762

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p 
_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^p, 
 x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 
 0]
 

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(x**n*(x**(2*n)*c + x**n*b + a)**p*b*e*n*p*x + x**n*(x**(2*n)*c + x**n*b + 
 a)**p*b*e*x + 2*(x**(2*n)*c + x**n*b + a)**p*a*e*n*p*x + 2*(x**(2*n)*c + 
x**n*b + a)**p*b*d*n*p*x + (x**(2*n)*c + x**n*b + a)**p*b*d*n*x + (x**(2*n 
)*c + x**n*b + a)**p*b*d*x - 4*int((x**(2*n)*c + x**n*b + a)**p/(2*x**(2*n 
)*c*n**2*p**2 + x**(2*n)*c*n**2*p + 3*x**(2*n)*c*n*p + x**(2*n)*c*n + x**( 
2*n)*c + 2*x**n*b*n**2*p**2 + x**n*b*n**2*p + 3*x**n*b*n*p + x**n*b*n + x* 
*n*b + 2*a*n**2*p**2 + a*n**2*p + 3*a*n*p + a*n + a),x)*a**2*e*n**3*p**3 - 
 2*int((x**(2*n)*c + x**n*b + a)**p/(2*x**(2*n)*c*n**2*p**2 + x**(2*n)*c*n 
**2*p + 3*x**(2*n)*c*n*p + x**(2*n)*c*n + x**(2*n)*c + 2*x**n*b*n**2*p**2 
+ x**n*b*n**2*p + 3*x**n*b*n*p + x**n*b*n + x**n*b + 2*a*n**2*p**2 + a*n** 
2*p + 3*a*n*p + a*n + a),x)*a**2*e*n**3*p**2 - 6*int((x**(2*n)*c + x**n*b 
+ a)**p/(2*x**(2*n)*c*n**2*p**2 + x**(2*n)*c*n**2*p + 3*x**(2*n)*c*n*p + x 
**(2*n)*c*n + x**(2*n)*c + 2*x**n*b*n**2*p**2 + x**n*b*n**2*p + 3*x**n*b*n 
*p + x**n*b*n + x**n*b + 2*a*n**2*p**2 + a*n**2*p + 3*a*n*p + a*n + a),x)* 
a**2*e*n**2*p**2 - 2*int((x**(2*n)*c + x**n*b + a)**p/(2*x**(2*n)*c*n**2*p 
**2 + x**(2*n)*c*n**2*p + 3*x**(2*n)*c*n*p + x**(2*n)*c*n + x**(2*n)*c + 2 
*x**n*b*n**2*p**2 + x**n*b*n**2*p + 3*x**n*b*n*p + x**n*b*n + x**n*b + 2*a 
*n**2*p**2 + a*n**2*p + 3*a*n*p + a*n + a),x)*a**2*e*n**2*p - 2*int((x**(2 
*n)*c + x**n*b + a)**p/(2*x**(2*n)*c*n**2*p**2 + x**(2*n)*c*n**2*p + 3*x** 
(2*n)*c*n*p + x**(2*n)*c*n + x**(2*n)*c + 2*x**n*b*n**2*p**2 + x**n*b*n...