\(\int \frac {a+b \log (c (d+e x)^n)}{\sqrt {f-g x} \sqrt {f+g x}} \, dx\) [278]

Optimal result

Integrand size = 34, antiderivative size = 510 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f-g x} \sqrt {f+g x}} \, dx=\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \arcsin \left (\frac {g x}{f}\right )^2}{2 g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \arcsin \left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \arcsin \left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \arcsin \left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \arcsin \left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \arcsin \left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \operatorname {PolyLog}\left (2,-\frac {e e^{i \arcsin \left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \operatorname {PolyLog}\left (2,-\frac {e e^{i \arcsin \left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}} \] Output:

1/2*I*b*f*n*(1-g^2*x^2/f^2)^(1/2)*arcsin(g*x/f)^2/g/(-g*x+f)^(1/2)/(g*x+f) 
^(1/2)-b*f*n*(1-g^2*x^2/f^2)^(1/2)*arcsin(g*x/f)*ln(1+e*(I*g*x/f+(1-g^2*x^ 
2/f^2)^(1/2))*f/(I*d*g-(-d^2*g^2+e^2*f^2)^(1/2)))/g/(-g*x+f)^(1/2)/(g*x+f) 
^(1/2)-b*f*n*(1-g^2*x^2/f^2)^(1/2)*arcsin(g*x/f)*ln(1+e*(I*g*x/f+(1-g^2*x^ 
2/f^2)^(1/2))*f/(I*d*g+(-d^2*g^2+e^2*f^2)^(1/2)))/g/(-g*x+f)^(1/2)/(g*x+f) 
^(1/2)+f*(1-g^2*x^2/f^2)^(1/2)*arcsin(g*x/f)*(a+b*ln(c*(e*x+d)^n))/g/(-g*x 
+f)^(1/2)/(g*x+f)^(1/2)+I*b*f*n*(1-g^2*x^2/f^2)^(1/2)*polylog(2,-e*(I*g*x/ 
f+(1-g^2*x^2/f^2)^(1/2))*f/(I*d*g-(-d^2*g^2+e^2*f^2)^(1/2)))/g/(-g*x+f)^(1 
/2)/(g*x+f)^(1/2)+I*b*f*n*(1-g^2*x^2/f^2)^(1/2)*polylog(2,-e*(I*g*x/f+(1-g 
^2*x^2/f^2)^(1/2))*f/(I*d*g+(-d^2*g^2+e^2*f^2)^(1/2)))/g/(-g*x+f)^(1/2)/(g 
*x+f)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 14.95 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.24 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f-g x} \sqrt {f+g x}} \, dx=\frac {\arctan \left (\frac {g x}{\sqrt {f-g x} \sqrt {f+g x}}\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b n \sqrt {f-g x} \left (2 f g (d+e x) \sqrt {\frac {f+g x}{f-g x}} \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right ) \log (d+e x)+(f+g x) \left (d g+e f \cos \left (2 \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )\right )\right ) \csc \left (2 \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )\right ) \left (2 i \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )^2-4 i \arcsin \left (\frac {\sqrt {1+\frac {d g}{e f}}}{\sqrt {2}}\right ) \arctan \left (\frac {-e f+d g}{\sqrt {-e^2 f^2+d^2 g^2} \sqrt {\frac {f+g x}{f-g x}}}\right )-2 \left (\arcsin \left (\frac {\sqrt {1+\frac {d g}{e f}}}{\sqrt {2}}\right )+\arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )\right ) \log \left (1+\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )} \left (d g-\sqrt {-e^2 f^2+d^2 g^2}\right )}{e f}\right )+2 \left (\arcsin \left (\frac {\sqrt {1+\frac {d g}{e f}}}{\sqrt {2}}\right )-\arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )\right ) \log \left (1+\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )} \left (d g+\sqrt {-e^2 f^2+d^2 g^2}\right )}{e f}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )} \left (-d g+\sqrt {-e^2 f^2+d^2 g^2}\right )}{e f}\right )+\operatorname {PolyLog}\left (2,-\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )} \left (d g+\sqrt {-e^2 f^2+d^2 g^2}\right )}{e f}\right )\right )\right )\right )}{f g^2 (d+e x) \sqrt {f+g x}} \] Input:

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

Output:

(ArcTan[(g*x)/(Sqrt[f - g*x]*Sqrt[f + g*x])]*(a - b*n*Log[d + e*x] + b*Log 
[c*(d + e*x)^n]))/g - (b*n*Sqrt[f - g*x]*(2*f*g*(d + e*x)*Sqrt[(f + g*x)/( 
f - g*x)]*ArcTan[1/Sqrt[(f + g*x)/(f - g*x)]]*Log[d + e*x] + (f + g*x)*(d* 
g + e*f*Cos[2*ArcTan[1/Sqrt[(f + g*x)/(f - g*x)]]])*Csc[2*ArcTan[1/Sqrt[(f 
 + g*x)/(f - g*x)]]]*((2*I)*ArcTan[1/Sqrt[(f + g*x)/(f - g*x)]]^2 - (4*I)* 
ArcSin[Sqrt[1 + (d*g)/(e*f)]/Sqrt[2]]*ArcTan[(-(e*f) + d*g)/(Sqrt[-(e^2*f^ 
2) + d^2*g^2]*Sqrt[(f + g*x)/(f - g*x)])] - 2*(ArcSin[Sqrt[1 + (d*g)/(e*f) 
]/Sqrt[2]] + ArcTan[1/Sqrt[(f + g*x)/(f - g*x)]])*Log[1 + (E^((2*I)*ArcTan 
[1/Sqrt[(f + g*x)/(f - g*x)]])*(d*g - Sqrt[-(e^2*f^2) + d^2*g^2]))/(e*f)] 
+ 2*(ArcSin[Sqrt[1 + (d*g)/(e*f)]/Sqrt[2]] - ArcTan[1/Sqrt[(f + g*x)/(f - 
g*x)]])*Log[1 + (E^((2*I)*ArcTan[1/Sqrt[(f + g*x)/(f - g*x)]])*(d*g + Sqrt 
[-(e^2*f^2) + d^2*g^2]))/(e*f)] + I*(PolyLog[2, (E^((2*I)*ArcTan[1/Sqrt[(f 
 + g*x)/(f - g*x)]])*(-(d*g) + Sqrt[-(e^2*f^2) + d^2*g^2]))/(e*f)] + PolyL 
og[2, -((E^((2*I)*ArcTan[1/Sqrt[(f + g*x)/(f - g*x)]])*(d*g + Sqrt[-(e^2*f 
^2) + d^2*g^2]))/(e*f))]))))/(f*g^2*(d + e*x)*Sqrt[f + g*x])
 

