\(\int \frac {a+b \log (c (d+e x)^n)}{(d+e x) (f+g x)^{5/2}} \, dx\) [203]

Optimal result

Integrand size = 31, antiderivative size = 406 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=-\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^{5/2}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{(e f-d g)^{5/2}}-\frac {2 b e^{3/2} n \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{(e f-d g)^{5/2}} \] Output:

-4/3*b*e*n/(-d*g+e*f)^2/(g*x+f)^(1/2)+16/3*b*e^(3/2)*n*arctanh(e^(1/2)*(g* 
x+f)^(1/2)/(-d*g+e*f)^(1/2))/(-d*g+e*f)^(5/2)+2*b*e^(3/2)*n*arctanh(e^(1/2 
)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))^2/(-d*g+e*f)^(5/2)+2/3*(a+b*ln(c*(e*x+d) 
^n))/(-d*g+e*f)/(g*x+f)^(3/2)+2*e*(a+b*ln(c*(e*x+d)^n))/(-d*g+e*f)^2/(g*x+ 
f)^(1/2)-2*e^(3/2)*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))*(a+b*ln 
(c*(e*x+d)^n))/(-d*g+e*f)^(5/2)-4*b*e^(3/2)*n*arctanh(e^(1/2)*(g*x+f)^(1/2 
)/(-d*g+e*f)^(1/2))*ln(2/(1-e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2)))/(-d*g 
+e*f)^(5/2)-2*b*e^(3/2)*n*polylog(2,1-2/(1-e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f 
)^(1/2)))/(-d*g+e*f)^(5/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.90 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.50 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=\frac {24 b e^{3/2} n (f+g x)^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )-8 b e \sqrt {e f-d g} n (f+g x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {e (f+g x)}{e f-d g}\right )+4 (e f-d g)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )+12 e \sqrt {e f-d g} (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )+6 e^{3/2} (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )-6 e^{3/2} (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )-3 b e^{3/2} n (f+g x)^{3/2} \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+3 b e^{3/2} n (f+g x)^{3/2} \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )}{6 (e f-d g)^{5/2} (f+g x)^{3/2}} \] Input:

Integrate[(a + b*Log[c*(d + e*x)^n])/((d + e*x)*(f + g*x)^(5/2)),x]
 

Output:

(24*b*e^(3/2)*n*(f + g*x)^(3/2)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - 
 d*g]] - 8*b*e*Sqrt[e*f - d*g]*n*(f + g*x)*Hypergeometric2F1[-1/2, 1, 1/2, 
 (e*(f + g*x))/(e*f - d*g)] + 4*(e*f - d*g)^(3/2)*(a + b*Log[c*(d + e*x)^n 
]) + 12*e*Sqrt[e*f - d*g]*(f + g*x)*(a + b*Log[c*(d + e*x)^n]) + 6*e^(3/2) 
*(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n])*Log[Sqrt[e*f - d*g] - Sqrt[e]* 
Sqrt[f + g*x]] - 6*e^(3/2)*(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n])*Log[ 
Sqrt[e*f - d*g] + Sqrt[e]*Sqrt[f + g*x]] - 3*b*e^(3/2)*n*(f + g*x)^(3/2)*( 
Log[Sqrt[e*f - d*g] - Sqrt[e]*Sqrt[f + g*x]]*(Log[Sqrt[e*f - d*g] - Sqrt[e 
]*Sqrt[f + g*x]] + 2*Log[(1 + (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])/2]) 
 + 2*PolyLog[2, 1/2 - (Sqrt[e]*Sqrt[f + g*x])/(2*Sqrt[e*f - d*g])]) + 3*b* 
e^(3/2)*n*(f + g*x)^(3/2)*(Log[Sqrt[e*f - d*g] + Sqrt[e]*Sqrt[f + g*x]]*(L 
og[Sqrt[e*f - d*g] + Sqrt[e]*Sqrt[f + g*x]] + 2*Log[1/2 - (Sqrt[e]*Sqrt[f 
+ g*x])/(2*Sqrt[e*f - d*g])]) + 2*PolyLog[2, (1 + (Sqrt[e]*Sqrt[f + g*x])/ 
Sqrt[e*f - d*g])/2]))/(6*(e*f - d*g)^(5/2)*(f + g*x)^(3/2))
 

Rubi [A] (verified)

Time = 5.05 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.65, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {2858, 2789, 2756, 61, 73, 221, 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.

\(\displaystyle \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx\)

\(\Big \downarrow \) 2858

\(\displaystyle \frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}d(d+e x)}{e}\)

\(\Big \downarrow \) 2789

\(\displaystyle \frac {\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{e f-d g}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}d(d+e x)}{e f-d g}}{e}\)

\(\Big \downarrow \) 2756

\(\displaystyle \frac {\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \int \frac {1}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{e f-d g}}{e}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \int \frac {1}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{e f-d g}+\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{e f-d g}}{e}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e^2 \int \frac {1}{d+\frac {e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}{g}-\frac {e f}{g}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{g (e f-d g)}+\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{e f-d g}}{e}\)

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 2790

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 2092

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

Int[(a + b*Log[c*(d + e*x)^n])/((d + e*x)*(f + g*x)^(5/2)),x]
 

Output:

