\(\int x^{-1-n (3+p)} (a+b x^n)^p (c+d x^n)^3 \, dx\) [513]

Optimal result

Integrand size = 29, antiderivative size = 225 \[ \int x^{-1-n (3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=-\frac {c \left (2 b^2 c^2-3 a b c d (3+p)+3 a^2 d^2 \left (6+5 p+p^2\right )\right ) x^{-n (1+p)} \left (a+b x^n\right )^{1+p}}{a^3 n (1+p) (2+p) (3+p)}+\frac {c^2 (2 b c-3 a d (3+p)) x^{-n (2+p)} \left (a+b x^n\right )^{1+p}}{a^2 n (2+p) (3+p)}-\frac {c^3 x^{-n (3+p)} \left (a+b x^n\right )^{1+p}}{a n (3+p)}-\frac {d^3 x^{-n p} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^n}{a}\right )}{n p} \] Output:

-c*(2*b^2*c^2-3*a*b*c*d*(3+p)+3*a^2*d^2*(p^2+5*p+6))*(a+b*x^n)^(p+1)/a^3/n 
/(p+1)/(2+p)/(3+p)/(x^(n*(p+1)))+c^2*(2*b*c-3*a*d*(3+p))*(a+b*x^n)^(p+1)/a 
^2/n/(2+p)/(3+p)/(x^(n*(2+p)))-c^3*(a+b*x^n)^(p+1)/a/n/(3+p)/(x^(n*(3+p))) 
-d^3*(a+b*x^n)^p*hypergeom([-p, -p],[1-p],-b*x^n/a)/n/p/(x^(n*p))/((1+b*x^ 
n/a)^p)
 

Mathematica [A] (verified)

Time = 5.38 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.84 \[ \int x^{-1-n (3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=\frac {x^{-n (3+p)} \left (a+b x^n\right )^p \left (-\frac {c^3 \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-3-p,-p,-2-p,-\frac {b x^n}{a}\right )}{3+p}+d x^n \left (-\frac {3 c^2 \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2-p,-p,-1-p,-\frac {b x^n}{a}\right )}{2+p}+d x^n \left (-\frac {3 c \left (a+b x^n\right )}{a (1+p)}-\frac {d x^n \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^n}{a}\right )}{p}\right )\right )\right )}{n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {1008}

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.

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

\(\Big \downarrow \) 1008

\(\displaystyle \text {Indeterminate}\)

Input:

Int[x^(-1 - n*(3 + 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int x^{-1-n (3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=\frac {-3 x^{3 n} \left (x^{n} b +a \right )^{p} a^{2} b c \,d^{2} p^{2}-15 x^{3 n} \left (x^{n} b +a \right )^{p} a^{2} b c \,d^{2} p -18 x^{3 n} \left (x^{n} b +a \right )^{p} a^{2} b c \,d^{2}+3 x^{3 n} \left (x^{n} b +a \right )^{p} a \,b^{2} c^{2} d p +9 x^{3 n} \left (x^{n} b +a \right )^{p} a \,b^{2} c^{2} d -2 x^{3 n} \left (x^{n} b +a \right )^{p} b^{3} c^{3}-3 x^{2 n} \left (x^{n} b +a \right )^{p} a^{3} c \,d^{2} p^{2}-15 x^{2 n} \left (x^{n} b +a \right )^{p} a^{3} c \,d^{2} p -18 x^{2 n} \left (x^{n} b +a \right )^{p} a^{3} c \,d^{2}-3 x^{2 n} \left (x^{n} b +a \right )^{p} a^{2} b \,c^{2} d \,p^{2}-9 x^{2 n} \left (x^{n} b +a \right )^{p} a^{2} b \,c^{2} d p +2 x^{2 n} \left (x^{n} b +a \right )^{p} a \,b^{2} c^{3} p -3 x^{n} \left (x^{n} b +a \right )^{p} a^{3} c^{2} d \,p^{2}-12 x^{n} \left (x^{n} b +a \right )^{p} a^{3} c^{2} d p -9 x^{n} \left (x^{n} b +a \right )^{p} a^{3} c^{2} d -x^{n} \left (x^{n} b +a \right )^{p} a^{2} b \,c^{3} p^{2}-x^{n} \left (x^{n} b +a \right )^{p} a^{2} b \,c^{3} p -\left (x^{n} b +a \right )^{p} a^{3} c^{3} p^{2}-3 \left (x^{n} b +a \right )^{p} a^{3} c^{3} p -2 \left (x^{n} b +a \right )^{p} a^{3} c^{3}+x^{n p +3 n} \left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n p} x}d x \right ) a^{3} d^{3} n \,p^{3}+6 x^{n p +3 n} \left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n p} x}d x \right ) a^{3} d^{3} n \,p^{2}+11 x^{n p +3 n} \left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n p} x}d x \right ) a^{3} d^{3} n p +6 x^{n p +3 n} \left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{n p} x}d x \right ) a^{3} d^{3} n}{x^{n p +3 n} a^{3} n \left (p^{3}+6 p^{2}+11 p +6\right )} \] Input:

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

Output:

( - 3*x**(3*n)*(x**n*b + a)**p*a**2*b*c*d**2*p**2 - 15*x**(3*n)*(x**n*b + 
a)**p*a**2*b*c*d**2*p - 18*x**(3*n)*(x**n*b + a)**p*a**2*b*c*d**2 + 3*x**( 
3*n)*(x**n*b + a)**p*a*b**2*c**2*d*p + 9*x**(3*n)*(x**n*b + a)**p*a*b**2*c 
**2*d - 2*x**(3*n)*(x**n*b + a)**p*b**3*c**3 - 3*x**(2*n)*(x**n*b + a)**p* 
a**3*c*d**2*p**2 - 15*x**(2*n)*(x**n*b + a)**p*a**3*c*d**2*p - 18*x**(2*n) 
*(x**n*b + a)**p*a**3*c*d**2 - 3*x**(2*n)*(x**n*b + a)**p*a**2*b*c**2*d*p* 
*2 - 9*x**(2*n)*(x**n*b + a)**p*a**2*b*c**2*d*p + 2*x**(2*n)*(x**n*b + a)* 
*p*a*b**2*c**3*p - 3*x**n*(x**n*b + a)**p*a**3*c**2*d*p**2 - 12*x**n*(x**n 
*b + a)**p*a**3*c**2*d*p - 9*x**n*(x**n*b + a)**p*a**3*c**2*d - x**n*(x**n 
*b + a)**p*a**2*b*c**3*p**2 - x**n*(x**n*b + a)**p*a**2*b*c**3*p - (x**n*b 
 + a)**p*a**3*c**3*p**2 - 3*(x**n*b + a)**p*a**3*c**3*p - 2*(x**n*b + a)** 
p*a**3*c**3 + x**(n*p + 3*n)*int((x**n*b + a)**p/(x**(n*p)*x),x)*a**3*d**3 
*n*p**3 + 6*x**(n*p + 3*n)*int((x**n*b + a)**p/(x**(n*p)*x),x)*a**3*d**3*n 
*p**2 + 11*x**(n*p + 3*n)*int((x**n*b + a)**p/(x**(n*p)*x),x)*a**3*d**3*n* 
p + 6*x**(n*p + 3*n)*int((x**n*b + a)**p/(x**(n*p)*x),x)*a**3*d**3*n)/(x** 
(n*p + 3*n)*a**3*n*(p**3 + 6*p**2 + 11*p + 6))