Integrand size = 25, antiderivative size = 328 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{3/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{3/2} \sqrt {d+e x^2}} \]
-x*(a+b*ln(c*x^n))/e/(e*x^2+d)^(1/2)+b*n*arcsinh(x*e^(1/2)/d^(1/2))*d^(1/2 )*(1+e*x^2/d)^(1/2)/e^(3/2)/(e*x^2+d)^(1/2)+1/2*b*n*arcsinh(x*e^(1/2)/d^(1 /2))^2*d^(1/2)*(1+e*x^2/d)^(1/2)/e^(3/2)/(e*x^2+d)^(1/2)-b*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)*d^(1/2)*(1+e*x ^2/d)^(1/2)/e^(3/2)/(e*x^2+d)^(1/2)+arcsinh(x*e^(1/2)/d^(1/2))*(a+b*ln(c*x ^n))*d^(1/2)*(1+e*x^2/d)^(1/2)/e^(3/2)/(e*x^2+d)^(1/2)-1/2*b*n*polylog(2,( x*e^(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*d^(1/2)*(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 = 0.32 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.66 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {b n \sqrt {1+\frac {e x^2}{d}} \left (e^{3/2} x^3 \left (d+e x^2\right ) \, _3F_2\left (\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {5}{2},\frac {5}{2};-\frac {e x^2}{d}\right )+9 d^2 \sqrt {e} x \sqrt {1+\frac {e x^2}{d}} \log (x)-9 d^{3/2} \left (d+e x^2\right ) \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)\right )}{9 d e^{3/2} \left (d+e x^2\right )^{3/2}}-\frac {x \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{e \sqrt {d+e x^2}}+\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{e^{3/2}} \]
-1/9*(b*n*Sqrt[1 + (e*x^2)/d]*(e^(3/2)*x^3*(d + e*x^2)*HypergeometricPFQ[{ 3/2, 3/2, 3/2}, {5/2, 5/2}, -((e*x^2)/d)] + 9*d^2*Sqrt[e]*x*Sqrt[1 + (e*x^ 2)/d]*Log[x] - 9*d^(3/2)*(d + e*x^2)*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[x])) /(d*e^(3/2)*(d + e*x^2)^(3/2)) - (x*(a - b*n*Log[x] + b*Log[c*x^n]))/(e*Sq rt[d + e*x^2]) + ((a - b*n*Log[x] + b*Log[c*x^n])*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/e^(3/2)
Time = 0.73 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.71, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2786, 2792, 25, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2786 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (\frac {e x^2}{d}+1\right )^{3/2}}dx}{d \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 2792 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (-b n \int -\frac {\frac {d x}{e \sqrt {\frac {e x^2}{d}+1}}-\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}}{x}dx+\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2}}-\frac {d x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {\frac {e x^2}{d}+1}}\right )}{d \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (b n \int \frac {\frac {d x}{e \sqrt {\frac {e x^2}{d}+1}}-\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}}{x}dx+\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2}}-\frac {d x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {\frac {e x^2}{d}+1}}\right )}{d \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (b n \int \left (\frac {d^2 \sqrt {\frac {e x^2}{d}+1}}{e \left (e x^2+d\right )}-\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} x}\right )dx+\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2}}-\frac {d x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {\frac {e x^2}{d}+1}}\right )}{d \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2}}-\frac {d x \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {\frac {e x^2}{d}+1}}+b n \left (-\frac {d^{3/2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{3/2}}+\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{3/2}}+\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}-\frac {d^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{e^{3/2}}\right )\right )}{d \sqrt {d+e x^2}}\) |
(Sqrt[1 + (e*x^2)/d]*(-((d*x*(a + b*Log[c*x^n]))/(e*Sqrt[1 + (e*x^2)/d])) + (d^(3/2)*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/e^(3/2) + b*n* ((d^(3/2)*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/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)))))/(d*Sqrt[d + e*x^2])
3.3.92.3.1 Defintions of rubi rules used
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])
\[\int \frac {x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
integral((sqrt(e*x^2 + d)*b*x^2*log(c*x^n) + sqrt(e*x^2 + d)*a*x^2)/(e^2*x ^4 + 2*d*e*x^2 + d^2), x)
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]