\(\int (e x)^m (a+b x^n)^p (A+B x^n) (c+d x^n)^q \, dx\) [62]

Optimal result

Integrand size = 31, antiderivative size = 214 \[ \int (e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right ) \left (c+d x^n\right )^q \, dx=\frac {A (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1+m}{n},-p,-q,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{e (1+m)}+\frac {B x^n (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1+m+n}{n},-p,-q,\frac {1+m+2 n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{e (1+m+n)} \] Output:

A*(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)+B*x^n*(e*x)^(1+m)* 
(a+b*x^n)^p*(c+d*x^n)^q*AppellF1((1+m+n)/n,-p,-q,(1+m+2*n)/n,-b*x^n/a,-d*x 
^n/c)/e/(1+m+n)/((1+b*x^n/a)^p)/((1+d*x^n/c)^q)
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.76 \[ \int (e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right ) \left (c+d x^n\right )^q \, dx=\frac {x (e x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \left (A (1+m+n) \operatorname {AppellF1}\left (\frac {1+m}{n},-p,-q,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+B (1+m) x^n \operatorname {AppellF1}\left (\frac {1+m+n}{n},-p,-q,\frac {1+m+2 n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}{(1+m) (1+m+n)} \] Input:

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

Output:

(x*(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(A*(1 + m + n)*AppellF1[(1 + m)/n, 
-p, -q, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)] + B*(1 + m)*x^n*AppellF 
1[(1 + m + n)/n, -p, -q, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)]))/(( 
1 + m)*(1 + m + n)*(1 + (b*x^n)/a)^p*(1 + (d*x^n)/c)^q)
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1068, 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.

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

\(\Big \downarrow \) 1068

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

\(\displaystyle \frac {A (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {m+1}{n},-p,-q,\frac {m+n+1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{e (m+1)}+\frac {B x^{n+1} (e x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {m+n+1}{n},-p,-q,\frac {m+2 n+1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{m+n+1}\)

Input:

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

Output:

(A*(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) + (B*x^(1 + n)*(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*Appell 
F1[(1 + m + n)/n, -p, -q, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)])/(( 
1 + m + n)*(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])
 

rule 1068

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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