\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{f+g x} \, dx\) [134]

Optimal result

Integrand size = 33, antiderivative size = 1442 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{f+g x} \, dx =\text {Too large to display} \] Output:

a^2*(-c^2*d*x^2+d)^(1/2)/g-2*b^2*(-c^2*d*x^2+d)^(1/2)/g-2*a*b*c*x*(-c^2*d* 
x^2+d)^(1/2)/g/(-c^2*x^2+1)^(1/2)+2*a*b*(-c^2*d*x^2+d)^(1/2)*arcsin(c*x)/g 
-2*b^2*c*x*(-c^2*d*x^2+d)^(1/2)*arcsin(c*x)/g/(-c^2*x^2+1)^(1/2)+b^2*(-c^2 
*d*x^2+d)^(1/2)*arcsin(c*x)^2/g+1/3*c*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c 
*x))^3/b/g/(-c^2*x^2+1)^(1/2)-1/3*(1-c^2*f^2/g^2)*(-c^2*d*x^2+d)^(1/2)*(a+ 
b*arcsin(c*x))^3/b/c/(g*x+f)/(-c^2*x^2+1)^(1/2)+1/3*(-c^2*x^2+1)^(1/2)*(-c 
^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^3/b/c/(g*x+f)-a^2*(c^2*f^2-g^2)^(1/2)* 
(-c^2*d*x^2+d)^(1/2)*arctan((c^2*f*x+g)/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^( 
1/2))/g^2/(-c^2*x^2+1)^(1/2)+2*I*b^2*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1 
/2)*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/g^ 
2/(-c^2*x^2+1)^(1/2)-I*b^2*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)*arcsin 
(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/g^2 
/(-c^2*x^2+1)^(1/2)+I*b^2*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)*arcsin( 
c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/g^2/ 
(-c^2*x^2+1)^(1/2)+2*I*a*b*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)*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)))/g^2/( 
-c^2*x^2+1)^(1/2)+2*a*b*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)*polylog(2 
,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/g^2/(-c^2*x^2+1 
)^(1/2)+2*b^2*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)*arcsin(c*x)*polylog 
(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/g^2/(-c^2*...
 

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 516, normalized size of antiderivative = 0.36 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (\left (c^2 f^2-g^2\right ) (a+b \arcsin (c x))^3+c^2 g x (f+g x) (a+b \arcsin (c x))^3+g^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3+3 b c (f+g x) \left (g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-2 b g \left (a c x+b \sqrt {1-c^2 x^2}+b c x \arcsin (c x)\right )+i \sqrt {c^2 f^2-g^2} \left ((a+b \arcsin (c x))^2 \log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-(a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )\right )}{3 b c g^2 (f+g x) \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(Sqrt[d - c^2*d*x^2]*((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^3 + c^2*g*x*(f + 
 g*x)*(a + b*ArcSin[c*x])^3 + g^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^3 + 3* 
b*c*(f + g*x)*(g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*g*(a*c*x + 
b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]) + I*Sqrt[c^2*f^2 - g^2]*((a + b*A 
rcSin[c*x])^2*Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2 
])] - (a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^ 
2*f^2 - g^2])] - (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x 
])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2 
, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + 2*b^2*PolyLog[3, 
(I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] - 2*b^2*PolyLog[3, (I 
*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))))/(3*b*c*g^2*(f + g*x 
)*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 3.58 (sec) , antiderivative size = 1018, normalized size of antiderivative = 0.71, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {5276, 5264, 25, 5256, 25, 5298, 2009}

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.

