Integrand size = 25, antiderivative size = 284 \[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {16 b d n \sqrt {d+e x^r}}{3 r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}+\frac {16 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{3 r^2}+\frac {2 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}+\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{r^2}-\frac {2 b d^{3/2} n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{r^2} \]
-4/9*b*n*(d+e*x^r)^(3/2)/r^2+16/3*b*d^(3/2)*n*arctanh((d+e*x^r)^(1/2)/d^(1 /2))/r^2+2*b*d^(3/2)*n*arctanh((d+e*x^r)^(1/2)/d^(1/2))^2/r^2-4*b*d^(3/2)* n*arctanh((d+e*x^r)^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)-(d+e*x^r)^(1/2))) /r^2-2*b*d^(3/2)*n*polylog(2,1-2*d^(1/2)/(d^(1/2)-(d+e*x^r)^(1/2)))/r^2-16 /3*b*d*n*(d+e*x^r)^(1/2)/r^2+2/3*(a+b*ln(c*x^n))*((d+e*x^r)^(3/2)/r-3*d^(3 /2)*arctanh((d+e*x^r)^(1/2)/d^(1/2))/r+3*d*(d+e*x^r)^(1/2)/r)
\[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx \]
Time = 0.68 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2790, 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.
\(\displaystyle \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 2790 |
\(\displaystyle \frac {2}{3} \left (-\frac {3 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}+\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \left (-\frac {2 \text {arctanh}\left (\frac {\sqrt {e x^r+d}}{\sqrt {d}}\right ) d^{3/2}}{r x}+\frac {2 \sqrt {e x^r+d} d}{r x}+\frac {2 \left (e x^r+d\right )^{3/2}}{3 r x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{3} \left (-\frac {3 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}+\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \left (-\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}-\frac {16 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{3 r^2}+\frac {4 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{r^2}+\frac {2 d^{3/2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {e x^r+d}}\right )}{r^2}+\frac {4 \left (d+e x^r\right )^{3/2}}{9 r^2}+\frac {16 d \sqrt {d+e x^r}}{3 r^2}\right )\) |
(2*((3*d*Sqrt[d + e*x^r])/r + (d + e*x^r)^(3/2)/r - (3*d^(3/2)*ArcTanh[Sqr t[d + e*x^r]/Sqrt[d]])/r)*(a + b*Log[c*x^n]))/3 - b*n*((16*d*Sqrt[d + e*x^ r])/(3*r^2) + (4*(d + e*x^r)^(3/2))/(9*r^2) - (16*d^(3/2)*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]])/(3*r^2) - (2*d^(3/2)*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]^2) /r^2 + (4*d^(3/2)*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d ] - Sqrt[d + e*x^r])])/r^2 + (2*d^(3/2)*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d ] - Sqrt[d + e*x^r])])/r^2)
3.5.34.3.1 Defintions of rubi rules used
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.)) /(x_), x_Symbol] :> With[{u = IntHide[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*L og[c*x^n]), x] - Simp[b*n Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, n , r}, x] && IntegerQ[q - 1/2]
\[\int \frac {\left (d +e \,x^{r}\right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x}d x\]
Exception generated. \[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x^{r}\right )^{\frac {3}{2}}}{x}\, dx \]
\[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
1/3*(3*d^(3/2)*log((sqrt(e*x^r + d) - sqrt(d))/(sqrt(e*x^r + d) + sqrt(d)) )/r + 2*((e*x^r + d)^(3/2) + 3*sqrt(e*x^r + d)*d)/r)*a + b*integrate((e*x^ r*log(c) + d*log(c) + (e*x^r + d)*log(x^n))*sqrt(e*x^r + d)/x, x)
\[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {{\left (d+e\,x^r\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]