Integrand size = 32, antiderivative size = 896 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b^2 d^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \arcsin (c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {d^2 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {c^2 d^2 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^2 x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {i d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 i b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \] Output:
2/3*b^2*d^2*(-c^2*x^2+1)^2/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)+2/3*b^2*d^2* x*(-c^2*x^2+1)^2/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)-1/3*b^2*d^2*(-c^2*x^2+1) ^(5/2)*arcsin(c*x)/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)-1/3*b*d^2*(-c^2*x^2+ 1)^(3/2)*(a+b*arcsin(c*x))/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)-2/3*b*d^2*x* (-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)-1/3* b*c*d^2*x^2*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))/(c*d*x+d)^(5/2)/(-c*e*x+e )^(5/2)+2/3*d^2*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/c/(c*d*x+d)^(5/2)/(-c*e*x +e)^(5/2)+1/3*d^2*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/(c*d*x+d)^(5/2)/(-c*e *x+e)^(5/2)+1/3*c^2*d^2*x^3*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/(c*d*x+d)^(5/ 2)/(-c*e*x+e)^(5/2)+2/3*d^2*x*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2/(c*d*x+d) ^(5/2)/(-c*e*x+e)^(5/2)-2/3*I*b^2*d^2*(-c^2*x^2+1)^(5/2)*polylog(2,-I*(I*c *x+(-c^2*x^2+1)^(1/2)))/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)-1/3*I*b^2*d^2*( -c^2*x^2+1)^(5/2)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/c/(c*d*x+d)^(5/ 2)/(-c*e*x+e)^(5/2)+2/3*b*d^2*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))*ln(1+(I *c*x+(-c^2*x^2+1)^(1/2))^2)/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)+2/3*I*b^2*d ^2*(-c^2*x^2+1)^(5/2)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c/(c*d*x+d)^ (5/2)/(-c*e*x+e)^(5/2)-1/3*I*d^2*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))^2/c/ (c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)+4/3*I*b*d^2*(-c^2*x^2+1)^(5/2)*(a+b*arcsi n(c*x))*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2 )
Time = 5.79 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.43 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (-\frac {2 a^2 (-2+c x)}{(-1+c x)^2}+\frac {2 a b \left (\cos \left (\frac {3}{2} \arcsin (c x)\right ) \left (\arcsin (c x)-2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )+\cos \left (\frac {1}{2} \arcsin (c x)\right ) \left (-2+3 \arcsin (c x)+6 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )+2 \left (1-\left (-1+\sqrt {1-c^2 x^2}\right ) \arcsin (c x)-2 \left (2+\sqrt {1-c^2 x^2}\right ) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right ) \sin \left (\frac {1}{2} \arcsin (c x)\right )\right )}{\sqrt {1-c^2 x^2} \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^3}+\frac {b^2 \left (-8 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+\arcsin (c x) \left (8 \log \left (1+i e^{i \arcsin (c x)}\right )-2 \sec ^2\left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+4 \tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )+\arcsin (c x)^2 \left (-2 i+\left (2+\sec ^2\left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right ) \tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )}{\sqrt {1-c^2 x^2}}\right )}{6 c d e^3} \] Input:
Integrate[(a + b*ArcSin[c*x])^2/(Sqrt[d + c*d*x]*(e - c*e*x)^(5/2)),x]
Output:
