\(\int x^{-1+3 n} (a+b x^n)^p (c+d x^n)^q \, dx\) [570]

Optimal result

Integrand size = 26, antiderivative size = 231 \[ \int x^{-1+3 n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=-\frac {(b c (2+p)+a d (4+p+2 q)) \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{1+q}}{b^2 d^2 n (2+p+q) (3+p+q)}+\frac {\left (a+b x^n\right )^{2+p} \left (c+d x^n\right )^{1+q}}{b^2 d n (3+p+q)}+\frac {\left (b^2 c^2 \left (2+3 p+p^2\right )+2 a b c d (1+p) (1+q)+a^2 d^2 \left (2+3 q+q^2\right )\right ) \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,2+p+q,2+p,-\frac {d \left (a+b x^n\right )}{b c-a d}\right )}{b^2 d^2 (b c-a d) n (1+p) (2+p+q) (3+p+q)} \] Output:

-(b*c*(2+p)+a*d*(4+p+2*q))*(a+b*x^n)^(p+1)*(c+d*x^n)^(1+q)/b^2/d^2/n/(2+p+ 
q)/(3+p+q)+(a+b*x^n)^(2+p)*(c+d*x^n)^(1+q)/b^2/d/n/(3+p+q)+(b^2*c^2*(p^2+3 
*p+2)+2*a*b*c*d*(p+1)*(1+q)+a^2*d^2*(q^2+3*q+2))*(a+b*x^n)^(p+1)*(c+d*x^n) 
^(1+q)*hypergeom([1, 2+p+q],[2+p],-d*(a+b*x^n)/(-a*d+b*c))/b^2/d^2/(-a*d+b 
*c)/n/(p+1)/(2+p+q)/(3+p+q)
 

Mathematica [A] (warning: unable to verify)

Time = 0.52 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.84 \[ \int x^{-1+3 n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\frac {\left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^q \left (-\frac {(b c (2+p)+a d (2+q)) \left (c+d x^n\right )}{b d (2+p+q)}+x^n \left (c+d x^n\right )+\frac {\left (b^2 c^2 \left (2+3 p+p^2\right )+2 a b c d (1+p) (1+q)+a^2 d^2 \left (2+3 q+q^2\right )\right ) \left (\frac {b \left (c+d x^n\right )}{b c-a d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (1+p,-q,2+p,\frac {d \left (a+b x^n\right )}{-b c+a d}\right )}{b^2 d (1+p) (2+p+q)}\right )}{b d n (3+p+q)} \] Input:

Integrate[x^(-1 + 3*n)*(a + b*x^n)^p*(c + d*x^n)^q,x]
 

Output:

((a + b*x^n)^(1 + p)*(c + d*x^n)^q*(-(((b*c*(2 + p) + a*d*(2 + q))*(c + d* 
x^n))/(b*d*(2 + p + q))) + x^n*(c + d*x^n) + ((b^2*c^2*(2 + 3*p + p^2) + 2 
*a*b*c*d*(1 + p)*(1 + q) + a^2*d^2*(2 + 3*q + q^2))*Hypergeometric2F1[1 + 
p, -q, 2 + p, (d*(a + b*x^n))/(-(b*c) + a*d)])/(b^2*d*(1 + p)*(2 + p + q)* 
((b*(c + d*x^n))/(b*c - a*d))^q)))/(b*d*n*(3 + p + q))
 

Rubi [A] (warning: unable to verify)

Time = 0.65 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {948, 101, 25, 90, 80, 79}

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.

\(\displaystyle \int x^{3 n-1} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle \frac {\int x^{2 n} \left (b x^n+a\right )^p \left (d x^n+c\right )^qdx^n}{n}\)

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

Int[x^(-1 + 3*n)*(a + b*x^n)^p*(c + d*x^n)^q,x]
 

Output:

((x^n*(a + b*x^n)^(1 + p)*(c + d*x^n)^(1 + q))/(b*d*(3 + p + q)) - (((b*c* 
(2 + p) + a*d*(2 + q))*(a + b*x^n)^(1 + p)*(c + d*x^n)^(1 + q))/(b*d*(2 + 
p + q)) + ((a*c - ((b*c*(1 + p) + a*d*(1 + q))*(b*c*(2 + p) + a*d*(2 + q)) 
)/(b*d*(2 + p + q)))*(a + b*x^n)^(1 + p)*(c + d*x^n)^q*Hypergeometric2F1[1 
 + p, -q, 2 + p, -((d*(a + b*x^n))/(b*c - a*d))])/(b*(1 + p)*((b*(c + d*x^ 
n))/(b*c - a*d))^q))/(b*d*(3 + p + q)))/n
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 948

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

Maple [F]

\[\int x^{-1+3 n} \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )^{q}d x\]

Input:

int(x^(-1+3*n)*(a+b*x^n)^p*(c+d*x^n)^q,x)
 

Output:

int(x^(-1+3*n)*(a+b*x^n)^p*(c+d*x^n)^q,x)
 

Fricas [F]

\[ \int x^{-1+3 n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} x^{3 \, n - 1} \,d x } \] Input:

integrate(x^(-1+3*n)*(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^(3*n - 1), x)
 

Sympy [F(-2)]

Exception generated. \[ \int x^{-1+3 n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(x**(-1+3*n)*(a+b*x**n)**p*(c+d*x**n)**q,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int x^{-1+3 n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} x^{3 \, n - 1} \,d x } \] Input:

integrate(x^(-1+3*n)*(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^(3*n - 1), x)
 

Giac [F]

\[ \int x^{-1+3 n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\int { {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} x^{3 \, n - 1} \,d x } \] Input:

integrate(x^(-1+3*n)*(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^(3*n - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{-1+3 n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\int x^{3\,n-1}\,{\left (a+b\,x^n\right )}^p\,{\left (c+d\,x^n\right )}^q \,d x \] Input:

int(x^(3*n - 1)*(a + b*x^n)^p*(c + d*x^n)^q,x)
 

Output:

int(x^(3*n - 1)*(a + b*x^n)^p*(c + d*x^n)^q, x)
 

Reduce [F]

\[ \int x^{-1+3 n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx=\text {too large to display} \] Input:

int(x^(-1+3*n)*(a+b*x^n)^p*(c+d*x^n)^q,x)
                                                                                    
                                                                                    
 

Output:

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