\(\int (f x)^m (d+e x^n) (a+b x^n+c x^{2 n})^p \, dx\) [185]

Optimal result

Integrand size = 29, antiderivative size = 326 \[ \int (f x)^m \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^p \, dx=\frac {d (f x)^{1+m} \left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \operatorname {AppellF1}\left (\frac {1+m}{n},-p,-p,\frac {1+m+n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{f (1+m)}+\frac {e x^n (f x)^{1+m} \left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \operatorname {AppellF1}\left (\frac {1+m+n}{n},-p,-p,\frac {1+m+2 n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{f (1+m+n)} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 1.21 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.84 \[ \int (f x)^m \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^p \, dx=\frac {x (f x)^m \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+x^n \left (b+c x^n\right )\right )^p \left (d (1+m+n) \operatorname {AppellF1}\left (\frac {1+m}{n},-p,-p,\frac {1+m+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )+e (1+m) x^n \operatorname {AppellF1}\left (\frac {1+m+n}{n},-p,-p,\frac {1+m+2 n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )}{(1+m) (1+m+n)} \] Input:

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

Output:

(x*(f*x)^m*(a + x^n*(b + c*x^n))^p*(d*(1 + m + n)*AppellF1[(1 + m)/n, -p, 
-p, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqr 
t[b^2 - 4*a*c])] + e*(1 + m)*x^n*AppellF1[(1 + m + n)/n, -p, -p, (1 + m + 
2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a 
*c])]))/((1 + m)*(1 + m + 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 = 0.56 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1884, 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.

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

\(\Big \downarrow \) 1884

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1884

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*( 
(d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d 
+ e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, 
 n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] &&  !RationalQ[n] && ( 
IGtQ[p, 0] || IGtQ[q, 0])
 

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((f*x)^m*(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*(f*x)^m, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((f*x)^m*(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*(f*x)^m, x)
 

Giac [F]

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

integrate((f*x)^m*(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*(f*x)^m, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(f**m*(x**(m + n)*(x**(2*n)*c + x**n*b + a)**p*b*e*m*x + x**(m + n)*(x**(2 
*n)*c + x**n*b + a)**p*b*e*n*p*x + x**(m + n)*(x**(2*n)*c + x**n*b + a)**p 
*b*e*x + 2*x**m*(x**(2*n)*c + x**n*b + a)**p*a*e*n*p*x + x**m*(x**(2*n)*c 
+ x**n*b + a)**p*b*d*m*x + 2*x**m*(x**(2*n)*c + x**n*b + a)**p*b*d*n*p*x + 
 x**m*(x**(2*n)*c + x**n*b + a)**p*b*d*n*x + x**m*(x**(2*n)*c + x**n*b + a 
)**p*b*d*x - 2*int((x**(m + 2*n)*(x**(2*n)*c + x**n*b + a)**p)/(x**(2*n)*c 
*m**2 + 3*x**(2*n)*c*m*n*p + x**(2*n)*c*m*n + 2*x**(2*n)*c*m + 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 + x**n*b*m**2 + 3*x**n*b*m*n*p + x**n*b*m*n + 2*x**n*b*m + 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 + a*m**2 + 3*a 
*m*n*p + a*m*n + 2*a*m + 2*a*n**2*p**2 + a*n**2*p + 3*a*n*p + a*n + a),x)* 
a*c*e*m**3*n*p - 10*int((x**(m + 2*n)*(x**(2*n)*c + x**n*b + a)**p)/(x**(2 
*n)*c*m**2 + 3*x**(2*n)*c*m*n*p + x**(2*n)*c*m*n + 2*x**(2*n)*c*m + 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 + x**n*b*m**2 + 3*x**n*b*m*n*p + x**n*b*m*n + 2*x**n*b*m + 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 + a*m**2 
+ 3*a*m*n*p + a*m*n + 2*a*m + 2*a*n**2*p**2 + a*n**2*p + 3*a*n*p + a*n + a 
),x)*a*c*e*m**2*n**2*p**2 - 2*int((x**(m + 2*n)*(x**(2*n)*c + x**n*b + a)* 
*p)/(x**(2*n)*c*m**2 + 3*x**(2*n)*c*m*n*p + x**(2*n)*c*m*n + 2*x**(2*n)*c* 
m + 2*x**(2*n)*c*n**2*p**2 + x**(2*n)*c*n**2*p + 3*x**(2*n)*c*n*p + x**...