Integrand size = 25, antiderivative size = 400 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {4 b e n \sqrt {d+e x^2}}{3 x}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 x^3}+\frac {4 b e^{3/2} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 \sqrt {d} \sqrt {1+\frac {e x^2}{d}}}+\frac {b e^{3/2} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {d} \sqrt {1+\frac {e x^2}{d}}}-\frac {b e^{3/2} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {d} \sqrt {1+\frac {e x^2}{d}}}-\frac {e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac {e^{3/2} \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {1+\frac {e x^2}{d}}}-\frac {b e^{3/2} n \sqrt {d+e x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {d} \sqrt {1+\frac {e x^2}{d}}} \]
-1/9*b*n*(e*x^2+d)^(3/2)/x^3-1/3*(e*x^2+d)^(3/2)*(a+b*ln(c*x^n))/x^3-4/3*b *e*n*(e*x^2+d)^(1/2)/x-e*(a+b*ln(c*x^n))*(e*x^2+d)^(1/2)/x+4/3*b*e^(3/2)*n *arcsinh(x*e^(1/2)/d^(1/2))*(e*x^2+d)^(1/2)/d^(1/2)/(1+e*x^2/d)^(1/2)+1/2* b*e^(3/2)*n*arcsinh(x*e^(1/2)/d^(1/2))^2*(e*x^2+d)^(1/2)/d^(1/2)/(1+e*x^2/ d)^(1/2)-b*e^(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)*(e*x^2+d)^(1/2)/d^(1/2)/(1+e*x^2/d)^(1/2)+e^(3/2)*arcs inh(x*e^(1/2)/d^(1/2))*(a+b*ln(c*x^n))*(e*x^2+d)^(1/2)/d^(1/2)/(1+e*x^2/d) ^(1/2)-1/2*b*e^(3/2)*n*polylog(2,(x*e^(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)* (e*x^2+d)^(1/2)/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 = 0.50 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.67 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\frac {b d n \sqrt {d+e x^2} \left (-\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {e x^2}{d}\right )-3 \left (1+\frac {e x^2}{d}\right )^{3/2} \log (x)\right )}{9 x^3 \sqrt {1+\frac {e x^2}{d}}}+\frac {b e n \sqrt {d+e x^2} \left (-\, _3F_2\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2};\frac {1}{2},\frac {1}{2};-\frac {e x^2}{d}\right )-\sqrt {1+\frac {e x^2}{d}} \log (x)+\frac {\sqrt {e} x \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)}{\sqrt {d}}\right )}{x \sqrt {1+\frac {e x^2}{d}}}-\frac {\sqrt {d+e x^2} \left (d+4 e x^2\right ) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{3 x^3}+e^{3/2} \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right ) \]
(b*d*n*Sqrt[d + e*x^2]*(-Hypergeometric2F1[-3/2, -3/2, -1/2, -((e*x^2)/d)] - 3*(1 + (e*x^2)/d)^(3/2)*Log[x]))/(9*x^3*Sqrt[1 + (e*x^2)/d]) + (b*e*n*S qrt[d + e*x^2]*(-HypergeometricPFQ[{-1/2, -1/2, -1/2}, {1/2, 1/2}, -((e*x^ 2)/d)] - Sqrt[1 + (e*x^2)/d]*Log[x] + (Sqrt[e]*x*ArcSinh[(Sqrt[e]*x)/Sqrt[ d]]*Log[x])/Sqrt[d]))/(x*Sqrt[1 + (e*x^2)/d]) - (Sqrt[d + e*x^2]*(d + 4*e* x^2)*(a - b*n*Log[x] + b*Log[c*x^n]))/(3*x^3) + e^(3/2)*(a - b*n*Log[x] + b*Log[c*x^n])*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]]
Time = 0.68 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.78, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2786, 2792, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 2786 |
| \(\displaystyle \frac {d \sqrt {d+e x^2} \int \frac {\left (\frac {e x^2}{d}+1\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^4}dx}{\sqrt {\frac {e x^2}{d}+1}}\) |
| \(\Big \downarrow \) 2792 |
| \(\displaystyle \frac {d \sqrt {d+e x^2} \left (-b n \int \left (\frac {e^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} x}-\frac {\left (4 e x^2+d\right ) \sqrt {\frac {e x^2}{d}+1}}{3 d x^4}\right )dx+\frac {e^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac {e \sqrt {\frac {e x^2}{d}+1} \left (a+b \log \left (c x^n\right )\right )}{d x}-\frac {\left (\frac {e x^2}{d}+1\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\right )}{\sqrt {\frac {e x^2}{d}+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {d+e x^2} \left (\frac {e^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac {e \sqrt {\frac {e x^2}{d}+1} \left (a+b \log \left (c x^n\right )\right )}{d x}-\frac {\left (\frac {e x^2}{d}+1\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-b n \left (\frac {e^{3/2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 d^{3/2}}-\frac {e^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 d^{3/2}}-\frac {4 e^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}+\frac {e^{3/2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{d^{3/2}}+\frac {4 e \sqrt {\frac {e x^2}{d}+1}}{3 d x}+\frac {\left (\frac {e x^2}{d}+1\right )^{3/2}}{9 x^3}\right )\right )}{\sqrt {\frac {e x^2}{d}+1}}\) |
(d*Sqrt[d + e*x^2]*(-((e*Sqrt[1 + (e*x^2)/d]*(a + b*Log[c*x^n]))/(d*x)) - ((1 + (e*x^2)/d)^(3/2)*(a + b*Log[c*x^n]))/(3*x^3) + (e^(3/2)*ArcSinh[(Sqr t[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/d^(3/2) - b*n*((4*e*Sqrt[1 + (e*x^2)/ d])/(3*d*x) + (1 + (e*x^2)/d)^(3/2)/(9*x^3) - (4*e^(3/2)*ArcSinh[(Sqrt[e]* x)/Sqrt[d]])/(3*d^(3/2)) - (e^(3/2)*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(2*d^( 3/2)) + (e^(3/2)*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e ]*x)/Sqrt[d]])])/d^(3/2) + (e^(3/2)*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sq rt[d]])])/(2*d^(3/2)))))/Sqrt[1 + (e*x^2)/d]
3.3.71.3.1 Defintions of rubi rules used
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])
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{4}}d x\]
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{4}} \,d x } \]
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^4, x)
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \]