\(\int (e x)^m (a+b x^n)^p (c+d x^n)^q \, dx\) [564]

Optimal result

Integrand size = 24, antiderivative size = 102 \[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\frac {(e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1+m}{n},-p,-q,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{e (1+m)} \] Output:

(e*x)^(1+m)*(a+b*x^n)^p*(c+d*x^n)^q*AppellF1((1+m)/n,-p,-q,(1+m+n)/n,-b*x^ 
n/a,-d*x^n/c)/e/(1+m)/((1+b*x^n/a)^p)/((1+d*x^n/c)^q)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.96 \[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\frac {x (e x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1+m}{n},-p,-q,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{1+m} \] Input:

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

Output:

(x*(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*AppellF1[(1 + m)/n, -p, -q, (1 + m 
+ n)/n, -((b*x^n)/a), -((d*x^n)/c)])/((1 + m)*(1 + (b*x^n)/a)^p*(1 + (d*x^ 
n)/c)^q)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1013, 1013, 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.

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

\(\displaystyle \frac {(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {m+1}{n},-p,-q,\frac {m+n+1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{e (m+1)}\)

Input:

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

Output:

((e*x)^(1 + m)*(a + b*x^n)^p*(c + d*x^n)^q*AppellF1[(1 + m)/n, -p, -q, (1 
+ m + n)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(1 + m)*(1 + (b*x^n)/a)^p*(1 + 
 (d*x^n)/c)^q)
 

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 1013

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ 
n/a))^FracPart[p])   Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], 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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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