\(\int (e x)^m (a+b x^n)^p (A+B x^n) (c+d x^n) \, dx\) [59]

Optimal result

Integrand size = 29, antiderivative size = 255 \[ \int (e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {(A b d n-a B d (1+m+n)+b B c (1+m+n (2+p))) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{b^2 e (1+m+n+n p) (1+m+n (2+p))}+\frac {d (e x)^{1+m} \left (a+b x^n\right )^{1+p} \left (A+B x^n\right )}{b e (1+m+n (2+p))}-\frac {\left (a A d-\frac {A b c (1+m+n (2+p))}{1+m}+\frac {a (A b d n-a B d (1+m+n)+b B c (1+m+n (2+p)))}{b (1+m+n+n p)}\right ) (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{b e (1+m+n (2+p))} \] Output:

(A*b*d*n-a*B*d*(1+m+n)+b*B*c*(1+m+n*(2+p)))*(e*x)^(1+m)*(a+b*x^n)^(p+1)/b^ 
2/e/(n*p+m+n+1)/(1+m+n*(2+p))+d*(e*x)^(1+m)*(a+b*x^n)^(p+1)*(A+B*x^n)/b/e/ 
(1+m+n*(2+p))-(A*a*d-A*b*c*(1+m+n*(2+p))/(1+m)+a*(A*b*d*n-a*B*d*(1+m+n)+b* 
B*c*(1+m+n*(2+p)))/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)/b/e/(1+m+n*(2+p))/((1+b*x^n/a)^p)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.64 \[ \int (e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=x (e x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (\frac {A c \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{1+m}+x^n \left (\frac {(B c+A d) \operatorname {Hypergeometric2F1}\left (\frac {1+m+n}{n},-p,\frac {1+m+2 n}{n},-\frac {b x^n}{a}\right )}{1+m+n}+\frac {B d x^n \operatorname {Hypergeometric2F1}\left (\frac {1+m+2 n}{n},-p,\frac {1+m+3 n}{n},-\frac {b x^n}{a}\right )}{1+m+2 n}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1066, 25, 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.

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

\(\Big \downarrow \) 1066

\(\displaystyle \frac {\int -(e x)^m \left (b x^n+a\right )^p \left (A (a d (m+1)-b c (m+n (p+2)+1))-(A b d n-a B d (m+n+1)+b B c (m+n (p+2)+1)) x^n\right )dx}{b (m+n (p+2)+1)}+\frac {d (e x)^{m+1} \left (A+B x^n\right ) \left (a+b x^n\right )^{p+1}}{b e (m+n (p+2)+1)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d (e x)^{m+1} \left (A+B x^n\right ) \left (a+b x^n\right )^{p+1}}{b e (m+n (p+2)+1)}-\frac {\int (e x)^m \left (b x^n+a\right )^p \left ((a B d (m+n+1)-b (A d n+B c (m+n (p+2)+1))) x^n+A (a d (m+1)-b c (m+n (p+2)+1))\right )dx}{b (m+n (p+2)+1)}\)

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

(d*(e*x)^(1 + m)*(a + b*x^n)^(1 + p)*(A + B*x^n))/(b*e*(1 + m + n*(2 + p)) 
) - (((a*B*d*(1 + m + n) - b*(A*d*n + B*c*(1 + m + n*(2 + p))))*(e*x)^(1 + 
 m)*(a + b*x^n)^(1 + p))/(b*e*(1 + m + n + n*p)) + ((a*A*d*(1 + m) - A*b*c 
*(1 + m + n*(2 + p)) - (a*(1 + m)*(a*B*d*(1 + m + n) - b*(A*d*n + B*c*(1 + 
 m + n*(2 + p)))))/(b*(1 + m + n + n*p)))*(e*x)^(1 + m)*(a + b*x^n)^p*Hype 
rgeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(e*(1 + m)*(1 + 
 (b*x^n)/a)^p))/(b*(1 + m + n*(2 + p)))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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]
 

rule 1066

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*(g*x)^(m + 1)*(a + 
b*x^n)^(p + 1)*((c + d*x^n)^q/(b*g*(m + n*(p + q + 1) + 1))), x] + Simp[1/( 
b*(m + n*(p + q + 1) + 1))   Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)* 
Simp[c*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + 
 f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c 
, d, e, f, g, m, n, p}, x] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[e + f*x 
^n, c + d*x^n])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate((e*x)^m*(a+b*x^n)^p*(A+B*x^n)*(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,[1,0,4,3,0,1,3,3,1,1,0,0]%%%}+%%%{-3,[1,0,4,3,0,1,2,3,1 
,1,0,0]%%
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(e**m*(x**(m + 2*n)*(x**n*b + a)**p*b**2*d*m**2*x + 2*x**(m + 2*n)*(x**n*b 
 + a)**p*b**2*d*m*n*p*x + x**(m + 2*n)*(x**n*b + a)**p*b**2*d*m*n*x + 2*x* 
*(m + 2*n)*(x**n*b + a)**p*b**2*d*m*x + x**(m + 2*n)*(x**n*b + a)**p*b**2* 
d*n**2*p**2*x + x**(m + 2*n)*(x**n*b + a)**p*b**2*d*n**2*p*x + 2*x**(m + 2 
*n)*(x**n*b + a)**p*b**2*d*n*p*x + x**(m + 2*n)*(x**n*b + a)**p*b**2*d*n*x 
 + x**(m + 2*n)*(x**n*b + a)**p*b**2*d*x + x**(m + n)*(x**n*b + a)**p*a*b* 
d*m**2*x + 3*x**(m + n)*(x**n*b + a)**p*a*b*d*m*n*p*x + 2*x**(m + n)*(x**n 
*b + a)**p*a*b*d*m*n*x + 2*x**(m + n)*(x**n*b + a)**p*a*b*d*m*x + 2*x**(m 
+ n)*(x**n*b + a)**p*a*b*d*n**2*p**2*x + 2*x**(m + n)*(x**n*b + a)**p*a*b* 
d*n**2*p*x + 3*x**(m + n)*(x**n*b + a)**p*a*b*d*n*p*x + 2*x**(m + n)*(x**n 
*b + a)**p*a*b*d*n*x + x**(m + n)*(x**n*b + a)**p*a*b*d*x + x**(m + n)*(x* 
*n*b + a)**p*b**2*c*m**2*x + 2*x**(m + n)*(x**n*b + a)**p*b**2*c*m*n*p*x + 
 2*x**(m + n)*(x**n*b + a)**p*b**2*c*m*n*x + 2*x**(m + n)*(x**n*b + a)**p* 
b**2*c*m*x + x**(m + n)*(x**n*b + a)**p*b**2*c*n**2*p**2*x + 2*x**(m + n)* 
(x**n*b + a)**p*b**2*c*n**2*p*x + 2*x**(m + n)*(x**n*b + a)**p*b**2*c*n*p* 
x + 2*x**(m + n)*(x**n*b + a)**p*b**2*c*n*x + x**(m + n)*(x**n*b + a)**p*b 
**2*c*x + x**m*(x**n*b + a)**p*a**2*d*n**2*p**2*x + x**m*(x**n*b + a)**p*a 
**2*d*n**2*p*x + x**m*(x**n*b + a)**p*a*b*c*m**2*x + 3*x**m*(x**n*b + a)** 
p*a*b*c*m*n*p*x + 3*x**m*(x**n*b + a)**p*a*b*c*m*n*x + 2*x**m*(x**n*b + a) 
**p*a*b*c*m*x + 2*x**m*(x**n*b + a)**p*a*b*c*n**2*p**2*x + 5*x**m*(x**n...