\(\int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx\) [205]

Optimal result

Integrand size = 23, antiderivative size = 323 \[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=-\frac {4 \sqrt {d+e x}}{b}+\frac {4 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}+\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{3/2}}+\frac {2 \sqrt {d+e x} \log (a+b x)}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{3/2}}-\frac {4 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{3/2}}-\frac {2 \sqrt {b d-a e} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{3/2}} \] Output:

-4*(e*x+d)^(1/2)/b+4*(-a*e+b*d)^(1/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+ 
b*d)^(1/2))/b^(3/2)+2*(-a*e+b*d)^(1/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e 
+b*d)^(1/2))^2/b^(3/2)+2*(e*x+d)^(1/2)*ln(b*x+a)/b-2*(-a*e+b*d)^(1/2)*arct 
anh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*ln(b*x+a)/b^(3/2)-4*(-a*e+b*d) 
^(1/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*ln(2/(1-b^(1/2)*(e* 
x+d)^(1/2)/(-a*e+b*d)^(1/2)))/b^(3/2)-2*(-a*e+b*d)^(1/2)*polylog(2,1-2/(1- 
b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2)))/b^(3/2)
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.65 \[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\frac {-8 \sqrt {b} \sqrt {d+e x}+8 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )+4 \sqrt {b} \sqrt {d+e x} \log (a+b x)+2 \sqrt {b d-a e} \log (a+b x) \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )-\sqrt {b d-a e} \log ^2\left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )-2 \sqrt {b d-a e} \log (a+b x) \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+\sqrt {b d-a e} \log ^2\left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+2 \sqrt {b d-a e} \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )-2 \sqrt {b d-a e} \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right ) \log \left (\frac {1}{2} \left (1+\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )-2 \sqrt {b d-a e} \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )+2 \sqrt {b d-a e} \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )}{2 b^{3/2}} \] Input:

Integrate[(Sqrt[d + e*x]*Log[a + b*x])/(a + b*x),x]
 

Output:

(-8*Sqrt[b]*Sqrt[d + e*x] + 8*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e* 
x])/Sqrt[b*d - a*e]] + 4*Sqrt[b]*Sqrt[d + e*x]*Log[a + b*x] + 2*Sqrt[b*d - 
 a*e]*Log[a + b*x]*Log[Sqrt[b*d - a*e] - Sqrt[b]*Sqrt[d + e*x]] - Sqrt[b*d 
 - a*e]*Log[Sqrt[b*d - a*e] - Sqrt[b]*Sqrt[d + e*x]]^2 - 2*Sqrt[b*d - a*e] 
*Log[a + b*x]*Log[Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]] + Sqrt[b*d - a* 
e]*Log[Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]]^2 + 2*Sqrt[b*d - a*e]*Log[ 
Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]]*Log[1/2 - (Sqrt[b]*Sqrt[d + e*x]) 
/(2*Sqrt[b*d - a*e])] - 2*Sqrt[b*d - a*e]*Log[Sqrt[b*d - a*e] - Sqrt[b]*Sq 
rt[d + e*x]]*Log[(1 + (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])/2] - 2*Sqrt 
[b*d - a*e]*PolyLog[2, 1/2 - (Sqrt[b]*Sqrt[d + e*x])/(2*Sqrt[b*d - a*e])] 
+ 2*Sqrt[b*d - a*e]*PolyLog[2, (1 + (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e 
])/2])/(2*b^(3/2))
 

Rubi [A] (verified)

Time = 3.50 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.54, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {2858, 2788, 2756, 60, 73, 221, 2790, 27, 7267, 2092, 6546, 6470, 2849, 2752}

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.

\(\displaystyle \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx\)

\(\Big \downarrow \) 2858

\(\displaystyle \frac {\int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}} \log (a+b x)}{a+b x}d(a+b x)}{b}\)

\(\Big \downarrow \) 2788

\(\displaystyle \frac {\frac {e \int \frac {\log (a+b x)}{\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)}{b}+\left (d-\frac {a e}{b}\right ) \int \frac {\log (a+b x)}{(a+b x) \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)}{b}\)

\(\Big \downarrow \) 2756

\(\displaystyle \frac {\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{a+b x}d(a+b x)}{e}\right )}{b}+\left (d-\frac {a e}{b}\right ) \int \frac {\log (a+b x)}{(a+b x) \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)}{b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \int \frac {1}{(a+b x) \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)+2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}\right )}{e}\right )}{b}+\left (d-\frac {a e}{b}\right ) \int \frac {\log (a+b x)}{(a+b x) \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)}{b}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (\frac {2 b \left (d-\frac {a e}{b}\right ) \int \frac {1}{a+\frac {b \left (d-\frac {a e}{b}+\frac {e (a+b x)}{b}\right )}{e}-\frac {b d}{e}}d\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{e}+2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}\right )}{e}\right )}{b}+\left (d-\frac {a e}{b}\right ) \int \frac {\log (a+b x)}{(a+b x) \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)}{b}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \int \frac {\log (a+b x)}{(a+b x) \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}}{b}\)

\(\Big \downarrow \) 2790

\(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (-\int -\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e} (a+b x)}d(a+b x)-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\frac {2 \sqrt {b} \int \frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}\right )}{a+b x}d(a+b x)}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}}{b}\)

\(\Big \downarrow \) 7267

\(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\frac {4 b^{3/2} \int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}\right )}{a e-b \left (\frac {a e}{b}-\frac {e (a+b x)}{b}\right )}d\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}}{b}\)

\(\Big \downarrow \) 2092

\(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\frac {4 b^{3/2} \int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}\right )}{-b d+a e+b \left (d-\frac {a e}{b}+\frac {e (a+b x)}{b}\right )}d\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}}{b}\)

\(\Big \downarrow \) 6546

\(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\frac {4 b^{3/2} \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )^2}{2 b}-\frac {\int \frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}\right )}{1-\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}}d\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b} \sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}}{b}\)

\(\Big \downarrow \) 6470

\(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\frac {4 b^{3/2} \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )^2}{2 b}-\frac {\frac {\sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}}\right )}{\sqrt {b}}-\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}}\right )}{1-\frac {b \left (d-\frac {a e}{b}+\frac {e (a+b x)}{b}\right )}{b d-a e}}d\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b} \sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}}{b}\)

\(\Big \downarrow \) 2849

\(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\frac {4 b^{3/2} \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )^2}{2 b}-\frac {\frac {\sqrt {b d-a e} \int \frac {\log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}}\right )}{1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}}}d\frac {1}{1-\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}}}{\sqrt {b}}+\frac {\sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}}\right )}{\sqrt {b}}}{\sqrt {b} \sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}}{b}\)

\(\Big \downarrow \) 2752

\(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\frac {4 b^{3/2} \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )^2}{2 b}-\frac {\frac {\sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}}\right )}{\sqrt {b}}+\frac {\sqrt {b d-a e} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}}\right )}{2 \sqrt {b}}}{\sqrt {b} \sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}}{b}\)

Input:

Int[(Sqrt[d + e*x]*Log[a + b*x])/(a + b*x),x]
 

Output:

((e*((-2*b*(2*Sqrt[d - (a*e)/b + (e*(a + b*x))/b] - (2*Sqrt[b]*(d - (a*e)/ 
b)*ArcTanh[(Sqrt[b]*Sqrt[d - (a*e)/b + (e*(a + b*x))/b])/Sqrt[b*d - a*e]]) 
/Sqrt[b*d - a*e]))/e + (2*b*Sqrt[d - (a*e)/b + (e*(a + b*x))/b]*Log[a + b* 
x])/e))/b + (d - (a*e)/b)*((-2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d - (a*e)/b + 
 (e*(a + b*x))/b])/Sqrt[b*d - a*e]]*Log[a + b*x])/Sqrt[b*d - a*e] + (4*b^( 
3/2)*(ArcTanh[(Sqrt[b]*Sqrt[d - (a*e)/b + (e*(a + b*x))/b])/Sqrt[b*d - a*e 
]]^2/(2*b) - ((Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d - (a*e)/b + (e*(a + 
 b*x))/b])/Sqrt[b*d - a*e]]*Log[2/(1 - (Sqrt[b]*Sqrt[d - (a*e)/b + (e*(a + 
 b*x))/b])/Sqrt[b*d - a*e])])/Sqrt[b] + (Sqrt[b*d - a*e]*PolyLog[2, 1 - 2/ 
(1 - (Sqrt[b]*Sqrt[d - (a*e)/b + (e*(a + b*x))/b])/Sqrt[b*d - a*e])])/(2*S 
qrt[b]))/(Sqrt[b]*Sqrt[b*d - a*e])))/Sqrt[b*d - a*e]))/b
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 2092

Int[(Px_)*(u_)^(p_.)*(z_)^(q_.), x_Symbol] :> Int[Px*ExpandToSum[z, x]^q*Ex 
pandToSum[u, x]^p, x] /; FreeQ[{p, q}, x] && BinomialQ[z, x] && BinomialQ[u 
, x] &&  !(BinomialMatchQ[z, x] && BinomialMatchQ[u, x])
 

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2756

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), 
x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] 
- Simp[b*n*(p/(e*(q + 1)))   Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 
 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, 
 -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] & 
& NeQ[q, 1]))
 

rule 2788

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.)) 
/(x_), x_Symbol] :> Simp[d   Int[(d + e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x) 
, x], x] + Simp[e   Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /; F 
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]
 

rule 2790

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]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

rule 6470

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 
2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 
, 0]
 

rule 6546

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ 
(c*d)   Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 4.24 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {2 \sqrt {e x +d}\, \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e}\right )}{b}-\frac {4 \sqrt {e x +d}}{b}-\frac {4 \left (-e a +b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (e a -b d \right ) b}}\right )}{b \sqrt {\left (e a -b d \right ) b}}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+e a -b d \right )}{\sum }\frac {\left (\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e}\right )-2 b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{4 \underline {\hspace {1.25 ex}}\alpha b}+\frac {\underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 e a -2 b d}+\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 e a -2 b d}\right )\right ) \left (-e a +b d \right )}{2 b^{2} \underline {\hspace {1.25 ex}}\alpha }\right )\)	\(256\)
default	\(\frac {2 \sqrt {e x +d}\, \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e}\right )}{b}-\frac {4 \sqrt {e x +d}}{b}-\frac {4 \left (-e a +b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (e a -b d \right ) b}}\right )}{b \sqrt {\left (e a -b d \right ) b}}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+e a -b d \right )}{\sum }\frac {\left (\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e}\right )-2 b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{4 \underline {\hspace {1.25 ex}}\alpha b}+\frac {\underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 e a -2 b d}+\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 e a -2 b d}\right )\right ) \left (-e a +b d \right )}{2 b^{2} \underline {\hspace {1.25 ex}}\alpha }\right )\)	\(256\)

Input:

int((e*x+d)^(1/2)*ln(b*x+a)/(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

2*(e*x+d)^(1/2)*ln(((e*x+d)*b+e*a-b*d)/e)/b-4*(e*x+d)^(1/2)/b-4*(-a*e+b*d) 
/b/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))+2*Sum(1 
/2*(ln((e*x+d)^(1/2)-_alpha)*ln(((e*x+d)*b+e*a-b*d)/e)-2*b*(1/4/_alpha/b*l 
n((e*x+d)^(1/2)-_alpha)^2+1/2*_alpha/(a*e-b*d)*ln((e*x+d)^(1/2)-_alpha)*ln 
(1/2*((e*x+d)^(1/2)+_alpha)/_alpha)+1/2*_alpha/(a*e-b*d)*dilog(1/2*((e*x+d 
)^(1/2)+_alpha)/_alpha)))*(-a*e+b*d)/b^2/_alpha,_alpha=RootOf(_Z^2*b+a*e-b 
*d))
 

Fricas [F]

\[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\int { \frac {\sqrt {e x + d} \log \left (b x + a\right )}{b x + a} \,d x } \] Input:

integrate((e*x+d)^(1/2)*log(b*x+a)/(b*x+a),x, algorithm="fricas")
 

Output:

integral(sqrt(e*x + d)*log(b*x + a)/(b*x + a), x)
 

Sympy [F]

\[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\int \frac {\sqrt {d + e x} \log {\left (a + b x \right )}}{a + b x}\, dx \] Input:

integrate((e*x+d)**(1/2)*ln(b*x+a)/(b*x+a),x)
 

Output:

Integral(sqrt(d + e*x)*log(a + b*x)/(a + b*x), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^(1/2)*log(b*x+a)/(b*x+a),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m 
ore detail
 

Giac [F]

\[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\int { \frac {\sqrt {e x + d} \log \left (b x + a\right )}{b x + a} \,d x } \] Input:

integrate((e*x+d)^(1/2)*log(b*x+a)/(b*x+a),x, algorithm="giac")
 

Output:

integrate(sqrt(e*x + d)*log(b*x + a)/(b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\int \frac {\ln \left (a+b\,x\right )\,\sqrt {d+e\,x}}{a+b\,x} \,d x \] Input:

int((log(a + b*x)*(d + e*x)^(1/2))/(a + b*x),x)
 

Output:

int((log(a + b*x)*(d + e*x)^(1/2))/(a + b*x), x)
 

Reduce [F]

\[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\frac {4 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) d +2 \sqrt {e x +d}\, \mathrm {log}\left (b x +a \right ) b d -4 \sqrt {e x +d}\, b d +\left (\int \frac {\sqrt {e x +d}\, \mathrm {log}\left (b x +a \right ) x}{b e \,x^{2}+a e x +b d x +a d}d x \right ) a b \,e^{2}-\left (\int \frac {\sqrt {e x +d}\, \mathrm {log}\left (b x +a \right ) x}{b e \,x^{2}+a e x +b d x +a d}d x \right ) b^{2} d e}{a b e} \] Input:

int((e*x+d)^(1/2)*log(b*x+a)/(b*x+a),x)
 

Output:

(4*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d) 
))*d + 2*sqrt(d + e*x)*log(a + b*x)*b*d - 4*sqrt(d + e*x)*b*d + int((sqrt( 
d + e*x)*log(a + b*x)*x)/(a*d + a*e*x + b*d*x + b*e*x**2),x)*a*b*e**2 - in 
t((sqrt(d + e*x)*log(a + b*x)*x)/(a*d + a*e*x + b*d*x + b*e*x**2),x)*b**2* 
d*e)/(a*b*e)