\(\int (c x)^{-1-4 n-n p} (a+b x^n)^p \, dx\) [699]

Optimal result

Integrand size = 23, antiderivative size = 199 \[ \int (c x)^{-1-4 n-n p} \left (a+b x^n\right )^p \, dx=-\frac {(c x)^{-n (4+p)} \left (a+b x^n\right )^{1+p}}{a c n (4+p)}+\frac {3 b x^n (c x)^{-n (4+p)} \left (a+b x^n\right )^{1+p}}{a^2 c n (3+p) (4+p)}-\frac {6 b^2 x^{2 n} (c x)^{-n (4+p)} \left (a+b x^n\right )^{1+p}}{a^3 c n (2+p) (3+p) (4+p)}+\frac {6 b^3 x^{3 n} (c x)^{-n (4+p)} \left (a+b x^n\right )^{1+p}}{a^4 c n (1+p) (2+p) (3+p) (4+p)} \] Output:

-(a+b*x^n)^(p+1)/a/c/n/(4+p)/((c*x)^(n*(4+p)))+3*b*x^n*(a+b*x^n)^(p+1)/a^2 
/c/n/(3+p)/(4+p)/((c*x)^(n*(4+p)))-6*b^2*x^(2*n)*(a+b*x^n)^(p+1)/a^3/c/n/( 
2+p)/(3+p)/(4+p)/((c*x)^(n*(4+p)))+6*b^3*x^(3*n)*(a+b*x^n)^(p+1)/a^4/c/n/( 
p+1)/(2+p)/(3+p)/(4+p)/((c*x)^(n*(4+p)))
 

Mathematica [C] (verified)

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

Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.35 \[ \int (c x)^{-1-4 n-n p} \left (a+b x^n\right )^p \, dx=-\frac {x (c x)^{-1-n (4+p)} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-4-p,-p,-3-p,-\frac {b x^n}{a}\right )}{n (4+p)} \] Input:

Integrate[(c*x)^(-1 - 4*n - n*p)*(a + b*x^n)^p,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {805, 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.

\(\displaystyle \int (c x)^{n (-p)-4 n-1} \left (a+b x^n\right )^p \, dx\)

\(\Big \downarrow \) 805

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

\(\Big \downarrow \) 805

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

\(\Big \downarrow \) 805

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

\(\Big \downarrow \) 796

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

Input:

Int[(c*x)^(-1 - 4*n - n*p)*(a + b*x^n)^p,x]
 

Output:

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

\[\int \left (c x \right )^{-n p -4 n -1} \left (a +b \,x^{n}\right )^{p}d x\]

Input:

int((c*x)^(-n*p-4*n-1)*(a+b*x^n)^p,x)
 

Output:

int((c*x)^(-n*p-4*n-1)*(a+b*x^n)^p,x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.48 \[ \int (c x)^{-1-4 n-n p} \left (a+b x^n\right )^p \, dx=-\frac {{\left (6 \, a b^{3} p x x^{3 \, n} e^{\left (-{\left (n p + 4 \, n + 1\right )} \log \left (c\right ) - {\left (n p + 4 \, n + 1\right )} \log \left (x\right )\right )} - 6 \, b^{4} x x^{4 \, n} e^{\left (-{\left (n p + 4 \, n + 1\right )} \log \left (c\right ) - {\left (n p + 4 \, n + 1\right )} \log \left (x\right )\right )} - 3 \, {\left (a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} x x^{2 \, n} e^{\left (-{\left (n p + 4 \, n + 1\right )} \log \left (c\right ) - {\left (n p + 4 \, n + 1\right )} \log \left (x\right )\right )} + {\left (a^{3} b p^{3} + 3 \, a^{3} b p^{2} + 2 \, a^{3} b p\right )} x x^{n} e^{\left (-{\left (n p + 4 \, n + 1\right )} \log \left (c\right ) - {\left (n p + 4 \, n + 1\right )} \log \left (x\right )\right )} + {\left (a^{4} p^{3} + 6 \, a^{4} p^{2} + 11 \, a^{4} p + 6 \, a^{4}\right )} x e^{\left (-{\left (n p + 4 \, n + 1\right )} \log \left (c\right ) - {\left (n p + 4 \, n + 1\right )} \log \left (x\right )\right )}\right )} {\left (b x^{n} + a\right )}^{p}}{a^{4} n p^{4} + 10 \, a^{4} n p^{3} + 35 \, a^{4} n p^{2} + 50 \, a^{4} n p + 24 \, a^{4} n} \] Input:

