Integrand size = 15, antiderivative size = 46 \[ \int \frac {\left (a+b (c x)^n\right )^p}{x} \, dx=-\frac {\left (a+b (c x)^n\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b (c x)^n}{a}\right )}{a n (1+p)} \] Output:
-(a+b*(c*x)^n)^(p+1)*hypergeom([1, p+1],[2+p],1+b*(c*x)^n/a)/a/n/(p+1)
Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b (c x)^n\right )^p}{x} \, dx=-\frac {\left (a+b (c x)^n\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b (c x)^n}{a}\right )}{a n (1+p)} \] Input:
Integrate[(a + b*(c*x)^n)^p/x,x]
Output:
-(((a + b*(c*x)^n)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*(c*x) ^n)/a])/(a*n*(1 + p)))
Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {891, 27, 798, 75}
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.
\(\displaystyle \int \frac {\left (a+b (c x)^n\right )^p}{x} \, dx\) |
\(\Big \downarrow \) 891 |
\(\displaystyle \frac {\int \frac {\left (b (c x)^n+a\right )^p}{x}d(c x)}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\left (a+b (c x)^n\right )^p}{c x}d(c x)\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int \frac {\left (b (c x)^n+a\right )^p}{c x}d(c x)^n}{n}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {\left (a+b (c x)^n\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b (c x)^n}{a}+1\right )}{a n (p+1)}\) |
Input:
Int[(a + b*(c*x)^n)^p/x,x]
Output:
-(((a + b*(c*x)^n)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*(c*x) ^n)/a])/(a*n*(1 + p)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_)*(x_))^(n_))^(p_.), x_Symbol] :> Simp[1/c Subst[Int[(d*(x/c))^m*(a + b*x^n)^p, x], x, c*x], x] /; FreeQ[{a , b, c, d, m, n, p}, x]
\[\int \frac {\left (a +b \left (c x \right )^{n}\right )^{p}}{x}d x\]
Input:
int((a+b*(c*x)^n)^p/x,x)
Output:
int((a+b*(c*x)^n)^p/x,x)
\[ \int \frac {\left (a+b (c x)^n\right )^p}{x} \, dx=\int { \frac {{\left (\left (c x\right )^{n} b + a\right )}^{p}}{x} \,d x } \] Input:
integrate((a+b*(c*x)^n)^p/x,x, algorithm="fricas")
Output:
integral(((c*x)^n*b + a)^p/x, x)
\[ \int \frac {\left (a+b (c x)^n\right )^p}{x} \, dx=\int \frac {\left (a + b \left (c x\right )^{n}\right )^{p}}{x}\, dx \] Input:
integrate((a+b*(c*x)**n)**p/x,x)
Output:
Integral((a + b*(c*x)**n)**p/x, x)
\[ \int \frac {\left (a+b (c x)^n\right )^p}{x} \, dx=\int { \frac {{\left (\left (c x\right )^{n} b + a\right )}^{p}}{x} \,d x } \] Input:
integrate((a+b*(c*x)^n)^p/x,x, algorithm="maxima")
Output:
integrate(((c*x)^n*b + a)^p/x, x)
\[ \int \frac {\left (a+b (c x)^n\right )^p}{x} \, dx=\int { \frac {{\left (\left (c x\right )^{n} b + a\right )}^{p}}{x} \,d x } \] Input:
integrate((a+b*(c*x)^n)^p/x,x, algorithm="giac")
Output:
integrate(((c*x)^n*b + a)^p/x, x)
Timed out. \[ \int \frac {\left (a+b (c x)^n\right )^p}{x} \, dx=\int \frac {{\left (a+b\,{\left (c\,x\right )}^n\right )}^p}{x} \,d x \] Input:
int((a + b*(c*x)^n)^p/x,x)
Output:
int((a + b*(c*x)^n)^p/x, x)
\[ \int \frac {\left (a+b (c x)^n\right )^p}{x} \, dx=\frac {\left (x^{n} c^{n} b +a \right )^{p}+\left (\int \frac {\left (x^{n} c^{n} b +a \right )^{p}}{x^{n} c^{n} b x +a x}d x \right ) a n p}{n p} \] Input:
int((a+b*(c*x)^n)^p/x,x)
Output:
((x**n*c**n*b + a)**p + int((x**n*c**n*b + a)**p/(x**n*c**n*b*x + a*x),x)* a*n*p)/(n*p)