3.1.47 \(\int \frac {a+b \arcsin (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx\) [47]

3.1.47.1 Optimal result

Integrand size = 31, antiderivative size = 380 \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=-\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}} \]

output

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

3.1.47.2 Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.61 \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-\log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )-b \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )+b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}} \]

input

Integrate[(a + b*ArcSin[c*x])/((f + g*x)*Sqrt[d - c^2*d*x^2]),x]
 

output

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

3.1.47.3 Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.76, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {5276, 5272, 3042, 3804, 2694, 27, 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.

\(\displaystyle \int \frac {a+b \arcsin (c x)}{\sqrt {d-c^2 d x^2} (f+g x)} \, dx\)

\(\Big \downarrow \) 5276

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

\(\Big \downarrow \) 5272

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{c f+g \sin (\arcsin (c x))}d\arcsin (c x)}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3804

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

Int[(a + b*ArcSin[c*x])/((f + g*x)*Sqrt[d - c^2*d*x^2]),x]
 

output

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

3.1.47.3.1 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 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && 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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3804

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

rule 5272

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sq 
rt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d])   Subst[In 
t[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c 
, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (G 
tQ[m, 0] || IGtQ[n, 0])
 

rule 5276

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ 
) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^ 
p]   Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ 
[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && Intege 
rQ[p - 1/2] &&  !GtQ[d, 0]
 

3.1.47.4 Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.34

method	result	size
default	\(-\frac {a \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} f^{2}+g^{2}}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (\frac {i c f +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) g -\sqrt {-c^{2} f^{2}+g^{2}}}{i c f -\sqrt {-c^{2} f^{2}+g^{2}}}\right )-i \arcsin \left (c x \right ) \ln \left (\frac {i c f +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) g +\sqrt {-c^{2} f^{2}+g^{2}}}{i c f +\sqrt {-c^{2} f^{2}+g^{2}}}\right )+\operatorname {dilog}\left (\frac {i c f +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) g -\sqrt {-c^{2} f^{2}+g^{2}}}{i c f -\sqrt {-c^{2} f^{2}+g^{2}}}\right )-\operatorname {dilog}\left (\frac {i c f +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) g +\sqrt {-c^{2} f^{2}+g^{2}}}{i c f +\sqrt {-c^{2} f^{2}+g^{2}}}\right )\right )}{d \left (c^{2} x^{2}-1\right ) \left (c^{2} f^{2}-g^{2}\right )}\)	\(508\)
parts	\(-\frac {a \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} f^{2}+g^{2}}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (\frac {i c f +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) g -\sqrt {-c^{2} f^{2}+g^{2}}}{i c f -\sqrt {-c^{2} f^{2}+g^{2}}}\right )-i \arcsin \left (c x \right ) \ln \left (\frac {i c f +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) g +\sqrt {-c^{2} f^{2}+g^{2}}}{i c f +\sqrt {-c^{2} f^{2}+g^{2}}}\right )+\operatorname {dilog}\left (\frac {i c f +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) g -\sqrt {-c^{2} f^{2}+g^{2}}}{i c f -\sqrt {-c^{2} f^{2}+g^{2}}}\right )-\operatorname {dilog}\left (\frac {i c f +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) g +\sqrt {-c^{2} f^{2}+g^{2}}}{i c f +\sqrt {-c^{2} f^{2}+g^{2}}}\right )\right )}{d \left (c^{2} x^{2}-1\right ) \left (c^{2} f^{2}-g^{2}\right )}\)	\(508\)

input

int((a+b*arcsin(c*x))/(g*x+f)/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE 
)
 

output

-a/g/(-d*(c^2*f^2-g^2)/g^2)^(1/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*( 
x+f/g)+2*(-d*(c^2*f^2-g^2)/g^2)^(1/2)*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g 
)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g))-I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*f^ 
2+g^2)^(1/2)*(-c^2*x^2+1)^(1/2)*(I*arcsin(c*x)*ln((I*c*f+(I*c*x+(-c^2*x^2+ 
1)^(1/2))*g-(-c^2*f^2+g^2)^(1/2))/(I*c*f-(-c^2*f^2+g^2)^(1/2)))-I*arcsin(c 
*x)*ln((I*c*f+(I*c*x+(-c^2*x^2+1)^(1/2))*g+(-c^2*f^2+g^2)^(1/2))/(I*c*f+(- 
c^2*f^2+g^2)^(1/2)))+dilog((I*c*f+(I*c*x+(-c^2*x^2+1)^(1/2))*g-(-c^2*f^2+g 
^2)^(1/2))/(I*c*f-(-c^2*f^2+g^2)^(1/2)))-dilog((I*c*f+(I*c*x+(-c^2*x^2+1)^ 
(1/2))*g+(-c^2*f^2+g^2)^(1/2))/(I*c*f+(-c^2*f^2+g^2)^(1/2))))/d/(c^2*x^2-1 
)/(c^2*f^2-g^2)
 

3.1.47.5 Fricas [F]

\[ \int \frac {a+b \arcsin (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}} \,d x } \]

input

integrate((a+b*arcsin(c*x))/(g*x+f)/(-c^2*d*x^2+d)^(1/2),x, algorithm="fri 
cas")
 

output

integral(-sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)/(c^2*d*g*x^3 + c^2*d*f* 
x^2 - d*g*x - d*f), x)
 

3.1.47.6 Sympy [F]

\[ \int \frac {a+b \arcsin (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (f + g x\right )}\, dx \]

input

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

output

Integral((a + b*asin(c*x))/(sqrt(-d*(c*x - 1)*(c*x + 1))*(f + g*x)), x)
 

3.1.47.7 Maxima [F]

\[ \int \frac {a+b \arcsin (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}} \,d x } \]

input

integrate((a+b*arcsin(c*x))/(g*x+f)/(-c^2*d*x^2+d)^(1/2),x, algorithm="max 
ima")
 

output

integrate((b*arcsin(c*x) + a)/(sqrt(-c^2*d*x^2 + d)*(g*x + f)), x)
 

3.1.47.8 Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate((a+b*arcsin(c*x))/(g*x+f)/(-c^2*d*x^2+d)^(1/2),x, algorithm="gia 
c")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

3.1.47.9 Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{\left (f+g\,x\right )\,\sqrt {d-c^2\,d\,x^2}} \,d x \]

input

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

output

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