\(\int (c x)^{-1-3 n-n p} (a+b x^n)^p \, dx\) [698]

Optimal result

Integrand size = 23, antiderivative size = 139 \[ \int (c x)^{-1-3 n-n p} \left (a+b x^n\right )^p \, dx=-\frac {(c x)^{-n (3+p)} \left (a+b x^n\right )^{1+p}}{a c n (3+p)}+\frac {2 b x^n (c x)^{-n (3+p)} \left (a+b x^n\right )^{1+p}}{a^2 c n (2+p) (3+p)}-\frac {2 b^2 x^{2 n} (c x)^{-n (3+p)} \left (a+b x^n\right )^{1+p}}{a^3 c n (1+p) (2+p) (3+p)} \] Output:

-(a+b*x^n)^(p+1)/a/c/n/(3+p)/((c*x)^(n*(3+p)))+2*b*x^n*(a+b*x^n)^(p+1)/a^2 
/c/n/(2+p)/(3+p)/((c*x)^(n*(3+p)))-2*b^2*x^(2*n)*(a+b*x^n)^(p+1)/a^3/c/n/( 
p+1)/(2+p)/(3+p)/((c*x)^(n*(3+p)))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.50 \[ \int (c x)^{-1-3 n-n p} \left (a+b x^n\right )^p \, dx=-\frac {x (c x)^{-1-n (3+p)} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-3-p,-p,-2-p,-\frac {b x^n}{a}\right )}{n (3+p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {805, 805, 796}

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.

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

\(\Big \downarrow \) 805

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

\(\Big \downarrow \) 805

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

\(\Big \downarrow \) 796

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

Input:

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

Output:

-((a + b*x^n)^(1 + p)/(a*c*n*(1 + p)*(c*x)^(n*(3 + p)))) - (2*(-((a + b*x^ 
n)^(2 + p)/(a*c*n*(2 + p)*(c*x)^(n*(3 + p)))) + (a + b*x^n)^(3 + p)/(a^2*c 
*n*(2 + p)*(3 + p)*(c*x)^(n*(3 + p)))))/(a*(1 + p))
 

Defintions of rubi rules used

rule 796

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

rule 805

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

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.55 \[ \int (c x)^{-1-3 n-n p} \left (a+b x^n\right )^p \, dx=\frac {{\left (2 \, a b^{2} p x x^{2 \, n} e^{\left (-{\left (n p + 3 \, n + 1\right )} \log \left (c\right ) - {\left (n p + 3 \, n + 1\right )} \log \left (x\right )\right )} - 2 \, b^{3} x x^{3 \, n} e^{\left (-{\left (n p + 3 \, n + 1\right )} \log \left (c\right ) - {\left (n p + 3 \, n + 1\right )} \log \left (x\right )\right )} - {\left (a^{2} b p^{2} + a^{2} b p\right )} x x^{n} e^{\left (-{\left (n p + 3 \, n + 1\right )} \log \left (c\right ) - {\left (n p + 3 \, n + 1\right )} \log \left (x\right )\right )} - {\left (a^{3} p^{2} + 3 \, a^{3} p + 2 \, a^{3}\right )} x e^{\left (-{\left (n p + 3 \, n + 1\right )} \log \left (c\right ) - {\left (n p + 3 \, n + 1\right )} \log \left (x\right )\right )}\right )} {\left (b x^{n} + a\right )}^{p}}{a^{3} n p^{3} + 6 \, a^{3} n p^{2} + 11 \, a^{3} n p + 6 \, a^{3} n} \] Input:

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

Output:

(2*a*b^2*p*x*x^(2*n)*e^(-(n*p + 3*n + 1)*log(c) - (n*p + 3*n + 1)*log(x)) 
- 2*b^3*x*x^(3*n)*e^(-(n*p + 3*n + 1)*log(c) - (n*p + 3*n + 1)*log(x)) - ( 
a^2*b*p^2 + a^2*b*p)*x*x^n*e^(-(n*p + 3*n + 1)*log(c) - (n*p + 3*n + 1)*lo 
g(x)) - (a^3*p^2 + 3*a^3*p + 2*a^3)*x*e^(-(n*p + 3*n + 1)*log(c) - (n*p + 
3*n + 1)*log(x)))*(b*x^n + a)^p/(a^3*n*p^3 + 6*a^3*n*p^2 + 11*a^3*n*p + 6* 
a^3*n)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (110) = 220\).

