Integrand size = 33, antiderivative size = 406 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {b n x \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b d^3 n \sqrt {1-\frac {e^2 x^2}{d^2}} \arcsin \left (\frac {e x}{d}\right )}{4 e^3 \sqrt {d-e x} \sqrt {d+e x}}+\frac {i b d^3 n \sqrt {1-\frac {e^2 x^2}{d^2}} \arcsin \left (\frac {e x}{d}\right )^2}{4 e^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b d^3 n \sqrt {1-\frac {e^2 x^2}{d^2}} \arcsin \left (\frac {e x}{d}\right ) \log \left (1-e^{2 i \arcsin \left (\frac {e x}{d}\right )}\right )}{2 e^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {x \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {d^3 \sqrt {1-\frac {e^2 x^2}{d^2}} \arcsin \left (\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {d-e x} \sqrt {d+e x}}+\frac {i b d^3 n \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {e x}{d}\right )}\right )}{4 e^3 \sqrt {d-e x} \sqrt {d+e x}} \]
1/4*b*n*x*(-e^2*x^2+d^2)/e^2/(-e*x+d)^(1/2)/(e*x+d)^(1/2)-1/2*x*(-e^2*x^2+ d^2)*(a+b*ln(c*x^n))/e^2/(-e*x+d)^(1/2)/(e*x+d)^(1/2)+1/4*b*d^3*n*arcsin(e *x/d)*(1-e^2*x^2/d^2)^(1/2)/e^3/(-e*x+d)^(1/2)/(e*x+d)^(1/2)+1/4*I*b*d^3*n *arcsin(e*x/d)^2*(1-e^2*x^2/d^2)^(1/2)/e^3/(-e*x+d)^(1/2)/(e*x+d)^(1/2)-1/ 2*b*d^3*n*arcsin(e*x/d)*ln(1-(I*e*x/d+(1-e^2*x^2/d^2)^(1/2))^2)*(1-e^2*x^2 /d^2)^(1/2)/e^3/(-e*x+d)^(1/2)/(e*x+d)^(1/2)+1/2*d^3*arcsin(e*x/d)*(a+b*ln (c*x^n))*(1-e^2*x^2/d^2)^(1/2)/e^3/(-e*x+d)^(1/2)/(e*x+d)^(1/2)+1/4*I*b*d^ 3*n*polylog(2,(I*e*x/d+(1-e^2*x^2/d^2)^(1/2))^2)*(1-e^2*x^2/d^2)^(1/2)/e^3 /(-e*x+d)^(1/2)/(e*x+d)^(1/2)
Time = 1.88 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.78 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {-2 e x \sqrt {d-e x} \sqrt {d+e x} \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+2 d^2 \arctan \left (\frac {e x}{\sqrt {d-e x} \sqrt {d+e x}}\right ) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+\frac {b n \left (d^3 \sqrt {1-\frac {e^2 x^2}{d^2}} \arcsin \left (\frac {e x}{d}\right )+e x \left (-d^2+e^2 x^2\right ) (-1+2 \log (x))+\frac {e^3 \sqrt {1-\frac {e^2 x^2}{d^2}} \left (\text {arcsinh}\left (\sqrt {-\frac {e^2}{d^2}} x\right )^2+2 \text {arcsinh}\left (\sqrt {-\frac {e^2}{d^2}} x\right ) \log \left (1-e^{-2 \text {arcsinh}\left (\sqrt {-\frac {e^2}{d^2}} x\right )}\right )-2 \log (x) \log \left (\sqrt {-\frac {e^2}{d^2}} x+\sqrt {1-\frac {e^2 x^2}{d^2}}\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\sqrt {-\frac {e^2}{d^2}} x\right )}\right )\right )}{\left (-\frac {e^2}{d^2}\right )^{3/2}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}}{4 e^3} \]
(-2*e*x*Sqrt[d - e*x]*Sqrt[d + e*x]*(a - b*n*Log[x] + b*Log[c*x^n]) + 2*d^ 2*ArcTan[(e*x)/(Sqrt[d - e*x]*Sqrt[d + e*x])]*(a - b*n*Log[x] + b*Log[c*x^ n]) + (b*n*(d^3*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d] + e*x*(-d^2 + e^2* x^2)*(-1 + 2*Log[x]) + (e^3*Sqrt[1 - (e^2*x^2)/d^2]*(ArcSinh[Sqrt[-(e^2/d^ 2)]*x]^2 + 2*ArcSinh[Sqrt[-(e^2/d^2)]*x]*Log[1 - E^(-2*ArcSinh[Sqrt[-(e^2/ d^2)]*x])] - 2*Log[x]*Log[Sqrt[-(e^2/d^2)]*x + Sqrt[1 - (e^2*x^2)/d^2]] - PolyLog[2, E^(-2*ArcSinh[Sqrt[-(e^2/d^2)]*x])]))/(-(e^2/d^2))^(3/2)))/(Sqr t[d - e*x]*Sqrt[d + e*x]))/(4*e^3)
Time = 0.82 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.53, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2787, 2792, 27, 2010, 2009}
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.
| \(\displaystyle \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 2787 |
| \(\displaystyle \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {1-\frac {e^2 x^2}{d^2}}}dx}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 2792 |
| \(\displaystyle \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \left (-b n \int -\frac {d^2 \left (e x \sqrt {\frac {d^2-e^2 x^2}{d^2}}-d \arcsin \left (\frac {e x}{d}\right )\right )}{2 e^3 x}dx+\frac {d^3 \arcsin \left (\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {d^2 x \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a+b \log \left (c x^n\right )\right )}{2 e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \left (\frac {b d^2 n \int \frac {e x \sqrt {\frac {d^2-e^2 x^2}{d^2}}-d \arcsin \left (\frac {e x}{d}\right )}{x}dx}{2 e^3}+\frac {d^3 \arcsin \left (\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {d^2 x \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a+b \log \left (c x^n\right )\right )}{2 e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \left (\frac {b d^2 n \int \left (e \sqrt {1-\frac {e^2 x^2}{d^2}}-\frac {d \arcsin \left (\frac {e x}{d}\right )}{x}\right )dx}{2 e^3}+\frac {d^3 \arcsin \left (\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {d^2 x \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a+b \log \left (c x^n\right )\right )}{2 e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \left (\frac {d^3 \arcsin \left (\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {d^2 x \sqrt {1-\frac {e^2 x^2}{d^2}} \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {b d^2 n \left (\frac {1}{2} i d \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {e x}{d}\right )}\right )+\frac {1}{2} i d \arcsin \left (\frac {e x}{d}\right )^2+\frac {1}{2} d \arcsin \left (\frac {e x}{d}\right )-d \arcsin \left (\frac {e x}{d}\right ) \log \left (1-e^{2 i \arcsin \left (\frac {e x}{d}\right )}\right )+\frac {1}{2} e x \sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{2 e^3}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
(Sqrt[1 - (e^2*x^2)/d^2]*(-1/2*(d^2*x*Sqrt[1 - (e^2*x^2)/d^2]*(a + b*Log[c *x^n]))/e^2 + (d^3*ArcSin[(e*x)/d]*(a + b*Log[c*x^n]))/(2*e^3) + (b*d^2*n* ((e*x*Sqrt[1 - (e^2*x^2)/d^2])/2 + (d*ArcSin[(e*x)/d])/2 + (I/2)*d*ArcSin[ (e*x)/d]^2 - d*ArcSin[(e*x)/d]*Log[1 - E^((2*I)*ArcSin[(e*x)/d])] + (I/2)* d*PolyLog[2, E^((2*I)*ArcSin[(e*x)/d])]))/(2*e^3)))/(Sqrt[d - e*x]*Sqrt[d + e*x])
3.4.13.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
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[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x] }, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2] ) || InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x ] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])
\[\int \frac {x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\sqrt {-e x +d}\, \sqrt {e x +d}}d x\]
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{\sqrt {e x + d} \sqrt {-e x + d}} \,d x } \]
integral(-(sqrt(e*x + d)*sqrt(-e*x + d)*b*x^2*log(c*x^n) + sqrt(e*x + d)*s qrt(-e*x + d)*a*x^2)/(e^2*x^2 - d^2), x)
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{\sqrt {e x + d} \sqrt {-e x + d}} \,d x } \]
1/2*a*(d^2*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^2) - sqrt(-e^2*x^2 + d ^2)*x/e^2) + b*integrate((x^2*log(c) + x^2*log(x^n))/(sqrt(e*x + d)*sqrt(- e*x + d)), x)
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{\sqrt {e x + d} \sqrt {-e x + d}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{\sqrt {d+e\,x}\,\sqrt {d-e\,x}} \,d x \]