Integrand size = 25, antiderivative size = 359 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x^2}} \, dx=-\frac {b n x \sqrt {d+e x^2}}{4 e}-\frac {b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 e^{3/2} \sqrt {d+e x^2}}-\frac {b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 e^{3/2} \sqrt {d+e x^2}}+\frac {b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{3/2} \sqrt {d+e x^2}}+\frac {x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {d^{3/2} \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2} \sqrt {d+e x^2}}+\frac {b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 e^{3/2} \sqrt {d+e x^2}} \]
-1/4*b*n*x*(e*x^2+d)^(1/2)/e+1/2*x*(a+b*ln(c*x^n))*(e*x^2+d)^(1/2)/e-1/4*b *d^(3/2)*n*arcsinh(x*e^(1/2)/d^(1/2))*(1+e*x^2/d)^(1/2)/e^(3/2)/(e*x^2+d)^ (1/2)-1/4*b*d^(3/2)*n*arcsinh(x*e^(1/2)/d^(1/2))^2*(1+e*x^2/d)^(1/2)/e^(3/ 2)/(e*x^2+d)^(1/2)+1/2*b*d^(3/2)*n*arcsinh(x*e^(1/2)/d^(1/2))*ln(1-(x*e^(1 /2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*(1+e*x^2/d)^(1/2)/e^(3/2)/(e*x^2+d)^(1/2 )-1/2*d^(3/2)*arcsinh(x*e^(1/2)/d^(1/2))*(a+b*ln(c*x^n))*(1+e*x^2/d)^(1/2) /e^(3/2)/(e*x^2+d)^(1/2)+1/4*b*d^(3/2)*n*polylog(2,(x*e^(1/2)/d^(1/2)+(1+e *x^2/d)^(1/2))^2)*(1+e*x^2/d)^(1/2)/e^(3/2)/(e*x^2+d)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.51 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.57 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x^2}} \, dx=\frac {\frac {b n \sqrt {1+\frac {e x^2}{d}} \left (2 e^2 x^3 \, _3F_2\left (\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {5}{2},\frac {5}{2};-\frac {e x^2}{d}\right )+9 d \sqrt {e} \left (\sqrt {e} x \sqrt {1+\frac {e x^2}{d}}-\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right ) (-1+2 \log (x))\right )}{\sqrt {d+e x^2}}+18 e x \sqrt {d+e x^2} \left (a-b n \log (x)+b \log \left (c x^n\right )\right )-18 d \sqrt {e} \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{36 e^2} \]
((b*n*Sqrt[1 + (e*x^2)/d]*(2*e^2*x^3*HypergeometricPFQ[{3/2, 3/2, 3/2}, {5 /2, 5/2}, -((e*x^2)/d)] + 9*d*Sqrt[e]*(Sqrt[e]*x*Sqrt[1 + (e*x^2)/d] - Sqr t[d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])*(-1 + 2*Log[x])))/Sqrt[d + e*x^2] + 18* e*x*Sqrt[d + e*x^2]*(a - b*n*Log[x] + b*Log[c*x^n]) - 18*d*Sqrt[e]*(a - b* n*Log[x] + b*Log[c*x^n])*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(36*e^2)
Time = 0.67 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.73, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2786, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x^2}} \, dx\) |
| \(\Big \downarrow \) 2786 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {\frac {e x^2}{d}+1}}dx}{\sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 2792 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (-b n \int \frac {\frac {d x \sqrt {\frac {e x^2}{d}+1}}{e}-\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}}{2 x}dx-\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2}}+\frac {d x \sqrt {\frac {e x^2}{d}+1} \left (a+b \log \left (c x^n\right )\right )}{2 e}\right )}{\sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (-\frac {1}{2} b n \int \frac {\frac {d x \sqrt {\frac {e x^2}{d}+1}}{e}-\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}}{x}dx-\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2}}+\frac {d x \sqrt {\frac {e x^2}{d}+1} \left (a+b \log \left (c x^n\right )\right )}{2 e}\right )}{\sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (-\frac {1}{2} b n \int \left (\frac {d \sqrt {\frac {e x^2}{d}+1}}{e}-\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} x}\right )dx-\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2}}+\frac {d x \sqrt {\frac {e x^2}{d}+1} \left (a+b \log \left (c x^n\right )\right )}{2 e}\right )}{\sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (-\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2}}+\frac {d x \sqrt {\frac {e x^2}{d}+1} \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {1}{2} b n \left (-\frac {d^{3/2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{3/2}}+\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{3/2}}+\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{3/2}}-\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{e^{3/2}}+\frac {d x \sqrt {\frac {e x^2}{d}+1}}{2 e}\right )\right )}{\sqrt {d+e x^2}}\) |
(Sqrt[1 + (e*x^2)/d]*((d*x*Sqrt[1 + (e*x^2)/d]*(a + b*Log[c*x^n]))/(2*e) - (d^(3/2)*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*e^(3/2)) - ( b*n*((d*x*Sqrt[1 + (e*x^2)/d])/(2*e) + (d^(3/2)*ArcSinh[(Sqrt[e]*x)/Sqrt[d ]])/(2*e^(3/2)) + (d^(3/2)*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(2*e^(3/2)) - ( d^(3/2)*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt [d]])])/e^(3/2) - (d^(3/2)*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])]) /(2*e^(3/2))))/2))/Sqrt[d + e*x^2]
3.3.81.3.1 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_.)*((d_) + (e_.)*(x_)^2)^ (q_), x_Symbol] :> Simp[d^IntPart[q]*((d + e*x^2)^FracPart[q]/(1 + (e/d)*x^ 2)^FracPart[q]) Int[x^m*(1 + (e/d)*x^2)^q*(a + b*Log[c*x^n]), x], x] /; F reeQ[{a, b, c, d, e, n}, x] && IntegerQ[m/2] && IntegerQ[q - 1/2] && !(LtQ [m + 2*q, -2] || GtQ[d, 0])
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])
\[\int \frac {x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\sqrt {e \,x^{2}+d}}d x\]
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{\sqrt {e x^{2} + d}} \,d x } \]
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]
Exception generated. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{\sqrt {e x^{2} + d}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]