\(\int x^{3 n} (a+b x^n+c x^{2 n})^p \, dx\) [257]

Optimal result

Integrand size = 22, antiderivative size = 152 \[ \int x^{3 n} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\frac {x^{1+3 n} \left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \operatorname {AppellF1}\left (3+\frac {1}{n},-p,-p,4+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{1+3 n} \] Output:

x^(1+3*n)*(a+b*x^n+c*x^(2*n))^p*AppellF1(3+1/n,-p,-p,4+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+3*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] (verified)

Time = 0.78 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.18 \[ \int x^{3 n} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\frac {x^{1+3 n} \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+x^n \left (b+c x^n\right )\right )^p \operatorname {AppellF1}\left (3+\frac {1}{n},-p,-p,4+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )}{1+3 n} \] Input:

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

Output:

(x^(1 + 3*n)*(a + x^n*(b + c*x^n))^p*AppellF1[3 + n^(-1), -p, -p, 4 + 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 + 3*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 = 0.33 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1721, 1012}

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.

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

\(\Big \downarrow \) 1721

\(\displaystyle \left (\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \int x^{3 n} \left (\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1\right )^p \left (\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}+1\right )^pdx\)

\(\Big \downarrow \) 1012

\(\displaystyle \frac {x^{3 n+1} \left (\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \operatorname {AppellF1}\left (3+\frac {1}{n},-p,-p,4+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{3 n+1}\)

Input:

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

Output:

(x^(1 + 3*n)*(a + b*x^n + c*x^(2*n))^p*AppellF1[3 + n^(-1), -p, -p, 4 + 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 + 3*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 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1721

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x 
_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n + c*x^(2*n))^FracPart[p]/((1 + 2* 
c*(x^n/(b + Rt[b^2 - 4*a*c, 2])))^FracPart[p]*(1 + 2*c*(x^n/(b - Rt[b^2 - 4 
*a*c, 2])))^FracPart[p]))   Int[(d*x)^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, x], x] /; FreeQ[{a, b, c, 
 d, m, n, p}, x] && EqQ[n2, 2*n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate(x^(3*n)*(a+b*x^n+c*x^(2*n))^p,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%%%{-256,[1,0,7,4,7,5,2,8]%%%}+%%%{-1280,[1,0,7,4,7,4,2,8]%%%} 
+%%%{-256
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(4*x**(3*n)*(x**(2*n)*c + x**n*b + a)**p*b*c**2*n**3*p**3*x + 6*x**(3*n)*( 
x**(2*n)*c + x**n*b + a)**p*b*c**2*n**3*p**2*x + 2*x**(3*n)*(x**(2*n)*c + 
x**n*b + a)**p*b*c**2*n**3*p*x + 8*x**(3*n)*(x**(2*n)*c + x**n*b + a)**p*b 
*c**2*n**2*p**2*x + 9*x**(3*n)*(x**(2*n)*c + x**n*b + a)**p*b*c**2*n**2*p* 
x + 2*x**(3*n)*(x**(2*n)*c + x**n*b + a)**p*b*c**2*n**2*x + 5*x**(3*n)*(x* 
*(2*n)*c + x**n*b + a)**p*b*c**2*n*p*x + 3*x**(3*n)*(x**(2*n)*c + x**n*b + 
 a)**p*b*c**2*n*x + x**(3*n)*(x**(2*n)*c + x**n*b + a)**p*b*c**2*x + 2*x** 
(2*n)*(x**(2*n)*c + x**n*b + a)**p*b**2*c*n**3*p**3*x + x**(2*n)*(x**(2*n) 
*c + x**n*b + a)**p*b**2*c*n**3*p**2*x + 3*x**(2*n)*(x**(2*n)*c + x**n*b + 
 a)**p*b**2*c*n**2*p**2*x + x**(2*n)*(x**(2*n)*c + x**n*b + a)**p*b**2*c*n 
**2*p*x + x**(2*n)*(x**(2*n)*c + x**n*b + a)**p*b**2*c*n*p*x + 4*x**n*(x** 
(2*n)*c + x**n*b + a)**p*a*b*c*n**3*p**3*x + 4*x**n*(x**(2*n)*c + x**n*b + 
 a)**p*a*b*c*n**3*p**2*x + 6*x**n*(x**(2*n)*c + x**n*b + a)**p*a*b*c*n**2* 
p**2*x + 4*x**n*(x**(2*n)*c + x**n*b + a)**p*a*b*c*n**2*p*x + 2*x**n*(x**( 
2*n)*c + x**n*b + a)**p*a*b*c*n*p*x - x**n*(x**(2*n)*c + x**n*b + a)**p*b* 
*3*n**3*p**3*x - 2*x**n*(x**(2*n)*c + x**n*b + a)**p*b**3*n**3*p**2*x - 2* 
x**n*(x**(2*n)*c + x**n*b + a)**p*b**3*n**2*p**2*x - 2*x**n*(x**(2*n)*c + 
x**n*b + a)**p*b**3*n**2*p*x - x**n*(x**(2*n)*c + x**n*b + a)**p*b**3*n*p* 
x - 4*(x**(2*n)*c + x**n*b + a)**p*a**2*c*n**3*p**2*x - 4*(x**(2*n)*c + x* 
*n*b + a)**p*a**2*c*n**3*p*x - 4*(x**(2*n)*c + x**n*b + a)**p*a**2*c*n*...