\(\int (a+b x^n)^p (c+d x^n)^q (e+f x^n) \, dx\) [8]

Optimal result

Integrand size = 26, antiderivative size = 176 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (e+f x^n\right ) \, dx=\frac {f x^{1+n} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (1+\frac {1}{n},-p,-q,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{1+n}+e x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{n},-p,-q,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right ) \] Output:

f*x^(1+n)*(a+b*x^n)^p*(c+d*x^n)^q*AppellF1(1+1/n,-p,-q,2+1/n,-b*x^n/a,-d*x 
^n/c)/(1+n)/((1+b*x^n/a)^p)/((1+d*x^n/c)^q)+e*x*(a+b*x^n)^p*(c+d*x^n)^q*Ap 
pellF1(1/n,-p,-q,1+1/n,-b*x^n/a,-d*x^n/c)/((1+b*x^n/a)^p)/((1+d*x^n/c)^q)
 

Mathematica [A] (warning: unable to verify)

Time = 0.54 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.52 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (e+f x^n\right ) \, dx=\frac {x \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (f x^n \left (1+\frac {b x^n}{a}\right )^{-p} \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (1+\frac {1}{n},-p,-q,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+\frac {a c e (1+n)^2 \operatorname {AppellF1}\left (\frac {1}{n},-p,-q,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{b c n p x^n \operatorname {AppellF1}\left (1+\frac {1}{n},1-p,-q,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a d n q x^n \operatorname {AppellF1}\left (1+\frac {1}{n},-p,1-q,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a c (1+n) \operatorname {AppellF1}\left (\frac {1}{n},-p,-q,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}\right )}{1+n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1027, 937, 937, 936, 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.

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

\(\Big \downarrow \) 1027

\(\displaystyle e \int \left (b x^n+a\right )^p \left (d x^n+c\right )^qdx+f \int x^n \left (b x^n+a\right )^p \left (d x^n+c\right )^qdx\)

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 936

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) 
], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] 
 && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 937

Int[((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[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q 
}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

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 1027

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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