Integrand size = 26, antiderivative size = 102 \[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=-b n x \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (1+\frac {e x^{-\frac {1}{1+q}}}{d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (-1-q,-1-q,-q,-\frac {e x^{-\frac {1}{1+q}}}{d}\right )+\frac {x \left (d+e x^{-\frac {1}{1+q}}\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d} \]
-b*n*x*(d+e/(x^(1/(1+q))))^q*hypergeom([-1-q, -1-q],[-q],-e/d/(x^(1/(1+q)) ))/((1+e/d/(x^(1/(1+q))))^q)+x*(d+e/(x^(1/(1+q))))^(1+q)*(a+b*ln(c*x^n))/d
Time = 0.51 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.40 \[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^{-\frac {1}{1+q}} \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (1+\frac {d x^{\frac {1}{1+q}}}{e}\right )^{-q} \left (-b d n (1+q)^2 x^{\frac {2+q}{1+q}} \, _3F_2\left (1,1,-q;2,2;-\frac {d x^{\frac {1}{1+q}}}{e}\right )-b e n x \log (x)+\left (1+\frac {d x^{\frac {1}{1+q}}}{e}\right )^q \left (e x+d x^{\frac {2+q}{1+q}}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d} \]
((d + e/x^(1 + q)^(-1))^q*(-(b*d*n*(1 + q)^2*x^((2 + q)/(1 + q))*Hypergeom etricPFQ[{1, 1, -q}, {2, 2}, -((d*x^(1 + q)^(-1))/e)]) - b*e*n*x*Log[x] + (1 + (d*x^(1 + q)^(-1))/e)^q*(e*x + d*x^((2 + q)/(1 + q)))*(a + b*Log[c*x^ n])))/(d*x^(1 + q)^(-1)*(1 + (d*x^(1 + q)^(-1))/e)^q)
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2751, 776, 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.
\(\displaystyle \int \left (d+e x^{-\frac {1}{q+1}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2751 |
\(\displaystyle \frac {x \left (d+e x^{-\frac {1}{q+1}}\right )^{q+1} \left (a+b \log \left (c x^n\right )\right )}{d}-\frac {b n \int \left (e x^{-\frac {1}{q+1}}+d\right )^{q+1}dx}{d}\) |
\(\Big \downarrow \) 776 |
\(\displaystyle \frac {b n (q+1) x \left (\frac {x^{-\frac {1}{q+1}}}{d+e x^{-\frac {1}{q+1}}}\right )^{q+1} \left (d+e x^{-\frac {1}{q+1}}\right )^{q+1} \int \frac {\left (\frac {x^{-\frac {1}{q+1}}}{e x^{-\frac {1}{q+1}}+d}\right )^{-q-2}}{1-\frac {e x^{-\frac {1}{q+1}}}{e x^{-\frac {1}{q+1}}+d}}d\frac {x^{-\frac {1}{q+1}}}{e x^{-\frac {1}{q+1}}+d}}{d}+\frac {x \left (d+e x^{-\frac {1}{q+1}}\right )^{q+1} \left (a+b \log \left (c x^n\right )\right )}{d}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {x \left (d+e x^{-\frac {1}{q+1}}\right )^{q+1} \left (a+b \log \left (c x^n\right )\right )}{d}-\frac {b n x \left (d+e x^{-\frac {1}{q+1}}\right )^{q+1} \operatorname {Hypergeometric2F1}\left (1,-q-1,-q,\frac {e x^{-\frac {1}{q+1}}}{e x^{-\frac {1}{q+1}}+d}\right )}{d}\) |
-((b*n*x*(d + e/x^(1 + q)^(-1))^(1 + q)*Hypergeometric2F1[1, -1 - q, -q, e /(x^(1 + q)^(-1)*(d + e/x^(1 + q)^(-1)))])/d) + (x*(d + e/x^(1 + q)^(-1))^ (1 + q)*(a + b*Log[c*x^n]))/d
3.5.46.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*(a + b*x^n)^p*((x^n/ (a + b*x^n))^p/n) Subst[Int[1/(x^(p + 1)*(1 - b*x)), x], x, x^n/(a + b*x^ n)], x] /; FreeQ[{a, b, n, p}, x] && EqQ[1/n + p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
\[\int \left (d +e \,x^{-\frac {1}{1+q}}\right )^{q} \left (a +b \ln \left (c \,x^{n}\right )\right )d x\]
\[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} {\left (d + \frac {e}{x^{\left (\frac {1}{q + 1}\right )}}\right )}^{q} \,d x } \]
Timed out. \[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
\[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} {\left (d + \frac {e}{x^{\left (\frac {1}{q + 1}\right )}}\right )}^{q} \,d x } \]
\[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} {\left (d + \frac {e}{x^{\left (\frac {1}{q + 1}\right )}}\right )}^{q} \,d x } \]
Timed out. \[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (d+\frac {e}{x^{\frac {1}{q+1}}}\right )}^q\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]