\(\int x^{-1-n (2+p)} (a+b x^n)^p (c+d x^n)^3 \, dx\) [519]

Optimal result

Integrand size = 29, antiderivative size = 186 \[ \int x^{-1-n (2+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=\frac {d^3 x^{-n p} \left (a+b x^n\right )^{1+p}}{b n}+\frac {c^2 (b c-3 a d (2+p)) x^{-n (1+p)} \left (a+b x^n\right )^{1+p}}{a^2 n (1+p) (2+p)}-\frac {c^3 x^{-n (2+p)} \left (a+b x^n\right )^{1+p}}{a n (2+p)}-\frac {d^2 (3 b c+a d p) x^{-n p} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^n}{a}\right )}{b n p} \] Output:

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

Mathematica [A] (verified)

Time = 5.27 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.04 \[ \int x^{-1-n (2+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=\frac {x^{-n (2+p)} \left (a+b x^n\right )^p \left (-\frac {3 c^2 d x^n \left (a+b x^n\right )}{a (1+p)}-\frac {c^3 \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2-p,-p,-1-p,-\frac {b x^n}{a}\right )}{2+p}-\frac {d^3 x^{3 n} \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^n}{a}\right )}{-1+p}-\frac {3 c d^2 x^{2 n} \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^n}{a}\right )}{p}\right )}{n} \] Input:

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

Output:

((a + b*x^n)^p*((-3*c^2*d*x^n*(a + b*x^n))/(a*(1 + p)) - (c^3*Hypergeometr 
ic2F1[-2 - p, -p, -1 - p, -((b*x^n)/a)])/((2 + p)*(1 + (b*x^n)/a)^p) - (d^ 
3*x^(3*n)*Hypergeometric2F1[1 - p, -p, 2 - p, -((b*x^n)/a)])/((-1 + p)*(1 
+ (b*x^n)/a)^p) - (3*c*d^2*x^(2*n)*Hypergeometric2F1[-p, -p, 1 - p, -((b*x 
^n)/a)])/(p*(1 + (b*x^n)/a)^p)))/(n*x^(n*(2 + p)))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1008, 1066}

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.

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

\(\Big \downarrow \) 1008

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

\(\Big \downarrow \) 1066

\(\displaystyle \text {Indeterminate}\)

Input:

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

Output:

Indeterminate
 

Defintions of rubi rules used

rule 1008

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n) 
^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Simp[1/(b*(m + n*(p + q) + 1)) 
Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + 
 c*b*n*(p + q)) + (d*(c*b - a*d)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d* 
n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c 
 - a*d, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
 

rule 1066

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*(g*x)^(m + 1)*(a + 
b*x^n)^(p + 1)*((c + d*x^n)^q/(b*g*(m + n*(p + q + 1) + 1))), x] + Simp[1/( 
b*(m + n*(p + q + 1) + 1))   Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)* 
Simp[c*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + 
 f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c 
, d, e, f, g, m, n, p}, x] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[e + f*x 
^n, c + d*x^n])
 

Maple [F]

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

Input:

int(x^(-1-n*(2+p))*(a+b*x^n)^p*(c+d*x^n)^3,x)
 

Output:

int(x^(-1-n*(2+p))*(a+b*x^n)^p*(c+d*x^n)^3,x)
 

Fricas [F]

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

integrate(x^(-1-n*(2+p))*(a+b*x^n)^p*(c+d*x^n)^3,x, algorithm="fricas")
 

Output:

integral((d^3*x^(-n*p - 2*n - 1)*x^(3*n) + 3*c*d^2*x^(-n*p - 2*n - 1)*x^(2 
*n) + 3*c^2*d*x^(-n*p - 2*n - 1)*x^n + c^3*x^(-n*p - 2*n - 1))*(b*x^n + a) 
^p, x)
 

Sympy [F(-1)]

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

integrate(x**(-1-n*(2+p))*(a+b*x**n)**p*(c+d*x**n)**3,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^(-1-n*(2+p))*(a+b*x^n)^p*(c+d*x^n)^3,x, algorithm="maxima")
 

Output:

integrate((d*x^n + c)^3*(b*x^n + a)^p*x^(-n*(p + 2) - 1), x)
 

Giac [F(-2)]

Exception generated. \[ \int x^{-1-n (2+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^(-1-n*(2+p))*(a+b*x^n)^p*(c+d*x^n)^3,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%%%{1,[2,0,6,4,2,4,4,3,0]%%%}+%%%{4,[2,0,6,4,2,4,3,3,0]%%%}+%% 
%{6,[2,0,
 

Mupad [F(-1)]

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

int(((a + b*x^n)^p*(c + d*x^n)^3)/x^(n*(p + 2) + 1),x)
 

Output:

int(((a + b*x^n)^p*(c + d*x^n)^3)/x^(n*(p + 2) + 1), x)
 

