Integrand size = 19, antiderivative size = 51 \[ \int \frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p}{x} \, dx=-\frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a}\right )}{a (1+p)} \] Output:
-(a+b*(c*x^n)^(1/n))^(p+1)*hypergeom([1, p+1],[2+p],1+b*(c*x^n)^(1/n)/a)/a /(p+1)
Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p}{x} \, dx=-\frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a}\right )}{a (1+p)} \] Input:
Integrate[(a + b*(c*x^n)^n^(-1))^p/x,x]
Output:
-(((a + b*(c*x^n)^n^(-1))^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + ( b*(c*x^n)^n^(-1))/a])/(a*(1 + p)))
Time = 0.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {892, 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.
\(\displaystyle \int \frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p}{x} \, dx\) |
\(\Big \downarrow \) 892 |
\(\displaystyle \int \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^pd\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a}+1\right )}{a (p+1)}\) |
Input:
Int[(a + b*(c*x^n)^n^(-1))^p/x,x]
Output:
-(((a + b*(c*x^n)^n^(-1))^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + ( b*(c*x^n)^n^(-1))/a])/(a*(1 + p)))
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[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo l] :> Simp[(d*x)^(m + 1)/(d*((c*x^q)^(1/q))^(m + 1)) Subst[Int[x^m*(a + b *x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x ] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
\[\int \frac {{\left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right )}^{p}}{x}d x\]
Input:
int((a+b*(c*x^n)^(1/n))^p/x,x)
Output:
int((a+b*(c*x^n)^(1/n))^p/x,x)
\[ \int \frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p}{x} \, dx=\int { \frac {{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p}}{x} \,d x } \] Input:
integrate((a+b*(c*x^n)^(1/n))^p/x,x, algorithm="fricas")
Output:
integral(((c*x^n)^(1/n)*b + a)^p/x, x)
\[ \int \frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p}{x} \, dx=\int \frac {\left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{p}}{x}\, dx \] Input:
integrate((a+b*(c*x**n)**(1/n))**p/x,x)
Output:
Integral((a + b*(c*x**n)**(1/n))**p/x, x)
\[ \int \frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p}{x} \, dx=\int { \frac {{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p}}{x} \,d x } \] Input:
integrate((a+b*(c*x^n)^(1/n))^p/x,x, algorithm="maxima")
Output:
integrate(((c*x^n)^(1/n)*b + a)^p/x, x)
\[ \int \frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p}{x} \, dx=\int { \frac {{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p}}{x} \,d x } \] Input:
integrate((a+b*(c*x^n)^(1/n))^p/x,x, algorithm="giac")
Output:
integrate(((c*x^n)^(1/n)*b + a)^p/x, x)
Timed out. \[ \int \frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p}{x} \, dx=\int \frac {{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^p}{x} \,d x \] Input:
int((a + b*(c*x^n)^(1/n))^p/x,x)
Output:
int((a + b*(c*x^n)^(1/n))^p/x, x)
\[ \int \frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p}{x} \, dx=\frac {\left (c^{\frac {1}{n}} b x +a \right )^{p}+\left (\int \frac {\left (c^{\frac {1}{n}} b x +a \right )^{p}}{c^{\frac {1}{n}} b \,x^{2}+a x}d x \right ) a p}{p} \] Input:
int((a+b*(c*x^n)^(1/n))^p/x,x)
Output:
((c**(1/n)*b*x + a)**p + int((c**(1/n)*b*x + a)**p/(c**(1/n)*b*x**2 + a*x) ,x)*a*p)/p