(-((g*((2*b*e*n*((2*e)/((e*f - d*g)*Sqrt[f - (d*g)/e + (g*(d + e*x))/e]) - 
 (2*e^(3/2)*ArcTanh[(Sqrt[e]*Sqrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f 
 - d*g]])/(e*f - d*g)^(3/2)))/(3*g) - (2*e*(a + b*Log[c*(d + e*x)^n]))/(3* 
g*(f - (d*g)/e + (g*(d + e*x))/e)^(3/2))))/(e*f - d*g)) + (e*(-((g*((-4*b* 
e^(3/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - 
 d*g]])/(g*Sqrt[e*f - d*g]) - (2*e*(a + b*Log[c*(d + e*x)^n]))/(g*Sqrt[f - 
 (d*g)/e + (g*(d + e*x))/e])))/(e*f - d*g)) + (e*((-2*Sqrt[e]*ArcTanh[(Sqr 
t[e]*Sqrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d 
 + e*x)^n]))/Sqrt[e*f - d*g] + (4*b*e^(3/2)*n*(ArcTanh[(Sqrt[e]*Sqrt[f - ( 
d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g]]^2/(2*e) - ((Sqrt[e*f - d*g]*Ar 
cTanh[(Sqrt[e]*Sqrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g]]*Log[2 
/(1 - (Sqrt[e]*Sqrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g])])/Sqr 
t[e] + (Sqrt[e*f - d*g]*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f - (d*g)/e + 
(g*(d + e*x))/e])/Sqrt[e*f - d*g])])/(2*Sqrt[e]))/(Sqrt[e]*Sqrt[e*f - d*g] 
)))/Sqrt[e*f - d*g]))/(e*f - d*g)))/(e*f - d*g))/e
 

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 61

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

\[\int \frac {a +b \ln \left (c \left (e x +d \right )^{n}\right )}{\left (e x +d \right ) \left (g x +f \right )^{\frac {5}{2}}}d x\]

Input:

int((a+b*ln(c*(e*x+d)^n))/(e*x+d)/(g*x+f)^(5/2),x)
 

Output:

int((a+b*ln(c*(e*x+d)^n))/(e*x+d)/(g*x+f)^(5/2),x)
 

Fricas [F]

\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (e x + d\right )} {\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*log(c*(e*x+d)^n))/(e*x+d)/(g*x+f)^(5/2),x, algorithm="frica 
s")
 

Output:

integral((sqrt(g*x + f)*b*log((e*x + d)^n*c) + sqrt(g*x + f)*a)/(e*g^3*x^4 
 + d*f^3 + (3*e*f*g^2 + d*g^3)*x^3 + 3*(e*f^2*g + d*f*g^2)*x^2 + (e*f^3 + 
3*d*f^2*g)*x), x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*ln(c*(e*x+d)**n))/(e*x+d)/(g*x+f)**(5/2),x)
 

Output:

Exception raised: TypeError >> Invalid comparison of non-real zoo
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*log(c*(e*x+d)^n))/(e*x+d)/(g*x+f)^(5/2),x, algorithm="maxim 
a")
 

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(e*(d*g-e*f)>0)', see `assume?` f 
or more de
 

Giac [F]

\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (e x + d\right )} {\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*log(c*(e*x+d)^n))/(e*x+d)/(g*x+f)^(5/2),x, algorithm="giac" 
)
                                                                                    
                                                                                    
 

Output:

integrate((b*log((e*x + d)^n*c) + a)/((e*x + d)*(g*x + f)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{{\left (f+g\,x\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \] Input:

int((a + b*log(c*(d + e*x)^n))/((f + g*x)^(5/2)*(d + e*x)),x)
 

Output:

int((a + b*log(c*(d + e*x)^n))/((f + g*x)^(5/2)*(d + e*x)), x)
 

Reduce [F]

\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx =\text {Too large to display} \] Input:

int((a+b*log(c*(e*x+d)^n))/(e*x+d)/(g*x+f)^(5/2),x)
 

Output:

(6*sqrt(e)*sqrt(f + g*x)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*s 
qrt(d*g - e*f)))*a*e*f + 6*sqrt(e)*sqrt(f + g*x)*sqrt(d*g - e*f)*atan((sqr 
t(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)))*a*e*g*x + 3*sqrt(f + g*x)*int((sq 
rt(f + g*x)*log((d + e*x)**n*c))/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 
+ d*g**3*x**3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g**3*x**4 
),x)*b*d**3*f*g**3 + 3*sqrt(f + g*x)*int((sqrt(f + g*x)*log((d + e*x)**n*c 
))/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + d*g**3*x**3 + e*f**3*x + 3*e 
*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g**3*x**4),x)*b*d**3*g**4*x - 9*sqrt(f 
+ g*x)*int((sqrt(f + g*x)*log((d + e*x)**n*c))/(d*f**3 + 3*d*f**2*g*x + 3* 
d*f*g**2*x**2 + d*g**3*x**3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 
 + e*g**3*x**4),x)*b*d**2*e*f**2*g**2 - 9*sqrt(f + g*x)*int((sqrt(f + g*x) 
*log((d + e*x)**n*c))/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + d*g**3*x* 
*3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g**3*x**4),x)*b*d**2 
*e*f*g**3*x + 9*sqrt(f + g*x)*int((sqrt(f + g*x)*log((d + e*x)**n*c))/(d*f 
**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + d*g**3*x**3 + e*f**3*x + 3*e*f**2*g 
*x**2 + 3*e*f*g**2*x**3 + e*g**3*x**4),x)*b*d*e**2*f**3*g + 9*sqrt(f + g*x 
)*int((sqrt(f + g*x)*log((d + e*x)**n*c))/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g 
**2*x**2 + d*g**3*x**3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e* 
g**3*x**4),x)*b*d*e**2*f**2*g**2*x - 3*sqrt(f + g*x)*int((sqrt(f + g*x)*lo 
g((d + e*x)**n*c))/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + d*g**3*x*...