\(\int (a+b x^n)^p (c+d x^n)^q \, dx\) [143]

Optimal result

Integrand size = 19, antiderivative size = 81 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{n},-p,-q,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right ) \] Output:

x*(a+b*x^n)^p*(c+d*x^n)^q*AppellF1(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 [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(190\) vs. \(2(81)=162\).

Time = 0.49 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.35 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\frac {a c (1+n) x \left (a+b x^n\right )^p \left (c+d x^n\right )^q \operatorname {AppellF1}\left (\frac {1}{n},-p,-q,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{b c n p x^n \operatorname {AppellF1}\left (1+\frac {1}{n},1-p,-q,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a d n q x^n \operatorname {AppellF1}\left (1+\frac {1}{n},-p,1-q,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a c (1+n) \operatorname {AppellF1}\left (\frac {1}{n},-p,-q,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )} \] Input:

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

Output:

(a*c*(1 + n)*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)])/(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)*Ap 
pellF1[n^(-1), -p, -q, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)])
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {937, 937, 936}

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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

(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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((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, x)
 

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

integrate((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, x)
 

Giac [F]

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

integrate((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, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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