\(\int x^{m+2 n} (a+b x^n)^p (c+d x^n) \, dx\) [502]

Optimal result

Integrand size = 24, antiderivative size = 128 \[ \int x^{m+2 n} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {d x^{1+m+2 n} \left (a+b x^n\right )^{1+p}}{b (1+m+n (3+p))}+\left (\frac {c}{1+m+2 n}-\frac {a d}{b (1+m+n (3+p))}\right ) x^{1+m+2 n} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m+2 n}{n},-p,\frac {1+m+3 n}{n},-\frac {b x^n}{a}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.05 \[ \int x^{m+2 n} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {x^{1+m+2 n} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c (1+m+3 n) \operatorname {Hypergeometric2F1}\left (\frac {1+m+2 n}{n},-p,\frac {1+m+3 n}{n},-\frac {b x^n}{a}\right )+d (1+m+2 n) x^n \operatorname {Hypergeometric2F1}\left (\frac {1+m+3 n}{n},-p,\frac {1+m+4 n}{n},-\frac {b x^n}{a}\right )\right )}{(1+m+2 n) (1+m+3 n)} \] Input:

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

Output:

(x^(1 + m + 2*n)*(a + b*x^n)^p*(c*(1 + m + 3*n)*Hypergeometric2F1[(1 + m + 
 2*n)/n, -p, (1 + m + 3*n)/n, -((b*x^n)/a)] + d*(1 + m + 2*n)*x^n*Hypergeo 
metric2F1[(1 + m + 3*n)/n, -p, (1 + m + 4*n)/n, -((b*x^n)/a)]))/((1 + m + 
2*n)*(1 + m + 3*n)*(1 + (b*x^n)/a)^p)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

(d*x^(1 + m + 2*n)*(a + b*x^n)^(1 + p))/(b*(1 + m + n*(3 + p))) + ((c - (a 
*d*(1 + m + 2*n))/(b*(1 + m + n*(3 + p))))*x^(1 + m + 2*n)*(a + b*x^n)^p*H 
ypergeometric2F1[(1 + m + 2*n)/n, -p, (1 + m + 3*n)/n, -((b*x^n)/a)])/((1 
+ m + 2*n)*(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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 27.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.29 \[ \int x^{m+2 n} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {a^{\frac {m}{n} + 2 + \frac {1}{n}} a^{- \frac {m}{n} + p - 2 - \frac {1}{n}} c x^{m + 2 n + 1} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + 2 + \frac {1}{n} \\ \frac {m}{n} + 3 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )} + \frac {a^{\frac {m}{n} + 3 + \frac {1}{n}} a^{- \frac {m}{n} + p - 3 - \frac {1}{n}} d x^{m + 3 n + 1} \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + 3 + \frac {1}{n} \\ \frac {m}{n} + 4 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 4 + \frac {1}{n}\right )} \] Input:

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

Output:

a**(m/n + 2 + 1/n)*a**(-m/n + p - 2 - 1/n)*c*x**(m + 2*n + 1)*gamma(m/n + 
2 + 1/n)*hyper((-p, m/n + 2 + 1/n), (m/n + 3 + 1/n,), b*x**n*exp_polar(I*p 
i)/a)/(n*gamma(m/n + 3 + 1/n)) + a**(m/n + 3 + 1/n)*a**(-m/n + p - 3 - 1/n 
)*d*x**(m + 3*n + 1)*gamma(m/n + 3 + 1/n)*hyper((-p, m/n + 3 + 1/n), (m/n 
+ 4 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/n + 4 + 1/n))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate(x^(m+2*n)*(a+b*x^n)^p*(c+d*x^n),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%%%{1,[0,0,2,2,1,2,0,1,0,1]%%%}+%%%{4,[0,0,2,2,1,1,0,1,0,1]%%% 
}+%%%{3,[
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(x**(m + 3*n)*(x**n*b + a)**p*b**3*d*m**3*x + 3*x**(m + 3*n)*(x**n*b + a)* 
*p*b**3*d*m**2*n*p*x + 3*x**(m + 3*n)*(x**n*b + a)**p*b**3*d*m**2*n*x + 3* 
x**(m + 3*n)*(x**n*b + a)**p*b**3*d*m**2*x + 3*x**(m + 3*n)*(x**n*b + a)** 
p*b**3*d*m*n**2*p**2*x + 6*x**(m + 3*n)*(x**n*b + a)**p*b**3*d*m*n**2*p*x 
+ 2*x**(m + 3*n)*(x**n*b + a)**p*b**3*d*m*n**2*x + 6*x**(m + 3*n)*(x**n*b 
+ a)**p*b**3*d*m*n*p*x + 6*x**(m + 3*n)*(x**n*b + a)**p*b**3*d*m*n*x + 3*x 
**(m + 3*n)*(x**n*b + a)**p*b**3*d*m*x + x**(m + 3*n)*(x**n*b + a)**p*b**3 
*d*n**3*p**3*x + 3*x**(m + 3*n)*(x**n*b + a)**p*b**3*d*n**3*p**2*x + 2*x** 
(m + 3*n)*(x**n*b + a)**p*b**3*d*n**3*p*x + 3*x**(m + 3*n)*(x**n*b + a)**p 
*b**3*d*n**2*p**2*x + 6*x**(m + 3*n)*(x**n*b + a)**p*b**3*d*n**2*p*x + 2*x 
**(m + 3*n)*(x**n*b + a)**p*b**3*d*n**2*x + 3*x**(m + 3*n)*(x**n*b + a)**p 
*b**3*d*n*p*x + 3*x**(m + 3*n)*(x**n*b + a)**p*b**3*d*n*x + x**(m + 3*n)*( 
x**n*b + a)**p*b**3*d*x + x**(m + 2*n)*(x**n*b + a)**p*a*b**2*d*m**2*n*p*x 
 + 2*x**(m + 2*n)*(x**n*b + a)**p*a*b**2*d*m*n**2*p**2*x + x**(m + 2*n)*(x 
**n*b + a)**p*a*b**2*d*m*n**2*p*x + 2*x**(m + 2*n)*(x**n*b + a)**p*a*b**2* 
d*m*n*p*x + x**(m + 2*n)*(x**n*b + a)**p*a*b**2*d*n**3*p**3*x + x**(m + 2* 
n)*(x**n*b + a)**p*a*b**2*d*n**3*p**2*x + 2*x**(m + 2*n)*(x**n*b + a)**p*a 
*b**2*d*n**2*p**2*x + x**(m + 2*n)*(x**n*b + a)**p*a*b**2*d*n**2*p*x + x** 
(m + 2*n)*(x**n*b + a)**p*a*b**2*d*n*p*x + x**(m + 2*n)*(x**n*b + a)**p*b* 
*3*c*m**3*x + 3*x**(m + 2*n)*(x**n*b + a)**p*b**3*c*m**2*n*p*x + 4*x**(...