\(\int (c x)^{-1-(1+n) p} (a x^n+b x^{1+n})^p \, dx\) [460]

Optimal result

Integrand size = 28, antiderivative size = 58 \[ \int (c x)^{-1-(1+n) p} \left (a x^n+b x^{1+n}\right )^p \, dx=-\frac {x^{-n} (c x)^{-((1+n) p)} \left (a x^n+b x^{1+n}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1,1-p,-\frac {b x}{a}\right )}{a c p} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03 \[ \int (c x)^{-1-(1+n) p} \left (a x^n+b x^{1+n}\right )^p \, dx=-\frac {(c x)^{-((1+n) p)} \left (x^n (a+b x)\right )^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x}{a}\right )}{c p} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1938, 76, 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.

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

\(\Big \downarrow \) 1938

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

\(\Big \downarrow \) 76

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

\(\Big \downarrow \) 74

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

Input:

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

Output:

-(((a*x^n + b*x^(1 + n))^p*Hypergeometric2F1[-p, -p, 1 - p, -((b*x)/a)])/( 
c*p*(c*x)^((1 + n)*p)*(1 + (b*x)/a)^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 76

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^IntPart 
[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n])   Int[(b*x)^m*(1 + d* 
(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integer 
Q[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2 
^(-1)] && EqQ[c^2 - d^2, 0])) ||  !RationalQ[n])
 

rule 1938

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F 
racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]))   Int[x^(m + j* 
p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !Inte 
gerQ[p] && NeQ[n, j] && PosQ[n - j]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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