(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*((-2*a^2*(-2 + c*x))/(-1 + c*x)^2 + (2*a* b*(Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*x] - 2*Log[Cos[ArcSin[c*x]/2] - Sin[Ar cSin[c*x]/2]]) + Cos[ArcSin[c*x]/2]*(-2 + 3*ArcSin[c*x] + 6*Log[Cos[ArcSin [c*x]/2] - Sin[ArcSin[c*x]/2]]) + 2*(1 - (-1 + Sqrt[1 - c^2*x^2])*ArcSin[c *x] - 2*(2 + Sqrt[1 - c^2*x^2])*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2 ]])*Sin[ArcSin[c*x]/2]))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcS in[c*x]/2])^3) + (b^2*((-8*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + ArcSin[ c*x]*(8*Log[1 + I*E^(I*ArcSin[c*x])] - 2*Sec[(Pi + 2*ArcSin[c*x])/4]^2) + 4*Tan[(Pi + 2*ArcSin[c*x])/4] + ArcSin[c*x]^2*(-2*I + (2 + Sec[(Pi + 2*Arc Sin[c*x])/4]^2)*Tan[(Pi + 2*ArcSin[c*x])/4])))/Sqrt[1 - c^2*x^2]))/(6*c*d* e^3)
Time = 1.65 (sec) , antiderivative size = 465, normalized size of antiderivative = 0.52, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5178, 27, 5262, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{\sqrt {c d x+d} (e-c e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5178 |
| \(\displaystyle \frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {d^2 (c x+1)^2 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}dx}{(c d x+d)^{5/2} (e-c e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (1-c^2 x^2\right )^{5/2} \int \frac {(c x+1)^2 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}dx}{(c d x+d)^{5/2} (e-c e x)^{5/2}}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {d^2 \left (1-c^2 x^2\right )^{5/2} \int \left (\frac {c^2 x^2 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac {2 c x (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}\right )dx}{(c d x+d)^{5/2} (e-c e x)^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \left (1-c^2 x^2\right )^{5/2} \left (\frac {4 i b \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 c}-\frac {b c x^2 (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )}+\frac {2 x (a+b \arcsin (c x))^2}{3 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arcsin (c x))^2}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )}+\frac {2 (a+b \arcsin (c x))^2}{3 c \left (1-c^2 x^2\right )^{3/2}}-\frac {b (a+b \arcsin (c x))}{3 c \left (1-c^2 x^2\right )}+\frac {c^2 x^3 (a+b \arcsin (c x))^2}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {i (a+b \arcsin (c x))^2}{3 c}+\frac {2 b \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 c}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c}+\frac {2 i b^2 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c}-\frac {i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c}-\frac {b^2 \arcsin (c x)}{3 c}+\frac {2 b^2 x}{3 \sqrt {1-c^2 x^2}}+\frac {2 b^2}{3 c \sqrt {1-c^2 x^2}}\right )}{(c d x+d)^{5/2} (e-c e x)^{5/2}}\) |
Input:
Int[(a + b*ArcSin[c*x])^2/(Sqrt[d + c*d*x]*(e - c*e*x)^(5/2)),x]
Output:
(d^2*(1 - c^2*x^2)^(5/2)*((2*b^2)/(3*c*Sqrt[1 - c^2*x^2]) + (2*b^2*x)/(3*S qrt[1 - c^2*x^2]) - (b^2*ArcSin[c*x])/(3*c) - (b*(a + b*ArcSin[c*x]))/(3*c *(1 - c^2*x^2)) - (2*b*x*(a + b*ArcSin[c*x]))/(3*(1 - c^2*x^2)) - (b*c*x^2 *(a + b*ArcSin[c*x]))/(3*(1 - c^2*x^2)) - ((I/3)*(a + b*ArcSin[c*x])^2)/c + (2*(a + b*ArcSin[c*x])^2)/(3*c*(1 - c^2*x^2)^(3/2)) + (x*(a + b*ArcSin[c *x])^2)/(3*(1 - c^2*x^2)^(3/2)) + (c^2*x^3*(a + b*ArcSin[c*x])^2)/(3*(1 - c^2*x^2)^(3/2)) + (2*x*(a + b*ArcSin[c*x])^2)/(3*Sqrt[1 - c^2*x^2]) + (((4 *I)/3)*b*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/c + (2*b*(a + b*Ar cSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(3*c) - (((2*I)/3)*b^2*PolyLog[ 2, (-I)*E^(I*ArcSin[c*x])])/c + (((2*I)/3)*b^2*PolyLog[2, I*E^(I*ArcSin[c* x])])/c - ((I/3)*b^2*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/c))/((d + c*d*x)^ (5/2)*(e - c*e*x)^(5/2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 - c^2*x^ 2)^q) Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] & & EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ [n, 0] && (m == 1 || p > 0 || (n == 1 && p > -1) || (m == 2 && p < -2))
Time = 3.07 (sec) , antiderivative size = 667, normalized size of antiderivative = 0.74
