\(\int x^{-1-n (-2+p)} (a+b x^n)^p (c+d x^n) \, dx\) [547]

Optimal result

Integrand size = 27, antiderivative size = 118 \[ \int x^{-1-n (-2+p)} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {d x^{n (2-p)} \left (a+b x^n\right )^{1+p}}{3 b n}+\frac {(3 b c-a d (2-p)) x^{n (2-p)} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,-\frac {b x^n}{a}\right )}{3 b n (2-p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.90 \[ \int x^{-1-n (-2+p)} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=-\frac {x^{-n (-2+p)} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c (-3+p) \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,-\frac {b x^n}{a}\right )+d (-2+p) x^n \operatorname {Hypergeometric2F1}\left (3-p,-p,4-p,-\frac {b x^n}{a}\right )\right )}{n (-3+p) (-2+p)} \] Input:

Integrate[x^(-1 - n*(-2 + p))*(a + b*x^n)^p*(c + d*x^n),x]
 

Output:

-(((a + b*x^n)^p*(c*(-3 + p)*Hypergeometric2F1[2 - p, -p, 3 - p, -((b*x^n) 
/a)] + d*(-2 + p)*x^n*Hypergeometric2F1[3 - p, -p, 4 - p, -((b*x^n)/a)]))/ 
(n*(-3 + p)*(-2 + p)*x^(n*(-2 + p))*(1 + (b*x^n)/a)^p))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {959, 882, 74}

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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 882

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

\(\Big \downarrow \) 74

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

Input:

Int[x^(-1 - n*(-2 + p))*(a + b*x^n)^p*(c + d*x^n),x]
 

Output:

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

Defintions of rubi rules used

rule 74

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

rule 882

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^Simplify[ 
(m + 1)/n + p]*x^m*(a + b*x^n)^p*((x^n/(a + b*x^n))^p/(n*x^Simplify[m + n*p 
]))   Subst[Int[x^((m + 1)/n - 1)/(1 - b*x)^(Simplify[(m + 1)/n + p] + 1), 
x], x, x^n/(a + b*x^n)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simpli 
fy[(m + 1)/n + p]]
 

rule 959

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

Maple [F]

\[\int x^{-1-n \left (-2+p \right )} \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )d x\]

Input:

int(x^(-1-n*(-2+p))*(a+b*x^n)^p*(c+d*x^n),x)
 

Output:

int(x^(-1-n*(-2+p))*(a+b*x^n)^p*(c+d*x^n),x)
 

Fricas [F]

\[ \int x^{-1-n (-2+p)} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\int { {\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 2\right )} - 1} \,d x } \] Input:

integrate(x^(-1-n*(-2+p))*(a+b*x^n)^p*(c+d*x^n),x, algorithm="fricas")
 

Output:

integral((d*x^(-n*p + 2*n - 1)*x^n + c*x^(-n*p + 2*n - 1))*(b*x^n + a)^p, 
x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 86.90 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.05 \[ \int x^{-1-n (-2+p)} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {a^{2 - p} a^{2 p - 2} b^{2 - p} b^{p - 2} c x^{- n p + 2 n} \Gamma \left (2 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 2 - p \\ 3 - p \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 - p\right )} + \frac {a^{3 - p} a^{2 p - 3} b^{3 - p} b^{p - 3} d x^{- n p + 3 n} \Gamma \left (3 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 3 - p \\ 4 - p \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (4 - p\right )} \] Input:

integrate(x**(-1-n*(-2+p))*(a+b*x**n)**p*(c+d*x**n),x)
 

Output:

a**(2 - p)*a**(2*p - 2)*b**(2 - p)*b**(p - 2)*c*x**(-n*p + 2*n)*gamma(2 - 
p)*hyper((-p, 2 - p), (3 - p,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(3 - p)) 
 + a**(3 - p)*a**(2*p - 3)*b**(3 - p)*b**(p - 3)*d*x**(-n*p + 3*n)*gamma(3 
 - p)*hyper((-p, 3 - p), (4 - p,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(4 - 
p))
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int x^{-1-n (-2+p)} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\int { {\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 2\right )} - 1} \,d x } \] Input:

integrate(x^(-1-n*(-2+p))*(a+b*x^n)^p*(c+d*x^n),x, algorithm="maxima")
 

Output:

integrate((d*x^n + c)*(b*x^n + a)^p*x^(-n*(p - 2) - 1), x)
 

Giac [F(-2)]

Exception generated. \[ \int x^{-1-n (-2+p)} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^(-1-n*(-2+p))*(a+b*x^n)^p*(c+d*x^n),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{3,[0,0,2,2,1,1,1,0,1]%%%}+%%%{3,[0,0,2,2,1,1,0,0,1]%%%}+%% 
%{1,[0,0,
 

Mupad [F(-1)]

Timed out. \[ \int x^{-1-n (-2+p)} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\int \frac {{\left (a+b\,x^n\right )}^p\,\left (c+d\,x^n\right )}{x^{n\,\left (p-2\right )+1}} \,d x \] Input:

int(((a + b*x^n)^p*(c + d*x^n))/x^(n*(p - 2) + 1),x)
 

Output:

int(((a + b*x^n)^p*(c + d*x^n))/x^(n*(p - 2) + 1), x)
 

Reduce [F]

\[ \int x^{-1-n (-2+p)} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {2 x^{3 n} \left (x^{n} b +a \right )^{p} b^{2} d +x^{2 n} \left (x^{n} b +a \right )^{p} a b d p +3 x^{2 n} \left (x^{n} b +a \right )^{p} b^{2} c +x^{n} \left (x^{n} b +a \right )^{p} a^{2} d \,p^{2}-2 x^{n} \left (x^{n} b +a \right )^{p} a^{2} d p +3 x^{n} \left (x^{n} b +a \right )^{p} a b c p +x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{3} d n \,p^{3}-3 x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{3} d n \,p^{2}+2 x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{3} d n p +3 x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{2} b c n \,p^{2}-3 x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{2} b c n p}{6 x^{n p} b^{2} n} \] Input:

int(x^(-1-n*(-2+p))*(a+b*x^n)^p*(c+d*x^n),x)
 

Output:

(2*x**(3*n)*(x**n*b + a)**p*b**2*d + x**(2*n)*(x**n*b + a)**p*a*b*d*p + 3* 
x**(2*n)*(x**n*b + a)**p*b**2*c + x**n*(x**n*b + a)**p*a**2*d*p**2 - 2*x** 
n*(x**n*b + a)**p*a**2*d*p + 3*x**n*(x**n*b + a)**p*a*b*c*p + x**(n*p)*int 
((x**n*(x**n*b + a)**p)/(x**(n*p + n)*b*x + x**(n*p)*a*x),x)*a**3*d*n*p**3 
 - 3*x**(n*p)*int((x**n*(x**n*b + a)**p)/(x**(n*p + n)*b*x + x**(n*p)*a*x) 
,x)*a**3*d*n*p**2 + 2*x**(n*p)*int((x**n*(x**n*b + a)**p)/(x**(n*p + n)*b* 
x + x**(n*p)*a*x),x)*a**3*d*n*p + 3*x**(n*p)*int((x**n*(x**n*b + a)**p)/(x 
**(n*p + n)*b*x + x**(n*p)*a*x),x)*a**2*b*c*n*p**2 - 3*x**(n*p)*int((x**n* 
(x**n*b + a)**p)/(x**(n*p + n)*b*x + x**(n*p)*a*x),x)*a**2*b*c*n*p)/(6*x** 
(n*p)*b**2*n)