integrate((c*x)^(-n*p-4*n-1)*(a+b*x^n)^p,x, algorithm="fricas")
 

Output:

-(6*a*b^3*p*x*x^(3*n)*e^(-(n*p + 4*n + 1)*log(c) - (n*p + 4*n + 1)*log(x)) 
 - 6*b^4*x*x^(4*n)*e^(-(n*p + 4*n + 1)*log(c) - (n*p + 4*n + 1)*log(x)) - 
3*(a^2*b^2*p^2 + a^2*b^2*p)*x*x^(2*n)*e^(-(n*p + 4*n + 1)*log(c) - (n*p + 
4*n + 1)*log(x)) + (a^3*b*p^3 + 3*a^3*b*p^2 + 2*a^3*b*p)*x*x^n*e^(-(n*p + 
4*n + 1)*log(c) - (n*p + 4*n + 1)*log(x)) + (a^4*p^3 + 6*a^4*p^2 + 11*a^4* 
p + 6*a^4)*x*e^(-(n*p + 4*n + 1)*log(c) - (n*p + 4*n + 1)*log(x)))*(b*x^n 
+ a)^p/(a^4*n*p^4 + 10*a^4*n*p^3 + 35*a^4*n*p^2 + 50*a^4*n*p + 24*a^4*n)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1168 vs. \(2 (160) = 320\).

Time = 6.71 (sec) , antiderivative size = 1168, normalized size of antiderivative = 5.87 \[ \int (c x)^{-1-4 n-n p} \left (a+b x^n\right )^p \, dx=\text {Too large to display} \] Input:

integrate((c*x)**(-n*p-4*n-1)*(a+b*x**n)**p,x)
 

Output:

-a**3*a**p*a**(-p - 4)*b**(p + 4)*c**(-n*p - 4*n - 1)*p**3*(a/(b*x**n) + 1 
)**(p + 4)*gamma(-p - 4)/(a**3*n*gamma(-p) + 3*a**2*b*n*x**n*gamma(-p) + 3 
*a*b**2*n*x**(2*n)*gamma(-p) + b**3*n*x**(3*n)*gamma(-p)) - 6*a**3*a**p*a* 
*(-p - 4)*b**(p + 4)*c**(-n*p - 4*n - 1)*p**2*(a/(b*x**n) + 1)**(p + 4)*ga 
mma(-p - 4)/(a**3*n*gamma(-p) + 3*a**2*b*n*x**n*gamma(-p) + 3*a*b**2*n*x** 
(2*n)*gamma(-p) + b**3*n*x**(3*n)*gamma(-p)) - 11*a**3*a**p*a**(-p - 4)*b* 
*(p + 4)*c**(-n*p - 4*n - 1)*p*(a/(b*x**n) + 1)**(p + 4)*gamma(-p - 4)/(a* 
*3*n*gamma(-p) + 3*a**2*b*n*x**n*gamma(-p) + 3*a*b**2*n*x**(2*n)*gamma(-p) 
 + b**3*n*x**(3*n)*gamma(-p)) - 6*a**3*a**p*a**(-p - 4)*b**(p + 4)*c**(-n* 
p - 4*n - 1)*(a/(b*x**n) + 1)**(p + 4)*gamma(-p - 4)/(a**3*n*gamma(-p) + 3 
*a**2*b*n*x**n*gamma(-p) + 3*a*b**2*n*x**(2*n)*gamma(-p) + b**3*n*x**(3*n) 
*gamma(-p)) + 3*a**2*a**p*a**(-p - 4)*b*b**(p + 4)*c**(-n*p - 4*n - 1)*p** 
2*x**n*(a/(b*x**n) + 1)**(p + 4)*gamma(-p - 4)/(a**3*n*gamma(-p) + 3*a**2* 
b*n*x**n*gamma(-p) + 3*a*b**2*n*x**(2*n)*gamma(-p) + b**3*n*x**(3*n)*gamma 
(-p)) + 9*a**2*a**p*a**(-p - 4)*b*b**(p + 4)*c**(-n*p - 4*n - 1)*p*x**n*(a 
/(b*x**n) + 1)**(p + 4)*gamma(-p - 4)/(a**3*n*gamma(-p) + 3*a**2*b*n*x**n* 
gamma(-p) + 3*a*b**2*n*x**(2*n)*gamma(-p) + b**3*n*x**(3*n)*gamma(-p)) + 6 
*a**2*a**p*a**(-p - 4)*b*b**(p + 4)*c**(-n*p - 4*n - 1)*x**n*(a/(b*x**n) + 
 1)**(p + 4)*gamma(-p - 4)/(a**3*n*gamma(-p) + 3*a**2*b*n*x**n*gamma(-p) + 
 3*a*b**2*n*x**(2*n)*gamma(-p) + b**3*n*x**(3*n)*gamma(-p)) - 6*a*a**p*...
 

Maxima [F]

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

integrate((c*x)^(-n*p-4*n-1)*(a+b*x^n)^p,x, algorithm="maxima")
 

Output:

integrate((b*x^n + a)^p*(c*x)^(-n*p - 4*n - 1), x)
 

Giac [F]

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

integrate((c*x)^(-n*p-4*n-1)*(a+b*x^n)^p,x, algorithm="giac")
 

Output:

integrate((b*x^n + a)^p*(c*x)^(-n*p - 4*n - 1), x)
 

Mupad [F(-1)]

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

int((a + b*x^n)^p/(c*x)^(4*n + n*p + 1),x)
 

Output:

int((a + b*x^n)^p/(c*x)^(4*n + n*p + 1), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.88 \[ \int (c x)^{-1-4 n-n p} \left (a+b x^n\right )^p \, dx=\frac {\left (x^{n} b +a \right )^{p} \left (6 x^{4 n} b^{4}-6 x^{3 n} a \,b^{3} p +3 x^{2 n} a^{2} b^{2} p^{2}+3 x^{2 n} a^{2} b^{2} p -x^{n} a^{3} b \,p^{3}-3 x^{n} a^{3} b \,p^{2}-2 x^{n} a^{3} b p -a^{4} p^{3}-6 a^{4} p^{2}-11 a^{4} p -6 a^{4}\right )}{x^{n p +4 n} c^{n p +4 n} a^{4} c n \left (p^{4}+10 p^{3}+35 p^{2}+50 p +24\right )} \] Input:

int((c*x)^(-n*p-4*n-1)*(a+b*x^n)^p,x)
 

Output:

((x**n*b + a)**p*(6*x**(4*n)*b**4 - 6*x**(3*n)*a*b**3*p + 3*x**(2*n)*a**2* 
b**2*p**2 + 3*x**(2*n)*a**2*b**2*p - x**n*a**3*b*p**3 - 3*x**n*a**3*b*p**2 
 - 2*x**n*a**3*b*p - a**4*p**3 - 6*a**4*p**2 - 11*a**4*p - 6*a**4))/(x**(n 
*p + 4*n)*c**(n*p + 4*n)*a**4*c*n*(p**4 + 10*p**3 + 35*p**2 + 50*p + 24))