Reduce [F]

\[ \int x^{-1-n (2+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=\frac {x^{3 n} \left (x^{n} b +a \right )^{p} a^{2} d^{3} p^{3}+3 x^{3 n} \left (x^{n} b +a \right )^{p} a^{2} d^{3} p^{2}+2 x^{3 n} \left (x^{n} b +a \right )^{p} a^{2} d^{3} p -3 x^{2 n} \left (x^{n} b +a \right )^{p} a^{2} c \,d^{2} p^{2}-9 x^{2 n} \left (x^{n} b +a \right )^{p} a^{2} c \,d^{2} p -6 x^{2 n} \left (x^{n} b +a \right )^{p} a^{2} c \,d^{2}-3 x^{2 n} \left (x^{n} b +a \right )^{p} a b \,c^{2} d \,p^{2}-6 x^{2 n} \left (x^{n} b +a \right )^{p} a b \,c^{2} d p +x^{2 n} \left (x^{n} b +a \right )^{p} b^{2} c^{3} p -3 x^{n} \left (x^{n} b +a \right )^{p} a^{2} c^{2} d \,p^{2}-6 x^{n} \left (x^{n} b +a \right )^{p} a^{2} c^{2} d p -x^{n} \left (x^{n} b +a \right )^{p} a b \,c^{3} p^{2}-\left (x^{n} b +a \right )^{p} a^{2} c^{3} p^{2}-\left (x^{n} b +a \right )^{p} a^{2} c^{3} p +x^{n p +2 n} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{3} d^{3} n \,p^{4}+3 x^{n p +2 n} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{3} d^{3} n \,p^{3}+2 x^{n p +2 n} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{3} d^{3} n \,p^{2}+3 x^{n p +2 n} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{2} b c \,d^{2} n \,p^{3}+9 x^{n p +2 n} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{2} b c \,d^{2} n \,p^{2}+6 x^{n p +2 n} \left (\int \frac {x^{n} \left (x^{n} b +a \right )^{p}}{x^{n p +n} b x +x^{n p} a x}d x \right ) a^{2} b c \,d^{2} n p}{x^{n p +2 n} a^{2} n p \left (p^{2}+3 p +2\right )} \] Input:

int(x^(-1-n*(2+p))*(a+b*x^n)^p*(c+d*x^n)^3,x)
 

Output:

(x**(3*n)*(x**n*b + a)**p*a**2*d**3*p**3 + 3*x**(3*n)*(x**n*b + a)**p*a**2 
*d**3*p**2 + 2*x**(3*n)*(x**n*b + a)**p*a**2*d**3*p - 3*x**(2*n)*(x**n*b + 
 a)**p*a**2*c*d**2*p**2 - 9*x**(2*n)*(x**n*b + a)**p*a**2*c*d**2*p - 6*x** 
(2*n)*(x**n*b + a)**p*a**2*c*d**2 - 3*x**(2*n)*(x**n*b + a)**p*a*b*c**2*d* 
p**2 - 6*x**(2*n)*(x**n*b + a)**p*a*b*c**2*d*p + x**(2*n)*(x**n*b + a)**p* 
b**2*c**3*p - 3*x**n*(x**n*b + a)**p*a**2*c**2*d*p**2 - 6*x**n*(x**n*b + a 
)**p*a**2*c**2*d*p - x**n*(x**n*b + a)**p*a*b*c**3*p**2 - (x**n*b + a)**p* 
a**2*c**3*p**2 - (x**n*b + a)**p*a**2*c**3*p + x**(n*p + 2*n)*int((x**n*(x 
**n*b + a)**p)/(x**(n*p + n)*b*x + x**(n*p)*a*x),x)*a**3*d**3*n*p**4 + 3*x 
**(n*p + 2*n)*int((x**n*(x**n*b + a)**p)/(x**(n*p + n)*b*x + x**(n*p)*a*x) 
,x)*a**3*d**3*n*p**3 + 2*x**(n*p + 2*n)*int((x**n*(x**n*b + a)**p)/(x**(n* 
p + n)*b*x + x**(n*p)*a*x),x)*a**3*d**3*n*p**2 + 3*x**(n*p + 2*n)*int((x** 
n*(x**n*b + a)**p)/(x**(n*p + n)*b*x + x**(n*p)*a*x),x)*a**2*b*c*d**2*n*p* 
*3 + 9*x**(n*p + 2*n)*int((x**n*(x**n*b + a)**p)/(x**(n*p + n)*b*x + x**(n 
*p)*a*x),x)*a**2*b*c*d**2*n*p**2 + 6*x**(n*p + 2*n)*int((x**n*(x**n*b + a) 
**p)/(x**(n*p + n)*b*x + x**(n*p)*a*x),x)*a**2*b*c*d**2*n*p)/(x**(n*p + 2* 
n)*a**2*n*p*(p**2 + 3*p + 2))