\(\int x^{-1-n (-3+p)} (a+b x^n)^p (c+d x^n) \, dx\) [555]

Optimal result

Integrand size = 27, antiderivative size = 118 \[ \int x^{-1-n (-3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {d x^{n (3-p)} \left (a+b x^n\right )^{1+p}}{4 b n}+\frac {(4 b c-a d (3-p)) x^{n (3-p)} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (3-p,-p,4-p,-\frac {b x^n}{a}\right )}{4 b n (3-p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.90 \[ \int x^{-1-n (-3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=-\frac {x^{-n (-3+p)} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c (-4+p) \operatorname {Hypergeometric2F1}\left (3-p,-p,4-p,-\frac {b x^n}{a}\right )+d (-3+p) x^n \operatorname {Hypergeometric2F1}\left (4-p,-p,5-p,-\frac {b x^n}{a}\right )\right )}{n (-4+p) (-3+p)} \] Input:

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

Output:

-(((a + b*x^n)^p*(c*(-4 + p)*Hypergeometric2F1[3 - p, -p, 4 - p, -((b*x^n) 
/a)] + d*(-3 + p)*x^n*Hypergeometric2F1[4 - p, -p, 5 - p, -((b*x^n)/a)]))/ 
(n*(-4 + p)*(-3 + p)*x^(n*(-3 + 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.

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

\(\Big \downarrow \) 959

\(\displaystyle \frac {(4 b c-a d (3-p)) \int x^{n (3-p)-1} \left (b x^n+a\right )^pdx}{4 b}+\frac {d x^{n (3-p)} \left (a+b x^n\right )^{p+1}}{4 b n}\)

\(\Big \downarrow \) 882

\(\displaystyle \frac {a^3 x^{-n p} (4 b c-a d (3-p)) \left (\frac {x^n}{a+b x^n}\right )^p \left (a+b x^n\right )^p \int \frac {\left (\frac {x^n}{b x^n+a}\right )^{2-p}}{\left (1-\frac {b x^n}{b x^n+a}\right )^4}d\frac {x^n}{b x^n+a}}{4 b n}+\frac {d x^{n (3-p)} \left (a+b x^n\right )^{p+1}}{4 b n}\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {a^3 x^{3 n-n p} (4 b c-a d (3-p)) \left (a+b x^n\right )^{p-3} \operatorname {Hypergeometric2F1}\left (4,3-p,4-p,\frac {b x^n}{b x^n+a}\right )}{4 b n (3-p)}+\frac {d x^{n (3-p)} \left (a+b x^n\right )^{p+1}}{4 b n}\)

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

integrate(x^(-1-n*(-3+p))*(a+b*x^n)^p*(c+d*x^n),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Giac [F(-2)]

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

integrate(x^(-1-n*(-3+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%%%{4,[0,0,2,2,1,1,1,0,1]%%%}+%%%{4,[0,0,2,2,1,1,0,0,1]%%%}+%% 
%{1,[0,0,
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^{-1-n (-3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {6 x^{4 n} \left (x^{n} b +a \right )^{p} b^{3} d +2 x^{3 n} \left (x^{n} b +a \right )^{p} a \,b^{2} d p +8 x^{3 n} \left (x^{n} b +a \right )^{p} b^{3} c +x^{2 n} \left (x^{n} b +a \right )^{p} a^{2} b d \,p^{2}-3 x^{2 n} \left (x^{n} b +a \right )^{p} a^{2} b d p +4 x^{2 n} \left (x^{n} b +a \right )^{p} a \,b^{2} c p +x^{n} \left (x^{n} b +a \right )^{p} a^{3} d \,p^{3}-5 x^{n} \left (x^{n} b +a \right )^{p} a^{3} d \,p^{2}+6 x^{n} \left (x^{n} b +a \right )^{p} a^{3} d p +4 x^{n} \left (x^{n} b +a \right )^{p} a^{2} b c \,p^{2}-8 x^{n} \left (x^{n} b +a \right )^{p} a^{2} b c p +x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{4} d n \,p^{4}-6 x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{4} d n \,p^{3}+11 x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{4} d n \,p^{2}-6 x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{4} d n p +4 x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{3} b c n \,p^{3}-12 x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{3} b c n \,p^{2}+8 x^{n p} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{3} b c n p}{24 x^{n p} b^{3} n} \] Input:

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

Output:

(6*x**(4*n)*(x**n*b + a)**p*b**3*d + 2*x**(3*n)*(x**n*b + a)**p*a*b**2*d*p 
 + 8*x**(3*n)*(x**n*b + a)**p*b**3*c + x**(2*n)*(x**n*b + a)**p*a**2*b*d*p 
**2 - 3*x**(2*n)*(x**n*b + a)**p*a**2*b*d*p + 4*x**(2*n)*(x**n*b + a)**p*a 
*b**2*c*p + x**n*(x**n*b + a)**p*a**3*d*p**3 - 5*x**n*(x**n*b + a)**p*a**3 
*d*p**2 + 6*x**n*(x**n*b + a)**p*a**3*d*p + 4*x**n*(x**n*b + a)**p*a**2*b* 
c*p**2 - 8*x**n*(x**n*b + a)**p*a**2*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**4*d*n*p**4 - 6*x**(n*p)*int 
((x**n*(x**n*b + a)**p)/(x**(n*p + n)*b*x + x**(n*p)*a*x),x)*a**4*d*n*p**3 
 + 11*x**(n*p)*int((x**n*(x**n*b + a)**p)/(x**(n*p + n)*b*x + x**(n*p)*a*x 
),x)*a**4*d*n*p**2 - 6*x**(n*p)*int((x**n*(x**n*b + a)**p)/(x**(n*p + n)*b 
*x + x**(n*p)*a*x),x)*a**4*d*n*p + 4*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*b*c*n*p**3 - 12*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*b*c*n*p**2 + 
8*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*b*c*n*p)/(24*x**(n*p)*b**3*n)