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

\(\Big \downarrow \) 5276

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

\(\Big \downarrow \) 5264

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5256

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5298

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3}{3 b c (f+g x)}+\frac {\frac {c^2 x (a+b \arcsin (c x))^3}{g}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) (a+b \arcsin (c x))^3}{f+g x}+3 b c \left (-\frac {\sqrt {c^2 f^2-g^2} \arctan \left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right ) a^2}{g^2}+\frac {\sqrt {1-c^2 x^2} a^2}{g}-\frac {2 b c x a}{g}+\frac {2 b \sqrt {1-c^2 x^2} \arcsin (c x) a}{g}+\frac {2 i b \sqrt {c^2 f^2-g^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) a}{g^2}-\frac {2 i b \sqrt {c^2 f^2-g^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) a}{g^2}+\frac {2 b \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) a}{g^2}-\frac {2 b \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) a}{g^2}+\frac {b^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{g}-\frac {2 b^2 c x \arcsin (c x)}{g}+\frac {i b^2 \sqrt {c^2 f^2-g^2} \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2}-\frac {i b^2 \sqrt {c^2 f^2-g^2} \arcsin (c x)^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2}+\frac {2 b^2 \sqrt {c^2 f^2-g^2} \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2}-\frac {2 b^2 \sqrt {c^2 f^2-g^2} \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2}+\frac {2 i b^2 \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2}-\frac {2 i b^2 \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2}-\frac {2 b^2 \sqrt {1-c^2 x^2}}{g}\right )}{3 b c}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

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

Output:

(Sqrt[d - c^2*d*x^2]*(((1 - c^2*x^2)*(a + b*ArcSin[c*x])^3)/(3*b*c*(f + g* 
x)) + ((c^2*x*(a + b*ArcSin[c*x])^3)/g - ((1 - (c^2*f^2)/g^2)*(a + b*ArcSi 
n[c*x])^3)/(f + g*x) + 3*b*c*((-2*a*b*c*x)/g + (a^2*Sqrt[1 - c^2*x^2])/g - 
 (2*b^2*Sqrt[1 - c^2*x^2])/g - (2*b^2*c*x*ArcSin[c*x])/g + (2*a*b*Sqrt[1 - 
 c^2*x^2]*ArcSin[c*x])/g + (b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/g - (a^2* 
Sqrt[c^2*f^2 - g^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2 
*x^2])])/g^2 + ((2*I)*a*b*Sqrt[c^2*f^2 - g^2]*ArcSin[c*x]*Log[1 - (I*E^(I* 
ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g^2 + (I*b^2*Sqrt[c^2*f^2 - 
g^2]*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g 
^2])])/g^2 - ((2*I)*a*b*Sqrt[c^2*f^2 - g^2]*ArcSin[c*x]*Log[1 - (I*E^(I*Ar 
cSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g^2 - (I*b^2*Sqrt[c^2*f^2 - g^ 
2]*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2 
])])/g^2 + (2*a*b*Sqrt[c^2*f^2 - g^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/( 
c*f - Sqrt[c^2*f^2 - g^2])])/g^2 + (2*b^2*Sqrt[c^2*f^2 - g^2]*ArcSin[c*x]* 
PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g^2 - (2* 
a*b*Sqrt[c^2*f^2 - g^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2 
*f^2 - g^2])])/g^2 - (2*b^2*Sqrt[c^2*f^2 - g^2]*ArcSin[c*x]*PolyLog[2, (I* 
E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g^2 + ((2*I)*b^2*Sqrt[c 
^2*f^2 - g^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2 
])])/g^2 - ((2*I)*b^2*Sqrt[c^2*f^2 - g^2]*PolyLog[3, (I*E^(I*ArcSin[c*x...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5256

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

rule 5264

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

rule 5298

Int[(ArcSin[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p 
_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSin[c*x]) 
^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGt 
Q[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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(g-c*f>0)', see `assume?` for mor 
e details)
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*(asin(c*x)*a**2*c*f - 2*sqrt(c**2*f**2 - g**2)*atan((tan(asin(c*x 
)/2)*c*f + g)/sqrt(c**2*f**2 - g**2))*a**2 + sqrt( - c**2*x**2 + 1)*a**2*g 
 + 2*int((sqrt( - c**2*x**2 + 1)*asin(c*x))/(f + g*x),x)*a*b*g**2 + int((s 
qrt( - c**2*x**2 + 1)*asin(c*x)**2)/(f + g*x),x)*b**2*g**2 - a**2*g))/g**2