Integrand size = 31, antiderivative size = 148 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {b n \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {arctanh}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}} \]
b*n*(-e^2*x^2+d^2)/e^2/(-e*x+d)^(1/2)/(e*x+d)^(1/2)-(-e^2*x^2+d^2)*(a+b*ln (c*x^n))/e^2/(-e*x+d)^(1/2)/(e*x+d)^(1/2)-b*d^2*n*arctanh((1-e^2*x^2/d^2)^ (1/2))*(1-e^2*x^2/d^2)^(1/2)/e^2/(-e*x+d)^(1/2)/(e*x+d)^(1/2)
Time = 0.15 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.76 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {b d n \log (x)}{e^2}-\frac {b n \sqrt {d-e x} \sqrt {d+e x} \log (x)}{e^2}-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (a-b n+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{e^2}-\frac {b d n \log \left (d+\sqrt {d-e x} \sqrt {d+e x}\right )}{e^2} \]
(b*d*n*Log[x])/e^2 - (b*n*Sqrt[d - e*x]*Sqrt[d + e*x]*Log[x])/e^2 - (Sqrt[ d - e*x]*Sqrt[d + e*x]*(a - b*n + b*(-(n*Log[x]) + Log[c*x^n])))/e^2 - (b* d*n*Log[d + Sqrt[d - e*x]*Sqrt[d + e*x]])/e^2
Time = 0.50 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2787, 2776, 243, 60, 73, 221}
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.
\(\displaystyle \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx\) |
\(\Big \downarrow \) 2787 |
\(\displaystyle \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {1-\frac {e^2 x^2}{d^2}}}dx}{\sqrt {d-e x} \sqrt {d+e x}}\) |
\(\Big \downarrow \) 2776 |
\(\displaystyle \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \left (\frac {b d^2 n \int \frac {\sqrt {1-\frac {e^2 x^2}{d^2}}}{x}dx}{e^2}-\frac {d^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a+b \log \left (c x^n\right )\right )}{e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \left (\frac {b d^2 n \int \frac {\sqrt {1-\frac {e^2 x^2}{d^2}}}{x^2}dx^2}{2 e^2}-\frac {d^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a+b \log \left (c x^n\right )\right )}{e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \left (\frac {b d^2 n \left (\int \frac {1}{x^2 \sqrt {1-\frac {e^2 x^2}{d^2}}}dx^2+2 \sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{2 e^2}-\frac {d^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a+b \log \left (c x^n\right )\right )}{e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \left (\frac {b d^2 n \left (2 \sqrt {1-\frac {e^2 x^2}{d^2}}-\frac {2 d^2 \int \frac {1}{\frac {d^2}{e^2}-\frac {d^2 x^4}{e^2}}d\sqrt {1-\frac {e^2 x^2}{d^2}}}{e^2}\right )}{2 e^2}-\frac {d^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a+b \log \left (c x^n\right )\right )}{e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \left (\frac {b d^2 n \left (2 \sqrt {1-\frac {e^2 x^2}{d^2}}-2 \text {arctanh}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )\right )}{2 e^2}-\frac {d^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a+b \log \left (c x^n\right )\right )}{e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
(Sqrt[1 - (e^2*x^2)/d^2]*((b*d^2*n*(2*Sqrt[1 - (e^2*x^2)/d^2] - 2*ArcTanh[ Sqrt[1 - (e^2*x^2)/d^2]]))/(2*e^2) - (d^2*Sqrt[1 - (e^2*x^2)/d^2]*(a + b*L og[c*x^n]))/e^2))/(Sqrt[d - e*x]*Sqrt[d + e*x])
3.4.10.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :> Simp[f^m*(d + e*x^r)^(q + 1)*((a + b*L og[c*x^n])^p/(e*r*(q + 1))), x] - Simp[b*f^m*n*(p/(e*r*(q + 1))) Int[(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d , e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || G tQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^ (q_)*((d2_) + (e2_.)*(x_))^(q_), x_Symbol] :> Simp[(d1 + e1*x)^q*((d2 + e2* x)^q/(1 + e1*(e2/(d1*d2))*x^2)^q) Int[x^m*(1 + e1*(e2/(d1*d2))*x^2)^q*(a + b*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2 *e1 + d1*e2, 0] && IntegerQ[m] && IntegerQ[q - 1/2]
\[\int \frac {x \left (a +b \ln \left (c \,x^{n}\right )\right )}{\sqrt {-e x +d}\, \sqrt {e x +d}}d x\]
Time = 0.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {b d n \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) - {\left (b n \log \left (x\right ) - b n + b \log \left (c\right ) + a\right )} \sqrt {e x + d} \sqrt {-e x + d}}{e^{2}} \]
(b*d*n*log((sqrt(e*x + d)*sqrt(-e*x + d) - d)/x) - (b*n*log(x) - b*n + b*l og(c) + a)*sqrt(e*x + d)*sqrt(-e*x + d))/e^2
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )}{\sqrt {d - e x} \sqrt {d + e x}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.71 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (d \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \sqrt {-e^{2} x^{2} + d^{2}}\right )} b n}{e^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b \log \left (c x^{n}\right )}{e^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{e^{2}} \]
-(d*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x)) - sqrt(-e^2*x^2 + d^2))*b*n/e^2 - sqrt(-e^2*x^2 + d^2)*b*log(c*x^n)/e^2 - sqrt(-e^2*x^2 + d^ 2)*a/e^2
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{\sqrt {e x + d} \sqrt {-e x + d}} \,d x } \]
Timed out. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{\sqrt {d+e\,x}\,\sqrt {d-e\,x}} \,d x \]