3.1.65 \(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \arcsin (c x))^2}{f+g x} \, dx\) [65]

3.1.65.1 Optimal result

Integrand size = 33, antiderivative size = 1992 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{f+g x} \, dx =\text {Too large to display} \]

output

-1/4*b^2*c^2*d*f*x*(-c^2*d*x^2+d)^(1/2)/g^2+1/2*c^2*d*f*x*(a+b*arcsin(c*x) 
)^2*(-c^2*d*x^2+d)^(1/2)/g^2+a^2*d*(c^2*f^2-g^2)^(3/2)*arctan((c^2*f*x+g)/ 
(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/g^4/(-c^2*x^2 
+1)^(1/2)-b^2*d*(c*f-g)*(c*f+g)*arcsin(c*x)^2*(-c^2*d*x^2+d)^(1/2)/g^3+I*b 
^2*d*(c^2*f^2-g^2)^(3/2)*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)))*(-c^2*d*x^2+d)^(1/2)/g^4/(-c^2*x^2+1)^(1/2)-4/ 
9*b^2*d*(-c^2*d*x^2+d)^(1/2)/g-a^2*d*(c*f-g)*(c*f+g)*(-c^2*d*x^2+d)^(1/2)/ 
g^3+2*I*a*b*d*(c^2*f^2-g^2)^(3/2)*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)))*(-c^2*d*x^2+d)^(1/2)/g^4/(-c^2*x^2+1)^( 
1/2)+2*I*b^2*d*(c^2*f^2-g^2)^(3/2)*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2))* 
g/(c*f+(c^2*f^2-g^2)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/g^4/(-c^2*x^2+1)^(1/2)-2 
*a*b*d*(c*f-g)*(c*f+g)*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/g^3+1/4*b^2*c*d*f* 
arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/g^2/(-c^2*x^2+1)^(1/2)-2/3*b*c*d*x*(a+b*a 
rcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/g/(-c^2*x^2+1)^(1/2)+2/9*b*c^3*d*x^3*(a+b 
*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/g/(-c^2*x^2+1)^(1/2)+1/6*c*d*f*(a+b*arc 
sin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/g^2/(-c^2*x^2+1)^(1/2)-2*a*b*d*(c^2*f^2 
-g^2)^(3/2)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1 
/2)))*(-c^2*d*x^2+d)^(1/2)/g^4/(-c^2*x^2+1)^(1/2)-2*b^2*d*(c^2*f^2-g^2)^(3 
/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)))*(-c^2*d*x^2+d)^(1/2)/g^4/(-c^2*x^2+1)^(1/2)+2*a*b*d*(c^2*f^2-...
 

3.1.65.2 Mathematica [A] (verified)

Time = 1.73 (sec) , antiderivative size = 740, normalized size of antiderivative = 0.37 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (54 c^2 f x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+36 g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {18 c f (a+b \arcsin (c x))^3}{b}+\frac {36 \left (c^2 f^2-g^2\right ) \left (-1+c^2 x^2\right ) (a+b \arcsin (c x))^3}{b c (f+g x)}-27 b c f \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (-1+2 c^2 x^2\right ) \arcsin (c x)\right )-8 b g \left (b \sqrt {1-c^2 x^2} \left (7-c^2 x^2\right )+9 c x (a+b \arcsin (c x))-3 c^3 x^3 (a+b \arcsin (c x))\right )-\frac {36 \left (c^2 f^2-g^2\right ) \left (\left (c^2 f^2-g^2\right ) (a+b \arcsin (c x))^3+c^2 g x (f+g x) (a+b \arcsin (c x))^3+3 b c (f+g x) \left (g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-2 b g \left (a c x+b \sqrt {1-c^2 x^2}+b c x \arcsin (c x)\right )+i \sqrt {c^2 f^2-g^2} \left ((a+b \arcsin (c x))^2 \log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-(a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )\right )}{b c g^2 (f+g x)}\right )}{108 g^2 \sqrt {1-c^2 x^2}} \]

input

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(f + g*x),x]
 

output

(d*Sqrt[d - c^2*d*x^2]*(54*c^2*f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 
 + 36*g*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2 + (18*c*f*(a + b*ArcSin[ 
c*x])^3)/b + (36*(c^2*f^2 - g^2)*(-1 + c^2*x^2)*(a + b*ArcSin[c*x])^3)/(b* 
c*(f + g*x)) - 27*b*c*f*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c 
^2*x^2)*ArcSin[c*x]) - 8*b*g*(b*Sqrt[1 - c^2*x^2]*(7 - c^2*x^2) + 9*c*x*(a 
 + b*ArcSin[c*x]) - 3*c^3*x^3*(a + b*ArcSin[c*x])) - (36*(c^2*f^2 - g^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 + 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*ArcSin[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*P 
olyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] - 2*b^2*Pol 
yLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))))/(b*c*g^2* 
(f + g*x))))/(108*g^2*Sqrt[1 - c^2*x^2])
 

3.1.65.3 Rubi [A] (verified)

Time = 3.77 (sec) , antiderivative size = 1369, normalized size of antiderivative = 0.69, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5276, 5266, 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.

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

\(\Big \downarrow \) 5276

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

\(\Big \downarrow \) 5266

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (\frac {2 b x^3 (a+b \arcsin (c x)) c^3}{9 g}-\frac {b f x^2 (a+b \arcsin (c x)) c^3}{2 g^2}+\frac {f x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 c^2}{2 g^2}-\frac {b^2 f x \sqrt {1-c^2 x^2} c^2}{4 g^2}-\frac {\left (c^2 f^2-g^2\right ) x (a+b \arcsin (c x))^3 c}{3 b g^3}+\frac {f (a+b \arcsin (c x))^3 c}{6 b g^2}+\frac {2 a b \left (c^2 f^2-g^2\right ) x c}{g^3}+\frac {2 b^2 \left (c^2 f^2-g^2\right ) x \arcsin (c x) c}{g^3}+\frac {b^2 f \arcsin (c x) c}{4 g^2}-\frac {2 b x (a+b \arcsin (c x)) c}{3 g}-\frac {b^2 (c f-g) (c f+g) \sqrt {1-c^2 x^2} \arcsin (c x)^2}{g^3}+\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 g}-\frac {2 b^2 \left (1-c^2 x^2\right )^{3/2}}{27 g}-\frac {2 a b (c f-g) (c f+g) \sqrt {1-c^2 x^2} \arcsin (c x)}{g^3}+\frac {a^2 \left (c^2 f^2-g^2\right )^{3/2} \arctan \left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4}-\frac {i b^2 \left (c^2 f^2-g^2\right )^{3/2} \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^4}-\frac {2 i a b \left (c^2 f^2-g^2\right )^{3/2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4}+\frac {i b^2 \left (c^2 f^2-g^2\right )^{3/2} \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^4}+\frac {2 i a b \left (c^2 f^2-g^2\right )^{3/2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4}-\frac {2 a b \left (c^2 f^2-g^2\right )^{3/2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4}-\frac {2 b^2 \left (c^2 f^2-g^2\right )^{3/2} \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^4}+\frac {2 a b \left (c^2 f^2-g^2\right )^{3/2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4}+\frac {2 b^2 \left (c^2 f^2-g^2\right )^{3/2} \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^4}-\frac {2 i b^2 \left (c^2 f^2-g^2\right )^{3/2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4}+\frac {2 i b^2 \left (c^2 f^2-g^2\right )^{3/2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4}-\frac {a^2 (c f-g) (c f+g) \sqrt {1-c^2 x^2}}{g^3}+\frac {2 b^2 (c f-g) (c f+g) \sqrt {1-c^2 x^2}}{g^3}-\frac {4 b^2 \sqrt {1-c^2 x^2}}{9 g}-\frac {\left (c^2 f^2-g^2\right ) \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3}{3 b g^2 (f+g x) c}-\frac {\left (c^2 f^2-g^2\right )^2 (a+b \arcsin (c x))^3}{3 b g^4 (f+g x) c}\right )}{\sqrt {1-c^2 x^2}}\)

input

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(f + g*x),x]
 

output

(d*Sqrt[d - c^2*d*x^2]*((2*a*b*c*(c^2*f^2 - g^2)*x)/g^3 - (4*b^2*Sqrt[1 - 
c^2*x^2])/(9*g) - (a^2*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2])/g^3 + (2*b^2 
*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2])/g^3 - (b^2*c^2*f*x*Sqrt[1 - c^2*x^ 
2])/(4*g^2) - (2*b^2*(1 - c^2*x^2)^(3/2))/(27*g) + (b^2*c*f*ArcSin[c*x])/( 
4*g^2) + (2*b^2*c*(c^2*f^2 - g^2)*x*ArcSin[c*x])/g^3 - (2*a*b*(c*f - g)*(c 
*f + g)*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/g^3 - (b^2*(c*f - g)*(c*f + g)*Sqrt 
[1 - c^2*x^2]*ArcSin[c*x]^2)/g^3 - (2*b*c*x*(a + b*ArcSin[c*x]))/(3*g) - ( 
b*c^3*f*x^2*(a + b*ArcSin[c*x]))/(2*g^2) + (2*b*c^3*x^3*(a + b*ArcSin[c*x] 
))/(9*g) + (c^2*f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*g^2) + ((1 
 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*g) + (c*f*(a + b*ArcSin[c*x])^ 
3)/(6*b*g^2) - (c*(c^2*f^2 - g^2)*x*(a + b*ArcSin[c*x])^3)/(3*b*g^3) - ((c 
^2*f^2 - g^2)^2*(a + b*ArcSin[c*x])^3)/(3*b*c*g^4*(f + g*x)) - ((c^2*f^2 - 
 g^2)*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^3)/(3*b*c*g^2*(f + g*x)) + (a^2*(c 
^2*f^2 - g^2)^(3/2)*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2 
*x^2])])/g^4 - ((2*I)*a*b*(c^2*f^2 - g^2)^(3/2)*ArcSin[c*x]*Log[1 - (I*E^( 
I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g^4 - (I*b^2*(c^2*f^2 - g^ 
2)^(3/2)*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 
 - g^2])])/g^4 + ((2*I)*a*b*(c^2*f^2 - g^2)^(3/2)*ArcSin[c*x]*Log[1 - (I*E 
^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g^4 + (I*b^2*(c^2*f^2 - 
g^2)^(3/2)*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^...
 

3.1.65.3.1 Defintions of rubi rules used

rule 2009

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

rule 5266

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

3.1.65.4 Maple [F]

\[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{g x +f}d x\]

input

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

output

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

3.1.65.5 Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{g x + f} \,d x } \]

input

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/(g*x+f),x, algorithm="f 
ricas")
 

output

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

3.1.65.6 Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{f + g x}\, dx \]

input

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x))**2/(g*x+f),x)
 

output

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x))**2/(f + g*x), x 
)
 

3.1.65.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\text {Exception raised: ValueError} \]

input

integrate((-c^2*d*x^2+d)^(3/2)*(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)
 

3.1.65.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\text {Exception raised: TypeError} \]

input

integrate((-c^2*d*x^2+d)^(3/2)*(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
 

3.1.65.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{f+g x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{f+g\,x} \,d x \]

input

int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(3/2))/(f + g*x),x)
 

output

int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(3/2))/(f + g*x), x)