Rubi [A] (verified)

Time = 1.90 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.66, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2854, 2851, 27, 5240, 5032, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 2854

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

\(\Big \downarrow \) 2851

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5240

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

\(\Big \downarrow \) 5032

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(Sqrt[1 - (g^2*x^2)/f^2]*((f*ArcSin[(g*x)/f]*(a + b*Log[c*(d + e*x)^n]))/g 
 - (b*e*f*n*(((-1/2*I)*ArcSin[(g*x)/f]^2)/e + I*(((-I)*ArcSin[(g*x)/f]*Log 
[1 + (e*E^(I*ArcSin[(g*x)/f])*f)/(I*d*g - Sqrt[e^2*f^2 - d^2*g^2])])/e - P 
olyLog[2, -((e*E^(I*ArcSin[(g*x)/f])*f)/(I*d*g - Sqrt[e^2*f^2 - d^2*g^2])) 
]/e) + I*(((-I)*ArcSin[(g*x)/f]*Log[1 + (e*E^(I*ArcSin[(g*x)/f])*f)/(I*d*g 
 + Sqrt[e^2*f^2 - d^2*g^2])])/e - PolyLog[2, -((e*E^(I*ArcSin[(g*x)/f])*f) 
/(I*d*g + Sqrt[e^2*f^2 - d^2*g^2]))]/e)))/g))/(Sqrt[f - g*x]*Sqrt[f + g*x] 
)
 

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 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

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

rule 2851

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)* 
(x_)^2], x_Symbol] :> With[{u = IntHide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + 
 b*Log[c*(d + e*x)^n]), x] - Simp[b*e*n   Int[SimplifyIntegrand[u/(d + e*x) 
, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]
 

rule 2854

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/(Sqrt[(f1_) + (g1_ 
.)*(x_)]*Sqrt[(f2_) + (g2_.)*(x_)]), x_Symbol] :> Simp[Sqrt[1 + g1*(g2/(f1* 
f2))*x^2]/(Sqrt[f1 + g1*x]*Sqrt[f2 + g2*x])   Int[(a + b*Log[c*(d + e*x)^n] 
)/Sqrt[1 + g1*(g2/(f1*f2))*x^2], x], x] /; FreeQ[{a, b, c, d, e, f1, g1, f2 
, g2, n}, x] && EqQ[f2*g1 + f1*g2, 0]
 

rule 5032

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[ 
(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1 
))), x] + (Simp[I   Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a - Rt[-a^2 + b^2, 
2] + b*E^(I*(c + d*x)))), x], x] + Simp[I   Int[(e + f*x)^m*(E^(I*(c + d*x) 
)/(I*a + Rt[-a^2 + b^2, 2] + b*E^(I*(c + d*x)))), x], x]) /; FreeQ[{a, b, c 
, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]
 

rule 5240

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] 
 :> Subst[Int[(a + b*x)^n*(Cos[x]/(c*d + e*Sin[x])), x], x, ArcSin[c*x]] /; 
 FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
 

Maple [F]

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

Input:

int((a+b*ln(c*(e*x+d)^n))/(-g*x+f)^(1/2)/(g*x+f)^(1/2),x)
 

Output:

int((a+b*ln(c*(e*x+d)^n))/(-g*x+f)^(1/2)/(g*x+f)^(1/2),x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

integrate((a+b*ln(c*(e*x+d)**n))/(-g*x+f)**(1/2)/(g*x+f)**(1/2),x)
 

Output:

Integral((a + b*log(c*(d + e*x)**n))/(sqrt(f - g*x)*sqrt(f + g*x)), x)
 

Maxima [F]

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

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

Output:

b*integrate((log((e*x + d)^n) + log(c))/(sqrt(g*x + f)*sqrt(-g*x + f)), x) 
 + a*arcsin(g*x/f)/g
                                                                                    
                                                                                    
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int((a + b*log(c*(d + e*x)^n))/((f + g*x)^(1/2)*(f - g*x)^(1/2)),x)
 

Output:

int((a + b*log(c*(d + e*x)^n))/((f + g*x)^(1/2)*(f - g*x)^(1/2)), x)
 

Reduce [F]

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

int((a+b*log(c*(e*x+d)^n))/(-g*x+f)^(1/2)/(g*x+f)^(1/2),x)
 

Output:

( - 2*asin(sqrt(f - g*x)/(sqrt(f)*sqrt(2)))*a + int(log((d + e*x)**n*c)/(s 
qrt(f + g*x)*sqrt(f - g*x)),x)*b*g)/g