\(\int (e x)^m (a+b x^n)^p (c+d x^n)^2 \, dx\) [497]

Optimal result

Integrand size = 24, antiderivative size = 225 \[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=-\frac {d (a d (1+m+n)-2 b c (1+m+n (2+p))) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{b^2 e (1+m+n+n p) (1+m+n (2+p))}+\frac {d^2 x^n (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{b e (1+m+n (2+p))}+\frac {\left (\frac {c^2}{1+m}+\frac {a d (a d (1+m+n)-2 b c (1+m+n (2+p)))}{b^2 (1+m+n+n p) (1+m+n (2+p))}\right ) (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{e} \] Output:

-d*(a*d*(1+m+n)-2*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^2*x^n*(e*x)^(1+m)*(a+b*x^n)^(p+1)/b/e/(1+m+n*(2+p 
))+(c^2/(1+m)+a*d*(a*d*(1+m+n)-2*b*c*(1+m+n*(2+p)))/b^2/(n*p+m+n+1)/(1+m+n 
*(2+p)))*(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 = 6.26 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.71 \[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=x (e x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (\frac {c^2 \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{1+m}+d x^n \left (\frac {2 c \operatorname {Hypergeometric2F1}\left (\frac {1+m+n}{n},-p,\frac {1+m+2 n}{n},-\frac {b x^n}{a}\right )}{1+m+n}+\frac {d x^n \operatorname {Hypergeometric2F1}\left (\frac {1+m+2 n}{n},-p,\frac {1+m+3 n}{n},-\frac {b x^n}{a}\right )}{1+m+2 n}\right )\right ) \] Input:

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

Output:

(x*(e*x)^m*(a + b*x^n)^p*((c^2*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n 
)/n, -((b*x^n)/a)])/(1 + m) + d*x^n*((2*c*Hypergeometric2F1[(1 + m + n)/n, 
 -p, (1 + m + 2*n)/n, -((b*x^n)/a)])/(1 + m + n) + (d*x^n*Hypergeometric2F 
1[(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 = 0.84 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1008, 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.

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

\(\Big \downarrow \) 1008

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

(d*(e*x)^(1 + m)*(a + b*x^n)^(1 + p)*(c + d*x^n))/(b*e*(1 + m + n*(2 + p)) 
) - ((d*(a*d*(1 + m + n) - b*c*(1 + m + n*(3 + p)))*(e*x)^(1 + m)*(a + b*x 
^n)^(1 + p))/(b*e*(1 + m + n + n*p)) + ((c*(a*d*(1 + m) - b*c*(1 + m + n*( 
2 + p))) - (a*d*(1 + m)*(a*d*(1 + m + n) - b*c*(1 + m + n*(3 + p))))/(b*(1 
 + m + n + n*p)))*(e*x)^(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(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 1008

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n) 
^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Simp[1/(b*(m + n*(p + q) + 1)) 
Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + 
 c*b*n*(p + q)) + (d*(c*b - a*d)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d* 
n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c 
 - a*d, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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