Integrand size = 17, antiderivative size = 53 \[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x} \, dx=-\frac {\left (a+b \left (c x^q\right )^n\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \left (c x^q\right )^n}{a}\right )}{a n (1+p) q} \] Output:
-(a+b*(c*x^q)^n)^(p+1)*hypergeom([1, p+1],[2+p],1+b*(c*x^q)^n/a)/a/n/(p+1) /q
Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x} \, dx=-\frac {\left (a+b \left (c x^q\right )^n\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \left (c x^q\right )^n}{a}\right )}{a n (1+p) q} \] Input:
Integrate[(a + b*(c*x^q)^n)^p/x,x]
Output:
-(((a + b*(c*x^q)^n)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*(c* x^q)^n)/a])/(a*n*(1 + p)*q))
Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {894, 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.
\(\displaystyle \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x} \, dx\) |
\(\Big \downarrow \) 894 |
\(\displaystyle \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int c^n \left (c x^q\right )^{-n} \left (b \left (c x^q\right )^n+a\right )^pd\left (c^{-n} \left (c x^q\right )^n\right )}{n q}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {\left (a+b \left (c x^q\right )^n\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b \left (c x^q\right )^n}{a}+1\right )}{a n (p+1) q}\) |
Input:
Int[(a + b*(c*x^q)^n)^p/x,x]
Output:
-(((a + b*(c*x^q)^n)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*(c* x^q)^n)/a])/(a*n*(1 + p)*q))
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_)^(q_))^(n_))^(p_.), x_Symbo l] :> Subst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(n*q), (c*x^q)^n/c^n] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && !RationalQ[n]
\[\int \frac {{\left (a +b \left (c \,x^{q}\right )^{n}\right )}^{p}}{x}d x\]
Input:
int((a+b*(c*x^q)^n)^p/x,x)
Output:
int((a+b*(c*x^q)^n)^p/x,x)
\[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x} \, dx=\int { \frac {{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x} \,d x } \] Input:
integrate((a+b*(c*x^q)^n)^p/x,x, algorithm="fricas")
Output:
integral(((c*x^q)^n*b + a)^p/x, x)
\[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x} \, dx=\int \frac {\left (a + b \left (c x^{q}\right )^{n}\right )^{p}}{x}\, dx \] Input:
integrate((a+b*(c*x**q)**n)**p/x,x)
Output:
Integral((a + b*(c*x**q)**n)**p/x, x)
\[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x} \, dx=\int { \frac {{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x} \,d x } \] Input:
integrate((a+b*(c*x^q)^n)^p/x,x, algorithm="maxima")
Output:
integrate(((c*x^q)^n*b + a)^p/x, x)
\[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x} \, dx=\int { \frac {{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x} \,d x } \] Input:
integrate((a+b*(c*x^q)^n)^p/x,x, algorithm="giac")
Output:
integrate(((c*x^q)^n*b + a)^p/x, x)
Timed out. \[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x} \, dx=\int \frac {{\left (a+b\,{\left (c\,x^q\right )}^n\right )}^p}{x} \,d x \] Input:
int((a + b*(c*x^q)^n)^p/x,x)
Output:
int((a + b*(c*x^q)^n)^p/x, x)
\[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x} \, dx=\frac {\left (x^{n q} c^{n} b +a \right )^{p}+\left (\int \frac {\left (x^{n q} c^{n} b +a \right )^{p}}{x^{n q} c^{n} b x +a x}d x \right ) a n p q}{n p q} \] Input:
int((a+b*(c*x^q)^n)^p/x,x)
Output:
((x**(n*q)*c**n*b + a)**p + int((x**(n*q)*c**n*b + a)**p/(x**(n*q)*c**n*b* x + a*x),x)*a*n*p*q)/(n*p*q)