Integrand size = 23, antiderivative size = 253 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^{3/2}} \, dx=-\frac {b n \sqrt {d+e x}}{d^2 x}-\frac {5 b e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{5/2}}-\frac {3 b e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{5/2}}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x}}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x}}+\frac {3 e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}+\frac {6 b e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{5/2}}+\frac {3 b e n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{5/2}} \]
-5*b*e*n*arctanh((e*x+d)^(1/2)/d^(1/2))/d^(5/2)-3*b*e*n*arctanh((e*x+d)^(1 /2)/d^(1/2))^2/d^(5/2)+3*e*arctanh((e*x+d)^(1/2)/d^(1/2))*(a+b*ln(c*x^n))/ d^(5/2)+6*b*e*n*arctanh((e*x+d)^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)-(e*x+ d)^(1/2)))/d^(5/2)+3*b*e*n*polylog(2,1-2*d^(1/2)/(d^(1/2)-(e*x+d)^(1/2)))/ d^(5/2)-3*e*(a+b*ln(c*x^n))/d^2/(e*x+d)^(1/2)+(-a-b*ln(c*x^n))/d/x/(e*x+d) ^(1/2)-b*n*(e*x+d)^(1/2)/d^2/x
Time = 0.30 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.00 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^{3/2}} \, dx=2 e \left (-\frac {2 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {b n \left (\frac {1}{\sqrt {d}-\sqrt {d+e x}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}\right )}{4 d^2}-\frac {b n \left (\frac {1}{\sqrt {d}+\sqrt {d+e x}}+\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}\right )}{4 d^2}-\frac {a+b \log \left (c x^n\right )}{d^2 \sqrt {d+e x}}+\frac {a+b \log \left (c x^n\right )}{4 d^2 \left (\sqrt {d}-\sqrt {d+e x}\right )}-\frac {a+b \log \left (c x^n\right )}{4 d^2 \left (\sqrt {d}+\sqrt {d+e x}\right )}-\frac {3 \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )}{4 d^{5/2}}+\frac {3 \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )}{4 d^{5/2}}+\frac {3 b n \left (\log ^2\left (\sqrt {d}-\sqrt {d+e x}\right )+2 \log \left (\sqrt {d}-\sqrt {d+e x}\right ) \log \left (\frac {\sqrt {d}+\sqrt {d+e x}}{2 \sqrt {d}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {d}-\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )}{8 d^{5/2}}-\frac {3 b n \left (2 \log \left (\frac {\sqrt {d}-\sqrt {d+e x}}{2 \sqrt {d}}\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )+\log ^2\left (\sqrt {d}+\sqrt {d+e x}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {d}+\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )}{8 d^{5/2}}\right ) \]
2*e*((-2*b*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(5/2) + (b*n*((Sqrt[d] - Sq rt[d + e*x])^(-1) - ArcTanh[Sqrt[d + e*x]/Sqrt[d]]/Sqrt[d]))/(4*d^2) - (b* n*((Sqrt[d] + Sqrt[d + e*x])^(-1) + ArcTanh[Sqrt[d + e*x]/Sqrt[d]]/Sqrt[d] ))/(4*d^2) - (a + b*Log[c*x^n])/(d^2*Sqrt[d + e*x]) + (a + b*Log[c*x^n])/( 4*d^2*(Sqrt[d] - Sqrt[d + e*x])) - (a + b*Log[c*x^n])/(4*d^2*(Sqrt[d] + Sq rt[d + e*x])) - (3*(a + b*Log[c*x^n])*Log[Sqrt[d] - Sqrt[d + e*x]])/(4*d^( 5/2)) + (3*(a + b*Log[c*x^n])*Log[Sqrt[d] + Sqrt[d + e*x]])/(4*d^(5/2)) + (3*b*n*(Log[Sqrt[d] - Sqrt[d + e*x]]^2 + 2*Log[Sqrt[d] - Sqrt[d + e*x]]*Lo g[(Sqrt[d] + Sqrt[d + e*x])/(2*Sqrt[d])] + 2*PolyLog[2, (Sqrt[d] - Sqrt[d + e*x])/(2*Sqrt[d])]))/(8*d^(5/2)) - (3*b*n*(2*Log[(Sqrt[d] - Sqrt[d + e*x ])/(2*Sqrt[d])]*Log[Sqrt[d] + Sqrt[d + e*x]] + Log[Sqrt[d] + Sqrt[d + e*x] ]^2 + 2*PolyLog[2, (Sqrt[d] + Sqrt[d + e*x])/(2*Sqrt[d])]))/(8*d^(5/2)))
Time = 0.71 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2792, 25, 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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2792 |
| \(\displaystyle -b n \int -\frac {\frac {\sqrt {d} (d+3 e x)}{\sqrt {d+e x}}-3 e x \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{5/2} x^2}dx+\frac {3 e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x}}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b n \int \frac {\frac {\sqrt {d} (d+3 e x)}{\sqrt {d+e x}}-3 e x \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{5/2} x^2}dx+\frac {3 e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x}}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b n \int \frac {\frac {\sqrt {d} (d+3 e x)}{\sqrt {d+e x}}-3 e x \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x^2}dx}{d^{5/2}}+\frac {3 e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x}}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {b n \int \left (-\frac {2 e^2}{\sqrt {d} \sqrt {d+e x}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) e}{x}+\frac {2 \sqrt {d+e x} e}{\sqrt {d} x}+\frac {\sqrt {d} \sqrt {d+e x}}{x^2}\right )dx}{d^{5/2}}+\frac {3 e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x}}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{5/2}}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt {d+e x}}-\frac {a+b \log \left (c x^n\right )}{d x \sqrt {d+e x}}+\frac {b n \left (-3 e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-5 e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+6 e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )+3 e \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-\frac {\sqrt {d} \sqrt {d+e x}}{x}\right )}{d^{5/2}}\) |
(-3*e*(a + b*Log[c*x^n]))/(d^2*Sqrt[d + e*x]) - (a + b*Log[c*x^n])/(d*x*Sq rt[d + e*x]) + (3*e*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]))/d^( 5/2) + (b*n*(-((Sqrt[d]*Sqrt[d + e*x])/x) - 5*e*ArcTanh[Sqrt[d + e*x]/Sqrt [d]] - 3*e*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2 + 6*e*ArcTanh[Sqrt[d + e*x]/Sq rt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])] + 3*e*PolyLog[2, 1 - (2* Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])]))/d^(5/2)
3.2.56.3.1 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])
\[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{2} \left (e x +d \right )^{\frac {3}{2}}}d x\]
\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^{3/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^{3/2}} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{x^{2} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^{3/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
-1/2*a*(2*(3*(e*x + d)*e - 2*d*e)/((e*x + d)^(3/2)*d^2 - sqrt(e*x + d)*d^3 ) + 3*e*log((sqrt(e*x + d) - sqrt(d))/(sqrt(e*x + d) + sqrt(d)))/d^(5/2)) + b*integrate((log(c) + log(x^n))/((e*x^3 + d*x^2)*sqrt(e*x + d)), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^{3/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^{3/2}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,{\left (d+e\,x\right )}^{3/2}} \,d x \]