Integrand size = 25, antiderivative size = 400 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {b d n \sqrt {d+e x^2}}{x}-\frac {1}{4} b e n x \sqrt {d+e x^2}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {1+\frac {e x^2}{d}}}+\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {1+\frac {e x^2}{d}}}-\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {1+\frac {e x^2}{d}}}+\frac {3}{2} e x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 \sqrt {d} \sqrt {e} \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {1+\frac {e x^2}{d}}}-\frac {3 b \sqrt {d} \sqrt {e} n \sqrt {d+e x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 \sqrt {1+\frac {e x^2}{d}}} \]
-(e*x^2+d)^(3/2)*(a+b*ln(c*x^n))/x-b*d*n*(e*x^2+d)^(1/2)/x-1/4*b*e*n*x*(e* x^2+d)^(1/2)+3/2*e*x*(a+b*ln(c*x^n))*(e*x^2+d)^(1/2)+3/4*b*n*arcsinh(x*e^( 1/2)/d^(1/2))*d^(1/2)*e^(1/2)*(e*x^2+d)^(1/2)/(1+e*x^2/d)^(1/2)+3/4*b*n*ar csinh(x*e^(1/2)/d^(1/2))^2*d^(1/2)*e^(1/2)*(e*x^2+d)^(1/2)/(1+e*x^2/d)^(1/ 2)-3/2*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)*e^(1/2)*(e*x^2+d)^(1/2)/(1+e*x^2/d)^(1/2)+3/2*arcsinh(x* e^(1/2)/d^(1/2))*(a+b*ln(c*x^n))*d^(1/2)*e^(1/2)*(e*x^2+d)^(1/2)/(1+e*x^2/ d)^(1/2)-3/4*b*n*polylog(2,(x*e^(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*d^(1/2 )*e^(1/2)*(e*x^2+d)^(1/2)/(1+e*x^2/d)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.63 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.82 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {b \sqrt {d} n \sqrt {d+e x^2} \left (\sqrt {d} \, _3F_2\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2};\frac {1}{2},\frac {1}{2};-\frac {e x^2}{d}\right )+\left (\sqrt {d} \sqrt {1+\frac {e x^2}{d}}-\sqrt {e} x \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right ) \log (x)\right )}{x \sqrt {1+\frac {e x^2}{d}}}+\frac {b \sqrt {e} n \sqrt {d+e x^2} \left (-2 \sqrt {e} x \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {e x^2}{d}\right )+\left (\sqrt {e} x \sqrt {1+\frac {e x^2}{d}}+\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right ) (-1+2 \log (x))\right )}{4 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (2 d-e x^2\right ) \sqrt {d+e x^2} \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{2 x}+\frac {3}{2} d \sqrt {e} \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right ) \]
-((b*Sqrt[d]*n*Sqrt[d + e*x^2]*(Sqrt[d]*HypergeometricPFQ[{-1/2, -1/2, -1/ 2}, {1/2, 1/2}, -((e*x^2)/d)] + (Sqrt[d]*Sqrt[1 + (e*x^2)/d] - Sqrt[e]*x*A rcSinh[(Sqrt[e]*x)/Sqrt[d]])*Log[x]))/(x*Sqrt[1 + (e*x^2)/d])) + (b*Sqrt[e ]*n*Sqrt[d + e*x^2]*(-2*Sqrt[e]*x*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, -((e*x^2)/d)] + (Sqrt[e]*x*Sqrt[1 + (e*x^2)/d] + Sqrt[d]*ArcSinh[(S qrt[e]*x)/Sqrt[d]])*(-1 + 2*Log[x])))/(4*Sqrt[1 + (e*x^2)/d]) - ((2*d - e* x^2)*Sqrt[d + e*x^2]*(a - b*n*Log[x] + b*Log[c*x^n]))/(2*x) + (3*d*Sqrt[e] *(a - b*n*Log[x] + b*Log[c*x^n])*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/2
Time = 0.73 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.78, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 2786 |
| \(\displaystyle \frac {d \sqrt {d+e x^2} \int \frac {\left (\frac {e x^2}{d}+1\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2}dx}{\sqrt {\frac {e x^2}{d}+1}}\) |
| \(\Big \downarrow \) 2792 |
