Integrand size = 26, antiderivative size = 60 \[ \int (c x)^{-((1+n) p)} \left (a x^n+b x^{1+n}\right )^p \, dx=\frac {x^{1-n} (c x)^{-((1+n) p)} \left (a x^n+b x^{1+n}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,2,2-p,-\frac {b x}{a}\right )}{a (1-p)} \] Output:
x^(1-n)*(a*x^n+b*x^(1+n))^(p+1)*hypergeom([1, 2],[2-p],-b*x/a)/a/(1-p)/((c *x)^((1+n)*p))
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.03 \[ \int (c x)^{-((1+n) p)} \left (a x^n+b x^{1+n}\right )^p \, dx=-\frac {x (c x)^{-((1+n) p)} \left (x^n (a+b x)\right )^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x}{a}\right )}{-1+p} \] Input:
Integrate[(a*x^n + b*x^(1 + n))^p/(c*x)^((1 + n)*p),x]
Output:
-((x*(x^n*(a + b*x))^p*Hypergeometric2F1[1 - p, -p, 2 - p, -((b*x)/a)])/(( -1 + p)*(c*x)^((1 + n)*p)*(1 + (b*x)/a)^p))
Time = 0.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, 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.
\(\displaystyle \int (c x)^{-((n+1) p)} \left (a x^n+b x^{n+1}\right )^p \, dx\) |
\(\Big \downarrow \) 1938 |
\(\displaystyle x^p (a+b x)^{-p} (c x)^{-((n+1) p)} \left (a x^n+b x^{n+1}\right )^p \int x^{-p} (a+b x)^pdx\) |
\(\Big \downarrow \) 76 |
\(\displaystyle x^p \left (\frac {b x}{a}+1\right )^{-p} (c x)^{-((n+1) p)} \left (a x^n+b x^{n+1}\right )^p \int x^{-p} \left (\frac {b x}{a}+1\right )^pdx\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {x \left (\frac {b x}{a}+1\right )^{-p} (c x)^{-((n+1) p)} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x}{a}\right ) \left (a x^n+b x^{n+1}\right )^p}{1-p}\) |
Input:
Int[(a*x^n + b*x^(1 + n))^p/(c*x)^((1 + n)*p),x]
Output:
(x*(a*x^n + b*x^(1 + n))^p*Hypergeometric2F1[1 - p, -p, 2 - p, -((b*x)/a)] )/((1 - p)*(c*x)^((1 + n)*p)*(1 + (b*x)/a)^p)
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])))
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])
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]
\[\int \left (a \,x^{n}+b \,x^{1+n}\right )^{p} \left (c x \right )^{-\left (1+n \right ) p}d x\]
Input:
int((a*x^n+b*x^(1+n))^p/((c*x)^((1+n)*p)),x)
Output:
int((a*x^n+b*x^(1+n))^p/((c*x)^((1+n)*p)),x)
\[ \int (c x)^{-((1+n) p)} \left (a x^n+b x^{1+n}\right )^p \, dx=\int { \frac {{\left (b x^{n + 1} + a x^{n}\right )}^{p}}{\left (c x\right )^{{\left (n + 1\right )} p}} \,d x } \] Input:
integrate((a*x^n+b*x^(1+n))^p/((c*x)^((1+n)*p)),x, algorithm="fricas")
Output:
integral((b*x^(n + 1) + a*x^n)^p/(c*x)^((n + 1)*p), x)
\[ \int (c x)^{-((1+n) p)} \left (a x^n+b x^{1+n}\right )^p \, dx=\int \left (c x\right )^{- p \left (n + 1\right )} \left (a x^{n} + b x^{n + 1}\right )^{p}\, dx \] Input:
integrate((a*x**n+b*x**(1+n))**p/((c*x)**((1+n)*p)),x)
Output:
Integral((a*x**n + b*x**(n + 1))**p/(c*x)**(p*(n + 1)), x)
\[ \int (c x)^{-((1+n) p)} \left (a x^n+b x^{1+n}\right )^p \, dx=\int { \frac {{\left (b x^{n + 1} + a x^{n}\right )}^{p}}{\left (c x\right )^{{\left (n + 1\right )} p}} \,d x } \] Input:
integrate((a*x^n+b*x^(1+n))^p/((c*x)^((1+n)*p)),x, algorithm="maxima")
Output:
integrate((b*x^(n + 1) + a*x^n)^p/(c*x)^((n + 1)*p), x)
\[ \int (c x)^{-((1+n) p)} \left (a x^n+b x^{1+n}\right )^p \, dx=\int { \frac {{\left (b x^{n + 1} + a x^{n}\right )}^{p}}{\left (c x\right )^{{\left (n + 1\right )} p}} \,d x } \] Input:
integrate((a*x^n+b*x^(1+n))^p/((c*x)^((1+n)*p)),x, algorithm="giac")
Output:
integrate((b*x^(n + 1) + a*x^n)^p/(c*x)^((n + 1)*p), x)
Timed out. \[ \int (c x)^{-((1+n) p)} \left (a x^n+b x^{1+n}\right )^p \, dx=\int \frac {{\left (a\,x^n+b\,x^{n+1}\right )}^p}{{\left (c\,x\right )}^{p\,\left (n+1\right )}} \,d x \] Input:
int((a*x^n + b*x^(n + 1))^p/(c*x)^(p*(n + 1)),x)
Output:
int((a*x^n + b*x^(n + 1))^p/(c*x)^(p*(n + 1)), x)
\[ \int (c x)^{-((1+n) p)} \left (a x^n+b x^{1+n}\right )^p \, dx=\frac {\left (x^{n} a +x^{n} b x \right )^{p} x +x^{n p +p} \left (\int \frac {\left (x^{n} a +x^{n} b x \right )^{p}}{x^{n p +p} a +x^{n p +p} b x}d x \right ) a p}{x^{n p +p} c^{n p +p}} \] Input:
int((a*x^n+b*x^(1+n))^p/((c*x)^((1+n)*p)),x)
Output:
((x**n*a + x**n*b*x)**p*x + x**(n*p + p)*int((x**n*a + x**n*b*x)**p/(x**(n *p + p)*a + x**(n*p + p)*b*x),x)*a*p)/(x**(n*p + p)*c**(n*p + p))