\(\int (e x)^m (a+b x^n)^p (c+d x^n) \, dx\) [498]

Optimal result

Integrand size = 22, antiderivative size = 122 \[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {d (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{b e (1+m+n+n p)}+\frac {\left (\frac {c}{1+m}-\frac {a d}{b (1+m+n+n p)}\right ) (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{e} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.94 \[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {x (e x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c (1+m+n) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )+d (1+m) x^n \operatorname {Hypergeometric2F1}\left (\frac {1+m+n}{n},-p,\frac {1+m+2 n}{n},-\frac {b x^n}{a}\right )\right )}{(1+m) (1+m+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

(d*(e*x)^(1 + m)*(a + b*x^n)^(1 + p))/(b*e*(1 + m + n + n*p)) + ((c - (a*d 
*(1 + m))/(b*(1 + m + n + n*p)))*(e*x)^(1 + m)*(a + b*x^n)^p*Hypergeometri 
c2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(e*(1 + m)*(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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 37.60 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.31 \[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {a^{\frac {m}{n} + \frac {1}{n}} a^{- \frac {m}{n} + p - \frac {1}{n}} c e^{m} x^{m + 1} \Gamma \left (\frac {m}{n} + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + \frac {1}{n} \\ \frac {m}{n} + 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {a^{\frac {m}{n} + 1 + \frac {1}{n}} a^{- \frac {m}{n} + p - 1 - \frac {1}{n}} d e^{m} x^{m + n + 1} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + 1 + \frac {1}{n} \\ \frac {m}{n} + 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} \] Input:

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

Output:

a**(m/n + 1/n)*a**(-m/n + p - 1/n)*c*e**m*x**(m + 1)*gamma(m/n + 1/n)*hype 
r((-p, m/n + 1/n), (m/n + 1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/ 
n + 1 + 1/n)) + a**(m/n + 1 + 1/n)*a**(-m/n + p - 1 - 1/n)*d*e**m*x**(m + 
n + 1)*gamma(m/n + 1 + 1/n)*hyper((-p, m/n + 1 + 1/n), (m/n + 2 + 1/n,), b 
*x**n*exp_polar(I*pi)/a)/(n*gamma(m/n + 2 + 1/n))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate((e*x)^m*(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,[0,0,2,2,0,1,2,1,0,1]%%%}+%%%{2,[0,0,2,2,0,1,1,1,0,1]%%% 
}+%%%{1,[
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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