Integrand size = 25, antiderivative size = 295 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{4} b \sqrt {d} e n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+\frac {3}{4} b \sqrt {d} e n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3}{2} \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{2} b \sqrt {d} e n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )-\frac {3}{4} b \sqrt {d} e n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right ) \]
-1/2*(e*x^2+d)^(3/2)*(a+b*ln(c*x^n))/x^2+3/4*b*e*n*arctanh((e*x^2+d)^(1/2) /d^(1/2))*d^(1/2)+3/4*b*e*n*arctanh((e*x^2+d)^(1/2)/d^(1/2))^2*d^(1/2)-3/2 *e*arctanh((e*x^2+d)^(1/2)/d^(1/2))*(a+b*ln(c*x^n))*d^(1/2)-3/2*b*e*n*arct anh((e*x^2+d)^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)-(e*x^2+d)^(1/2)))*d^(1/ 2)-3/4*b*e*n*polylog(2,1-2*d^(1/2)/(d^(1/2)-(e*x^2+d)^(1/2)))*d^(1/2)-b*e* n*(e*x^2+d)^(1/2)-1/4*b*d*n*(e*x^2+d)^(1/2)/x^2+3/2*e*(a+b*ln(c*x^n))*(e*x ^2+d)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.59 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.18 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {b e n \sqrt {d+e x^2} \left (-\, _3F_2\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2};\frac {1}{2},\frac {1}{2};-\frac {d}{e x^2}\right )+\sqrt {1+\frac {d}{e x^2}} \log (x)-\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {d}}{\sqrt {e} x}\right ) \log (x)}{\sqrt {e} x}\right )}{\sqrt {1+\frac {d}{e x^2}}}-\frac {b \sqrt {d} n \sqrt {d+e x^2} \left (2 \sqrt {d} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {d}{e x^2}\right )+\left (\sqrt {d} \sqrt {1+\frac {d}{e x^2}}+\sqrt {e} x \text {arcsinh}\left (\frac {\sqrt {d}}{\sqrt {e} x}\right )\right ) (1+2 \log (x))\right )}{4 \sqrt {1+\frac {d}{e x^2}} x^2}-\frac {\left (d-2 e x^2\right ) \sqrt {d+e x^2} \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {3}{2} \sqrt {d} e \log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )-\frac {3}{2} \sqrt {d} e \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right ) \]
(b*e*n*Sqrt[d + e*x^2]*(-HypergeometricPFQ[{-1/2, -1/2, -1/2}, {1/2, 1/2}, -(d/(e*x^2))] + Sqrt[1 + d/(e*x^2)]*Log[x] - (Sqrt[d]*ArcSinh[Sqrt[d]/(Sq rt[e]*x)]*Log[x])/(Sqrt[e]*x)))/Sqrt[1 + d/(e*x^2)] - (b*Sqrt[d]*n*Sqrt[d + e*x^2]*(2*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, -(d/(e* x^2))] + (Sqrt[d]*Sqrt[1 + d/(e*x^2)] + Sqrt[e]*x*ArcSinh[Sqrt[d]/(Sqrt[e] *x)])*(1 + 2*Log[x])))/(4*Sqrt[1 + d/(e*x^2)]*x^2) - ((d - 2*e*x^2)*Sqrt[d + e*x^2]*(a - b*n*Log[x] + b*Log[c*x^n]))/(2*x^2) + (3*Sqrt[d]*e*Log[x]*( a - b*n*Log[x] + b*Log[c*x^n]))/2 - (3*Sqrt[d]*e*(a - b*n*Log[x] + b*Log[c *x^n])*Log[d + Sqrt[d]*Sqrt[d + e*x^2]])/2
Time = 0.75 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 2792 |
| \(\displaystyle -b n \int -\frac {3 \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {e x^2+d}}{\sqrt {d}}\right ) x^2+\left (d-2 e x^2\right ) \sqrt {e x^2+d}}{2 x^3}dx-\frac {3}{2} \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} b n \int \frac {3 \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {e x^2+d}}{\sqrt {d}}\right ) x^2+\left (d-2 e x^2\right ) \sqrt {e x^2+d}}{x^3}dx-\frac {3}{2} \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {1}{2} b n \int \left (\frac {\sqrt {e x^2+d} d}{x^3}+\frac {3 e \text {arctanh}\left (\frac {\sqrt {e x^2+d}}{\sqrt {d}}\right ) \sqrt {d}}{x}-\frac {2 e \sqrt {e x^2+d}}{x}\right )dx-\frac {3}{2} \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{2} \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} b n \left (\frac {3}{2} \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{2} \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )-3 \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )-\frac {3}{2} \sqrt {d} e \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {e x^2+d}}\right )-2 e \sqrt {d+e x^2}-\frac {d \sqrt {d+e x^2}}{2 x^2}\right )\) |
(3*e*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/2 - ((d + e*x^2)^(3/2)*(a + b*Log [c*x^n]))/(2*x^2) - (3*Sqrt[d]*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*(a + b*L og[c*x^n]))/2 + (b*n*(-2*e*Sqrt[d + e*x^2] - (d*Sqrt[d + e*x^2])/(2*x^2) + (3*Sqrt[d]*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/2 + (3*Sqrt[d]*e*ArcTanh[S qrt[d + e*x^2]/Sqrt[d]]^2)/2 - 3*Sqrt[d]*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d] ]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])] - (3*Sqrt[d]*e*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/2))/2
3.3.67.3.1 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_.))*((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])
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{3}}d x\]
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}} \,d x } \]
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^3, x)
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \]