\(\int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx\) [207]

Optimal result

Integrand size = 23, antiderivative size = 316 \[ \int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx=\frac {4 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}+\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{(b d-a e)^{3/2}}+\frac {2 \log (a+b x)}{(b d-a e) \sqrt {d+e x}}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{(b d-a e)^{3/2}}-\frac {4 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{(b d-a e)^{3/2}}-\frac {2 \sqrt {b} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{(b d-a e)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.73 \[ \int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx=2 \left (\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}+\frac {\log (a+b x)}{(b d-a e) \sqrt {d+e x}}+\frac {\sqrt {b} \log (a+b x) \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )}{2 (b d-a e)^{3/2}}-\frac {\sqrt {b} \log (a+b x) \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )}{2 (b d-a e)^{3/2}}-\frac {\sqrt {b} \left (\log ^2\left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )+2 \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right ) \log \left (\frac {\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )\right )}{4 (b d-a e)^{3/2}}+\frac {\sqrt {b} \left (2 \log \left (\frac {\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right ) \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+\log ^2\left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )\right )}{4 (b d-a e)^{3/2}}\right ) \] Input:

Integrate[Log[a + b*x]/((a + b*x)*(d + e*x)^(3/2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 3.47 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.49, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {2858, 2789, 2756, 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.

\(\displaystyle \int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx\)

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 2790

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 2092

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

Int[Log[a + b*x]/((a + b*x)*(d + e*x)^(3/2)),x]
 

Output:

(-((e*((-4*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d - (a*e)/b + (e*(a + b*x))/b])/S 
qrt[b*d - a*e]])/(e*Sqrt[b*d - a*e]) - (2*b*Log[a + b*x])/(e*Sqrt[d - (a*e 
)/b + (e*(a + b*x))/b])))/(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*Sqrt[b]))/(Sqrt[b]*Sqrt[b*d - a*e])))/Sqrt[b*d - a*e]))/( 
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 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 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && 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 = 4.32 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+e a -b d \right )}{\sum }\left (-\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e}\right )-2 b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{4 \underline {\hspace {1.25 ex}}\alpha b}+\frac {\underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 e a -2 b d}+\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 e a -2 b d}\right )}{2 \left (e a -b d \right ) \underline {\hspace {1.25 ex}}\alpha }\right )\right )+\frac {-\frac {2 \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e}\right )}{\sqrt {e x +d}}+\frac {4 b \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (e a -b d \right ) b}}\right )}{\sqrt {\left (e a -b d \right ) b}}}{e a -b d}\)	\(243\)
default	\(2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+e a -b d \right )}{\sum }\left (-\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e}\right )-2 b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{4 \underline {\hspace {1.25 ex}}\alpha b}+\frac {\underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 e a -2 b d}+\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 e a -2 b d}\right )}{2 \left (e a -b d \right ) \underline {\hspace {1.25 ex}}\alpha }\right )\right )+\frac {-\frac {2 \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e}\right )}{\sqrt {e x +d}}+\frac {4 b \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (e a -b d \right ) b}}\right )}{\sqrt {\left (e a -b d \right ) b}}}{e a -b d}\)	\(243\)

Input:

int(ln(b*x+a)/(b*x+a)/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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/_a 
lpha/b*ln((e*x+d)^(1/2)-_alpha)^2+1/2*_alpha/(a*e-b*d)*ln((e*x+d)^(1/2)-_a 
lpha)*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)/_alpha,_alpha=RootOf(_Z^2*b+a* 
e-b*d))+2*(-1/(e*x+d)^(1/2)*ln(((e*x+d)*b+e*a-b*d)/e)+2*b/((a*e-b*d)*b)^(1 
/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)))/(a*e-b*d)
 

Fricas [F]

\[ \int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx=\int { \frac {\log \left (b x + a\right )}{{\left (b x + a\right )} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(log(b*x+a)/(b*x+a)/(e*x+d)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(e*x + d)*log(b*x + a)/(b*e^2*x^3 + a*d^2 + (2*b*d*e + a*e^2) 
*x^2 + (b*d^2 + 2*a*d*e)*x), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(ln(b*x+a)/(b*x+a)/(e*x+d)**(3/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(log(b*x+a)/(b*x+a)/(e*x+d)^(3/2),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]

\[ \int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx=\int { \frac {\log \left (b x + a\right )}{{\left (b x + a\right )} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(log(b*x+a)/(b*x+a)/(e*x+d)^(3/2),x, algorithm="giac")
 

Output:

integrate(log(b*x + a)/((b*x + a)*(e*x + d)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx=\int \frac {\ln \left (a+b\,x\right )}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{3/2}} \,d x \] Input:

int(log(a + b*x)/((a + b*x)*(d + e*x)^(3/2)),x)
 

Output:

int(log(a + b*x)/((a + b*x)*(d + e*x)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx=\int \frac {\sqrt {e x +d}\, \mathrm {log}\left (b x +a \right )}{b \,e^{2} x^{3}+a \,e^{2} x^{2}+2 b d e \,x^{2}+2 a d e x +b \,d^{2} x +a \,d^{2}}d x \] Input:

int(log(b*x+a)/(b*x+a)/(e*x+d)^(3/2),x)
 

Output:

int((sqrt(d + e*x)*log(a + b*x))/(a*d**2 + 2*a*d*e*x + a*e**2*x**2 + b*d** 
2*x + 2*b*d*e*x**2 + b*e**2*x**3),x)