| \(\displaystyle \frac {d \sqrt {d+e x^2} \left (-b n \int -\frac {\left (2 d-e x^2\right ) \sqrt {\frac {e x^2}{d}+1}-3 \sqrt {d} \sqrt {e} x \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d x^2}dx+\frac {3 \sqrt {e} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {d}}-\frac {\left (\frac {e x^2}{d}+1\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 e x \sqrt {\frac {e x^2}{d}+1} \left (a+b \log \left (c x^n\right )\right )}{2 d}\right )}{\sqrt {\frac {e x^2}{d}+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \sqrt {d+e x^2} \left (\frac {b n \int \frac {\left (2 d-e x^2\right ) \sqrt {\frac {e x^2}{d}+1}-3 \sqrt {d} \sqrt {e} x \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x^2}dx}{2 d}+\frac {3 \sqrt {e} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {d}}-\frac {\left (\frac {e x^2}{d}+1\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 e x \sqrt {\frac {e x^2}{d}+1} \left (a+b \log \left (c x^n\right )\right )}{2 d}\right )}{\sqrt {\frac {e x^2}{d}+1}}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {d \sqrt {d+e x^2} \left (\frac {b n \int \left (\frac {2 \sqrt {\frac {e x^2}{d}+1} d}{x^2}-\frac {3 \sqrt {e} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \sqrt {d}}{x}-e \sqrt {\frac {e x^2}{d}+1}\right )dx}{2 d}+\frac {3 \sqrt {e} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {d}}-\frac {\left (\frac {e x^2}{d}+1\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 e x \sqrt {\frac {e x^2}{d}+1} \left (a+b \log \left (c x^n\right )\right )}{2 d}\right )}{\sqrt {\frac {e x^2}{d}+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {d+e x^2} \left (\frac {3 \sqrt {e} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {d}}-\frac {\left (\frac {e x^2}{d}+1\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 e x \sqrt {\frac {e x^2}{d}+1} \left (a+b \log \left (c x^n\right )\right )}{2 d}+\frac {b n \left (-\frac {3}{2} \sqrt {d} \sqrt {e} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )+\frac {3}{2} \sqrt {d} \sqrt {e} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2+\frac {3}{2} \sqrt {d} \sqrt {e} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-3 \sqrt {d} \sqrt {e} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )-\frac {1}{2} e x \sqrt {\frac {e x^2}{d}+1}-\frac {2 d \sqrt {\frac {e x^2}{d}+1}}{x}\right )}{2 d}\right )}{\sqrt {\frac {e x^2}{d}+1}}\) |
(d*Sqrt[d + e*x^2]*((3*e*x*Sqrt[1 + (e*x^2)/d]*(a + b*Log[c*x^n]))/(2*d) - ((1 + (e*x^2)/d)^(3/2)*(a + b*Log[c*x^n]))/x + (3*Sqrt[e]*ArcSinh[(Sqrt[e ]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*Sqrt[d]) + (b*n*((-2*d*Sqrt[1 + (e*x^ 2)/d])/x - (e*x*Sqrt[1 + (e*x^2)/d])/2 + (3*Sqrt[d]*Sqrt[e]*ArcSinh[(Sqrt[ e]*x)/Sqrt[d]])/2 + (3*Sqrt[d]*Sqrt[e]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/2 - 3*Sqrt[d]*Sqrt[e]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt [e]*x)/Sqrt[d]])] - (3*Sqrt[d]*Sqrt[e]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x) /Sqrt[d]])])/2))/(2*d)))/Sqrt[1 + (e*x^2)/d]
3.3.70.3.1 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])
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{2}}d x\]
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{2}} \,d x } \]
integral(((b*e*x^2 + b*d)*sqrt(e*x^2 + d)*log(c*x^n) + (a*e*x^2 + a*d)*sqr t(e*x^2 + d))/x^2, x)
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \]