Integrand size = 33, antiderivative size = 1589 \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx =\text {Too large to display} \]
1/3*I*b^2*(c*f+g)^3*polylog(2,I/(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^( 1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+1/12*I*(c*f+g)^3*(a+b*arcsin(c*x))^2*(-c ^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+1/4*I*(c*f-2*g)*(c*f+g)^2*(a+ b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/12*I*(c *f-g)^3*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2 )-1/6*b^2*(c*f-g)^3*cot(1/4*Pi+1/2*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^4/d^2 /(-c^2*d*x^2+d)^(1/2)-1/12*(c*f-g)^3*(a+b*arcsin(c*x))^2*cot(1/4*Pi+1/2*ar csin(c*x))*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/4*(c*f-g)^2*( c*f+2*g)*(a+b*arcsin(c*x))^2*cot(1/4*Pi+1/2*arcsin(c*x))*(-c^2*x^2+1)^(1/2 )/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/12*b*(c*f-g)^3*(a+b*arcsin(c*x))*csc(1/4* Pi+1/2*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/24 *(c*f-g)^3*(a+b*arcsin(c*x))^2*cot(1/4*Pi+1/2*arcsin(c*x))*csc(1/4*Pi+1/2* arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+b*(c*f-2*g) *(c*f+g)^2*(a+b*arcsin(c*x))*ln(1-I/(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+ 1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+1/3*b*(c*f+g)^3*(a+b*arcsin(c*x))*ln (1-I/(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d) ^(1/2)+1/3*b*(c*f-g)^3*(a+b*arcsin(c*x))*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)) )*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+b*(c*f-g)^2*(c*f+2*g)*(a +b*arcsin(c*x))*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^4/ d^2/(-c^2*d*x^2+d)^(1/2)+I*b^2*(c*f-2*g)*(c*f+g)^2*polylog(2,I/(I*c*x+(...
Time = 6.26 (sec) , antiderivative size = 715, normalized size of antiderivative = 0.45 \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {1-c^2 x^2} \left (\frac {(c f-g)^2 (c f+2 g) \left (i b \left (\frac {(a+b \arcsin (c x))^2}{b}-4 \left (i (a+b \arcsin (c x)) \log \left (1+e^{\frac {1}{2} i (\pi -2 \arcsin (c x))}\right )-b \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (\pi -2 \arcsin (c x))}\right )\right )\right )-(a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}-\frac {1}{2} \arcsin (c x)\right )\right )}{4 c^4}-\frac {(c f-g)^3 \left (2 b (a+b \arcsin (c x)) \sec ^2\left (\frac {\pi }{4}-\frac {1}{2} \arcsin (c x)\right )+4 b^2 \tan \left (\frac {\pi }{4}-\frac {1}{2} \arcsin (c x)\right )+(a+b \arcsin (c x))^2 \sec ^2\left (\frac {\pi }{4}-\frac {1}{2} \arcsin (c x)\right ) \tan \left (\frac {\pi }{4}-\frac {1}{2} \arcsin (c x)\right )-2 \left (i b \left (\frac {(a+b \arcsin (c x))^2}{b}-4 \left (i (a+b \arcsin (c x)) \log \left (1+e^{\frac {1}{2} i (\pi -2 \arcsin (c x))}\right )-b \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (\pi -2 \arcsin (c x))}\right )\right )\right )-(a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}-\frac {1}{2} \arcsin (c x)\right )\right )\right )}{24 c^4}-\frac {(c f-2 g) (c f+g)^2 \left (i b \left (\frac {(a+b \arcsin (c x))^2}{b}+4 \left (i (a+b \arcsin (c x)) \log \left (1+e^{\frac {1}{2} i (\pi +2 \arcsin (c x))}\right )+b \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (\pi +2 \arcsin (c x))}\right )\right )\right )-(a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{4 c^4}-\frac {(c f+g)^3 \left (2 b (a+b \arcsin (c x)) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )-4 b^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )-(a+b \arcsin (c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )+2 \left (i b \left (\frac {(a+b \arcsin (c x))^2}{b}+4 \left (i (a+b \arcsin (c x)) \log \left (1+e^{\frac {1}{2} i (\pi +2 \arcsin (c x))}\right )+b \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (\pi +2 \arcsin (c x))}\right )\right )\right )-(a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )\right )}{24 c^4}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
(Sqrt[1 - c^2*x^2]*(((c*f - g)^2*(c*f + 2*g)*(I*b*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi - 2*ArcSin[c*x]))] - b*Po lyLog[2, -E^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi /4 - ArcSin[c*x]/2]))/(4*c^4) - ((c*f - g)^3*(2*b*(a + b*ArcSin[c*x])*Sec[ Pi/4 - ArcSin[c*x]/2]^2 + 4*b^2*Tan[Pi/4 - ArcSin[c*x]/2] + (a + b*ArcSin[ c*x])^2*Sec[Pi/4 - ArcSin[c*x]/2]^2*Tan[Pi/4 - ArcSin[c*x]/2] - 2*(I*b*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi - 2* ArcSin[c*x]))] - b*PolyLog[2, -E^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b* ArcSin[c*x])^2*Tan[Pi/4 - ArcSin[c*x]/2])))/(24*c^4) - ((c*f - 2*g)*(c*f + g)^2*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^( (I/2)*(Pi + 2*ArcSin[c*x]))] + b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]) )])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2]))/(4*c^4) - ((c*f + g)^3*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2 - 4*b^2*Tan[Pi/ 4 + ArcSin[c*x]/2] - (a + b*ArcSin[c*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan [Pi/4 + ArcSin[c*x]/2] + 2*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*(a + b*Arc Sin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] + b*PolyLog[2, -E^((I/2) *(Pi + 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2 ])))/(24*c^4)))/(d^2*Sqrt[d - c^2*d*x^2])
Time = 2.34 (sec) , antiderivative size = 918, normalized size of antiderivative = 0.58, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5276, 5274, 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.
| \(\displaystyle \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}dx}{d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5274 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \left (\frac {(a+b \arcsin (c x))^2 (c f-g)^3}{4 c^3 (c x+1)^2 \sqrt {1-c^2 x^2}}+\frac {(c f+2 g) (a+b \arcsin (c x))^2 (c f-g)^2}{4 c^3 (c x+1) \sqrt {1-c^2 x^2}}-\frac {(c f-2 g) (c f+g)^2 (a+b \arcsin (c x))^2}{4 c^3 (c x-1) \sqrt {1-c^2 x^2}}+\frac {(c f+g)^3 (a+b \arcsin (c x))^2}{4 c^3 (c x-1)^2 \sqrt {1-c^2 x^2}}\right )dx}{d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-\frac {i (a+b \arcsin (c x))^2 (c f-g)^3}{12 c^4}-\frac {b (a+b \arcsin (c x)) \csc ^2\left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (c f-g)^3}{12 c^4}-\frac {(a+b \arcsin (c x))^2 \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) \csc ^2\left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (c f-g)^3}{24 c^4}-\frac {(a+b \arcsin (c x))^2 \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (c f-g)^3}{12 c^4}-\frac {b^2 \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (c f-g)^3}{6 c^4}+\frac {b (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right ) (c f-g)^3}{3 c^4}-\frac {i b^2 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (c f-g)^3}{3 c^4}-\frac {i (c f+2 g) (a+b \arcsin (c x))^2 (c f-g)^2}{4 c^4}-\frac {(c f+2 g) (a+b \arcsin (c x))^2 \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (c f-g)^2}{4 c^4}+\frac {b (c f+2 g) (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right ) (c f-g)^2}{c^4}-\frac {i b^2 (c f+2 g) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (c f-g)^2}{c^4}+\frac {i (c f+g)^3 (a+b \arcsin (c x))^2}{12 c^4}+\frac {i (c f-2 g) (c f+g)^2 (a+b \arcsin (c x))^2}{4 c^4}-\frac {b (c f+g)^3 (a+b \arcsin (c x)) \sec ^2\left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{12 c^4}+\frac {b (c f+g)^3 (a+b \arcsin (c x)) \log \left (1-i e^{-i \arcsin (c x)}\right )}{3 c^4}+\frac {b (c f-2 g) (c f+g)^2 (a+b \arcsin (c x)) \log \left (1-i e^{-i \arcsin (c x)}\right )}{c^4}+\frac {i b^2 (c f+g)^3 \operatorname {PolyLog}\left (2,i e^{-i \arcsin (c x)}\right )}{3 c^4}+\frac {i b^2 (c f-2 g) (c f+g)^2 \operatorname {PolyLog}\left (2,i e^{-i \arcsin (c x)}\right )}{c^4}+\frac {b^2 (c f+g)^3 \tan \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{6 c^4}+\frac {(c f+g)^3 (a+b \arcsin (c x))^2 \tan \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{12 c^4}+\frac {(c f-2 g) (c f+g)^2 (a+b \arcsin (c x))^2 \tan \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{4 c^4}+\frac {(c f+g)^3 (a+b \arcsin (c x))^2 \sec ^2\left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) \tan \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{24 c^4}\right )}{d^2 \sqrt {d-c^2 d x^2}}\) |
(Sqrt[1 - c^2*x^2]*(((-1/12*I)*(c*f - g)^3*(a + b*ArcSin[c*x])^2)/c^4 + (( I/4)*(c*f - 2*g)*(c*f + g)^2*(a + b*ArcSin[c*x])^2)/c^4 + ((I/12)*(c*f + g )^3*(a + b*ArcSin[c*x])^2)/c^4 - ((I/4)*(c*f - g)^2*(c*f + 2*g)*(a + b*Arc Sin[c*x])^2)/c^4 - (b^2*(c*f - g)^3*Cot[Pi/4 + ArcSin[c*x]/2])/(6*c^4) - ( (c*f - g)^3*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2])/(12*c^4) - (( c*f - g)^2*(c*f + 2*g)*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2])/(4 *c^4) - (b*(c*f - g)^3*(a + b*ArcSin[c*x])*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(1 2*c^4) - ((c*f - g)^3*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2]*Csc[ Pi/4 + ArcSin[c*x]/2]^2)/(24*c^4) + (b*(c*f - 2*g)*(c*f + g)^2*(a + b*ArcS in[c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/c^4 + (b*(c*f + g)^3*(a + b*ArcSin[ c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/(3*c^4) + (b*(c*f - g)^3*(a + b*ArcSin [c*x])*Log[1 - I*E^(I*ArcSin[c*x])])/(3*c^4) + (b*(c*f - g)^2*(c*f + 2*g)* (a + b*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])])/c^4 + (I*b^2*(c*f - 2*g) *(c*f + g)^2*PolyLog[2, I/E^(I*ArcSin[c*x])])/c^4 + ((I/3)*b^2*(c*f + g)^3 *PolyLog[2, I/E^(I*ArcSin[c*x])])/c^4 - ((I/3)*b^2*(c*f - g)^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/c^4 - (I*b^2*(c*f - g)^2*(c*f + 2*g)*PolyLog[2, I*E^ (I*ArcSin[c*x])])/c^4 - (b*(c*f + g)^3*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcS in[c*x]/2]^2)/(12*c^4) + (b^2*(c*f + g)^3*Tan[Pi/4 + ArcSin[c*x]/2])/(6*c^ 4) + ((c*f - 2*g)*(c*f + g)^2*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x] /2])/(4*c^4) + ((c*f + g)^3*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x...
3.1.79.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x] )^n/Sqrt[d + e*x^2], (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] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 13139 vs. \(2 (1473 ) = 2946\).
Time = 1.40 (sec) , antiderivative size = 13140, normalized size of antiderivative = 8.27
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(13140\) |
| parts | \(\text {Expression too large to display}\) | \(13140\) |
\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
integral(-(a^2*g^3*x^3 + 3*a^2*f*g^2*x^2 + 3*a^2*f^2*g*x + a^2*f^3 + (b^2* g^3*x^3 + 3*b^2*f*g^2*x^2 + 3*b^2*f^2*g*x + b^2*f^3)*arcsin(c*x)^2 + 2*(a* b*g^3*x^3 + 3*a*b*f*g^2*x^2 + 3*a*b*f^2*g*x + a*b*f^3)*arcsin(c*x))*sqrt(- c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{3}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
1/3*a*b*c*f^3*(1/(c^4*d^(5/2)*x^2 - c^2*d^(5/2)) + 2*log(c*x + 1)/(c^2*d^( 5/2)) + 2*log(c*x - 1)/(c^2*d^(5/2))) + 2/3*a*b*f^3*(2*x/(sqrt(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d))*arcsin(c*x) + 1/3*a^2*f^3*(2*x/( sqrt(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d)) + 1/3*a^2*g^3*(3 *x^2/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 2/((-c^2*d*x^2 + d)^(3/2)*c^4*d)) - a^2*f*g^2*(x/(sqrt(-c^2*d*x^2 + d)*c^2*d^2) - x/((-c^2*d*x^2 + d)^(3/2)*c^ 2*d)) + sqrt(d)*integrate(((b^2*g^3*x^3 + 3*b^2*f*g^2*x^2 + 3*b^2*f^2*g*x + b^2*f^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*g^3*x^3 + 3*a*b*f*g^2*x^2 + 3*a*b*f^2*g*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1 )))/((c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + a^2*f^2*g/((-c^2*d*x^2 + d)^(3/2)*c^2*d)
Exception generated. \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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
Timed out. \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]