Integrand size = 23, antiderivative size = 298 \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {b n \sqrt {d+e x}}{4 x^2}-\frac {3 b e n \sqrt {d+e x}}{8 d x}-\frac {b e^2 n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{8 d^{3/2}}-\frac {b e^2 n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{4 d^{3/2}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}+\frac {e^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}+\frac {b e^2 n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{2 d^{3/2}}+\frac {b e^2 n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{4 d^{3/2}} \]
-1/8*b*e^2*n*arctanh((e*x+d)^(1/2)/d^(1/2))/d^(3/2)-1/4*b*e^2*n*arctanh((e *x+d)^(1/2)/d^(1/2))^2/d^(3/2)+1/4*e^2*arctanh((e*x+d)^(1/2)/d^(1/2))*(a+b *ln(c*x^n))/d^(3/2)+1/2*b*e^2*n*arctanh((e*x+d)^(1/2)/d^(1/2))*ln(2*d^(1/2 )/(d^(1/2)-(e*x+d)^(1/2)))/d^(3/2)+1/4*b*e^2*n*polylog(2,1-2*d^(1/2)/(d^(1 /2)-(e*x+d)^(1/2)))/d^(3/2)-1/4*b*n*(e*x+d)^(1/2)/x^2-3/8*b*e*n*(e*x+d)^(1 /2)/d/x-1/2*(a+b*ln(c*x^n))*(e*x+d)^(1/2)/x^2-1/4*e*(a+b*ln(c*x^n))*(e*x+d )^(1/2)/d/x
Time = 0.34 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.68 \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {8 a d^{3/2} \sqrt {d+e x}+4 b d^{3/2} n \sqrt {d+e x}+4 a \sqrt {d} e x \sqrt {d+e x}+6 b \sqrt {d} e n x \sqrt {d+e x}+2 b e^2 n x^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+8 b d^{3/2} \sqrt {d+e x} \log \left (c x^n\right )+4 b \sqrt {d} e x \sqrt {d+e x} \log \left (c x^n\right )+2 a e^2 x^2 \log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 b e^2 x^2 \log \left (c x^n\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )-b e^2 n x^2 \log ^2\left (\sqrt {d}-\sqrt {d+e x}\right )-2 a e^2 x^2 \log \left (\sqrt {d}+\sqrt {d+e x}\right )-2 b e^2 x^2 \log \left (c x^n\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )+b e^2 n x^2 \log ^2\left (\sqrt {d}+\sqrt {d+e x}\right )+2 b e^2 n x^2 \log \left (\sqrt {d}+\sqrt {d+e x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )-2 b e^2 n x^2 \log \left (\sqrt {d}-\sqrt {d+e x}\right ) \log \left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )-2 b e^2 n x^2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )+2 b e^2 n x^2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{16 d^{3/2} x^2} \]
-1/16*(8*a*d^(3/2)*Sqrt[d + e*x] + 4*b*d^(3/2)*n*Sqrt[d + e*x] + 4*a*Sqrt[ d]*e*x*Sqrt[d + e*x] + 6*b*Sqrt[d]*e*n*x*Sqrt[d + e*x] + 2*b*e^2*n*x^2*Arc Tanh[Sqrt[d + e*x]/Sqrt[d]] + 8*b*d^(3/2)*Sqrt[d + e*x]*Log[c*x^n] + 4*b*S qrt[d]*e*x*Sqrt[d + e*x]*Log[c*x^n] + 2*a*e^2*x^2*Log[Sqrt[d] - Sqrt[d + e *x]] + 2*b*e^2*x^2*Log[c*x^n]*Log[Sqrt[d] - Sqrt[d + e*x]] - b*e^2*n*x^2*L og[Sqrt[d] - Sqrt[d + e*x]]^2 - 2*a*e^2*x^2*Log[Sqrt[d] + Sqrt[d + e*x]] - 2*b*e^2*x^2*Log[c*x^n]*Log[Sqrt[d] + Sqrt[d + e*x]] + b*e^2*n*x^2*Log[Sqr t[d] + Sqrt[d + e*x]]^2 + 2*b*e^2*n*x^2*Log[Sqrt[d] + Sqrt[d + e*x]]*Log[1 /2 - Sqrt[d + e*x]/(2*Sqrt[d])] - 2*b*e^2*n*x^2*Log[Sqrt[d] - Sqrt[d + e*x ]]*Log[(1 + Sqrt[d + e*x]/Sqrt[d])/2] - 2*b*e^2*n*x^2*PolyLog[2, 1/2 - Sqr t[d + e*x]/(2*Sqrt[d])] + 2*b*e^2*n*x^2*PolyLog[2, (1 + Sqrt[d + e*x]/Sqrt [d])/2])/(d^(3/2)*x^2)
Time = 0.62 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, 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.
| \(\displaystyle \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 2792 |
| \(\displaystyle -b n \int -\frac {\sqrt {d} \sqrt {d+e x} (2 d+e x)-e^2 x^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 d^{3/2} x^3}dx+\frac {e^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b n \int \frac {\sqrt {d} \sqrt {d+e x} (2 d+e x)-e^2 x^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x^3}dx}{4 d^{3/2}}+\frac {e^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {b n \int \left (-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) e^2}{x}+\frac {\sqrt {d} \sqrt {d+e x} e}{x^2}+\frac {2 d^{3/2} \sqrt {d+e x}}{x^3}\right )dx}{4 d^{3/2}}+\frac {e^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^{3/2}}-\frac {e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{4 d x}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {b n \left (e^2 \left (-\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2\right )-\frac {1}{2} e^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 e^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-\frac {d^{3/2} \sqrt {d+e x}}{x^2}+e^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-\frac {3 \sqrt {d} e \sqrt {d+e x}}{2 x}\right )}{4 d^{3/2}}\) |
-1/2*(Sqrt[d + e*x]*(a + b*Log[c*x^n]))/x^2 - (e*Sqrt[d + e*x]*(a + b*Log[ c*x^n]))/(4*d*x) + (e^2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n])) /(4*d^(3/2)) + (b*n*(-((d^(3/2)*Sqrt[d + e*x])/x^2) - (3*Sqrt[d]*e*Sqrt[d + e*x])/(2*x) - (e^2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/2 - e^2*ArcTanh[Sqrt[ d + e*x]/Sqrt[d]]^2 + 2*e^2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d]) /(Sqrt[d] - Sqrt[d + e*x])] + e^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sq rt[d + e*x])]))/(4*d^(3/2))
3.2.36.3.1 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])
\[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e x +d}}{x^{3}}d x\]
\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}} \,d x } \]
\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x}}{x^{3}}\, dx \]
\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}} \,d x } \]
-1/8*(e^2*log((sqrt(e*x + d) - sqrt(d))/(sqrt(e*x + d) + sqrt(d)))/d^(3/2) + 2*((e*x + d)^(3/2)*e^2 + sqrt(e*x + d)*d*e^2)/((e*x + d)^2*d - 2*(e*x + d)*d^2 + d^3))*a + b*integrate(sqrt(e*x + d)*(log(c) + log(x^n))/x^3, x)
\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\left (a+b\,\ln \left (c\,x^n\right )\right )\,\sqrt {d+e\,x}}{x^3} \,d x \]