\(\int (c x)^{-1+n-n p} (a+b x^n)^p \, dx\) [694]

Optimal result

Integrand size = 21, antiderivative size = 70 \[ \int (c x)^{-1+n-n p} \left (a+b x^n\right )^p \, dx=\frac {(c x)^{n-n p} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^n}{a}\right )}{c n (1-p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.97 \[ \int (c x)^{-1+n-n p} \left (a+b x^n\right )^p \, dx=\frac {(c x)^{n-n p} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^n}{a}\right )}{c n-c n p} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {883, 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.

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

\(\Big \downarrow \) 883

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

\(\Big \downarrow \) 882

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

\(\Big \downarrow \) 74

\(\displaystyle \frac {a x^{-(n (1-p))-n p+n} (c x)^{n-n p} \left (a+b x^n\right )^{p-1} \operatorname {Hypergeometric2F1}\left (2,1-p,2-p,\frac {b x^n}{b x^n+a}\right )}{c n (1-p)}\)

Input:

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

Output:

(a*x^(n - n*(1 - p) - n*p)*(c*x)^(n - n*p)*(a + b*x^n)^(-1 + p)*Hypergeome 
tric2F1[2, 1 - p, 2 - p, (b*x^n)/(a + b*x^n)])/(c*n*(1 - 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 883

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int (c x)^{-1+n-n p} \left (a+b x^n\right )^p \, dx=\frac {a^{1 - p} a^{2 p - 1} b^{1 - p} b^{p - 1} c^{- n p + n - 1} x^{- n p + n} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 - p\right )} \] Input:

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

Output:

a**(1 - p)*a**(2*p - 1)*b**(1 - p)*b**(p - 1)*c**(-n*p + n - 1)*x**(-n*p + 
 n)*gamma(1 - p)*hyper((-p, 1 - p), (2 - p,), b*x**n*exp_polar(I*pi)/a)/(n 
*gamma(2 - p))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(c**n*(x**n*(x**n*b + a)**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*n*p))/(x**(n*p)*c**(n*p)*c*n)