Time = 6.25 (sec) , antiderivative size = 566, normalized size of antiderivative = 4.07 \[ \int (c x)^{-1-3 n-n p} \left (a+b x^n\right )^p \, dx=\frac {a^{2} a^{p} a^{- p - 3} b^{p + 3} c^{- n p - 3 n - 1} p^{2} \left (\frac {a x^{- n}}{b} + 1\right )^{p + 3} \Gamma \left (- p - 3\right )}{a^{2} n \Gamma \left (- p\right ) + 2 a b n x^{n} \Gamma \left (- p\right ) + b^{2} n x^{2 n} \Gamma \left (- p\right )} + \frac {3 a^{2} a^{p} a^{- p - 3} b^{p + 3} c^{- n p - 3 n - 1} p \left (\frac {a x^{- n}}{b} + 1\right )^{p + 3} \Gamma \left (- p - 3\right )}{a^{2} n \Gamma \left (- p\right ) + 2 a b n x^{n} \Gamma \left (- p\right ) + b^{2} n x^{2 n} \Gamma \left (- p\right )} + \frac {2 a^{2} a^{p} a^{- p - 3} b^{p + 3} c^{- n p - 3 n - 1} \left (\frac {a x^{- n}}{b} + 1\right )^{p + 3} \Gamma \left (- p - 3\right )}{a^{2} n \Gamma \left (- p\right ) + 2 a b n x^{n} \Gamma \left (- p\right ) + b^{2} n x^{2 n} \Gamma \left (- p\right )} - \frac {2 a a^{p} a^{- p - 3} b b^{p + 3} c^{- n p - 3 n - 1} p x^{n} \left (\frac {a x^{- n}}{b} + 1\right )^{p + 3} \Gamma \left (- p - 3\right )}{a^{2} n \Gamma \left (- p\right ) + 2 a b n x^{n} \Gamma \left (- p\right ) + b^{2} n x^{2 n} \Gamma \left (- p\right )} - \frac {2 a a^{p} a^{- p - 3} b b^{p + 3} c^{- n p - 3 n - 1} x^{n} \left (\frac {a x^{- n}}{b} + 1\right )^{p + 3} \Gamma \left (- p - 3\right )}{a^{2} n \Gamma \left (- p\right ) + 2 a b n x^{n} \Gamma \left (- p\right ) + b^{2} n x^{2 n} \Gamma \left (- p\right )} + \frac {2 a^{p} a^{- p - 3} b^{2} b^{p + 3} c^{- n p - 3 n - 1} x^{2 n} \left (\frac {a x^{- n}}{b} + 1\right )^{p + 3} \Gamma \left (- p - 3\right )}{a^{2} n \Gamma \left (- p\right ) + 2 a b n x^{n} \Gamma \left (- p\right ) + b^{2} n x^{2 n} \Gamma \left (- p\right )} \] Input:

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

Output:

a**2*a**p*a**(-p - 3)*b**(p + 3)*c**(-n*p - 3*n - 1)*p**2*(a/(b*x**n) + 1) 
**(p + 3)*gamma(-p - 3)/(a**2*n*gamma(-p) + 2*a*b*n*x**n*gamma(-p) + b**2* 
n*x**(2*n)*gamma(-p)) + 3*a**2*a**p*a**(-p - 3)*b**(p + 3)*c**(-n*p - 3*n 
- 1)*p*(a/(b*x**n) + 1)**(p + 3)*gamma(-p - 3)/(a**2*n*gamma(-p) + 2*a*b*n 
*x**n*gamma(-p) + b**2*n*x**(2*n)*gamma(-p)) + 2*a**2*a**p*a**(-p - 3)*b** 
(p + 3)*c**(-n*p - 3*n - 1)*(a/(b*x**n) + 1)**(p + 3)*gamma(-p - 3)/(a**2* 
n*gamma(-p) + 2*a*b*n*x**n*gamma(-p) + b**2*n*x**(2*n)*gamma(-p)) - 2*a*a* 
*p*a**(-p - 3)*b*b**(p + 3)*c**(-n*p - 3*n - 1)*p*x**n*(a/(b*x**n) + 1)**( 
p + 3)*gamma(-p - 3)/(a**2*n*gamma(-p) + 2*a*b*n*x**n*gamma(-p) + b**2*n*x 
**(2*n)*gamma(-p)) - 2*a*a**p*a**(-p - 3)*b*b**(p + 3)*c**(-n*p - 3*n - 1) 
*x**n*(a/(b*x**n) + 1)**(p + 3)*gamma(-p - 3)/(a**2*n*gamma(-p) + 2*a*b*n* 
x**n*gamma(-p) + b**2*n*x**(2*n)*gamma(-p)) + 2*a**p*a**(-p - 3)*b**2*b**( 
p + 3)*c**(-n*p - 3*n - 1)*x**(2*n)*(a/(b*x**n) + 1)**(p + 3)*gamma(-p - 3 
)/(a**2*n*gamma(-p) + 2*a*b*n*x**n*gamma(-p) + b**2*n*x**(2*n)*gamma(-p))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.86 \[ \int (c x)^{-1-3 n-n p} \left (a+b x^n\right )^p \, dx=\frac {\left (x^{n} b +a \right )^{p} \left (-2 x^{3 n} b^{3}+2 x^{2 n} a \,b^{2} p -x^{n} a^{2} b \,p^{2}-x^{n} a^{2} b p -a^{3} p^{2}-3 a^{3} p -2 a^{3}\right )}{x^{n p +3 n} c^{n p +3 n} a^{3} c n \left (p^{3}+6 p^{2}+11 p +6\right )} \] Input:

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

Output:

((x**n*b + a)**p*( - 2*x**(3*n)*b**3 + 2*x**(2*n)*a*b**2*p - x**n*a**2*b*p 
**2 - x**n*a**2*b*p - a**3*p**2 - 3*a**3*p - 2*a**3))/(x**(n*p + 3*n)*c**( 
n*p + 3*n)*a**3*c*n*(p**3 + 6*p**2 + 11*p + 6))