| method | result | size |
| default | \(a^{2} \left (\frac {\sqrt {c d x +d}}{3 c d e \left (-c x e +e \right )^{\frac {3}{2}}}+\frac {\sqrt {c d x +d}}{3 c d \,e^{2} \sqrt {-c x e +e}}\right )+\frac {i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (2 i \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+4 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\arcsin \left (c x \right )^{2} x^{2} c^{2}+2 i \arcsin \left (c x \right ) x c +4 \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}-8 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x c +2 i \sqrt {-c^{2} x^{2}+1}-2 i \sqrt {-c^{2} x^{2}+1}\, c x -2 \arcsin \left (c x \right )^{2} c x -2 c^{2} x^{2}+4 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}-8 \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x c -i \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, x c -2 i \arcsin \left (c x \right )+\arcsin \left (c x \right )^{2}+4 c x +4 \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2\right )}{3 d \,e^{3} c \left (c^{4} x^{4}-2 c^{3} x^{3}+2 c x -1\right )}+\frac {2 a b \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) x^{2} c^{2}-2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right ) x^{2} c^{2}+\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -2 i \arcsin \left (c x \right ) x c +4 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right ) x c -2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+i \arcsin \left (c x \right )-c x -2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )+1\right )}{3 d \,e^{3} c \left (c^{4} x^{4}-2 c^{3} x^{3}+2 c x -1\right )}\) | \(667\) |
| parts | \(a^{2} \left (\frac {\sqrt {c d x +d}}{3 c d e \left (-c x e +e \right )^{\frac {3}{2}}}+\frac {\sqrt {c d x +d}}{3 c d \,e^{2} \sqrt {-c x e +e}}\right )+\frac {i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (2 i \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+4 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\arcsin \left (c x \right )^{2} x^{2} c^{2}+2 i \arcsin \left (c x \right ) x c +4 \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}-8 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x c +2 i \sqrt {-c^{2} x^{2}+1}-2 i \sqrt {-c^{2} x^{2}+1}\, c x -2 \arcsin \left (c x \right )^{2} c x -2 c^{2} x^{2}+4 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}-8 \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x c -i \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, x c -2 i \arcsin \left (c x \right )+\arcsin \left (c x \right )^{2}+4 c x +4 \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2\right )}{3 d \,e^{3} c \left (c^{4} x^{4}-2 c^{3} x^{3}+2 c x -1\right )}+\frac {2 a b \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) x^{2} c^{2}-2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right ) x^{2} c^{2}+\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -2 i \arcsin \left (c x \right ) x c +4 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right ) x c -2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+i \arcsin \left (c x \right )-c x -2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )+1\right )}{3 d \,e^{3} c \left (c^{4} x^{4}-2 c^{3} x^{3}+2 c x -1\right )}\) | \(667\) |
Input:
int((a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(5/2),x,method=_RETURNV ERBOSE)
Output:
a^2*(1/3/c/d/e/(-c*e*x+e)^(3/2)*(c*d*x+d)^(1/2)+1/3/c/d/e^2/(-c*e*x+e)^(1/ 2)*(c*d*x+d)^(1/2))+1/3*I*b^2*(-c^2*x^2+1)^(1/2)*(d*(c*x+1))^(1/2)*(-e*(c* x-1))^(1/2)*(2*I*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)+4*I*arcsin(c*x)*ln(1+I*( I*c*x+(-c^2*x^2+1)^(1/2)))+arcsin(c*x)^2*x^2*c^2+2*I*arcsin(c*x)*c*x+4*pol ylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))*x^2*c^2-8*I*arcsin(c*x)*ln(1+I*(I*c* x+(-c^2*x^2+1)^(1/2)))*x*c+2*I*(-c^2*x^2+1)^(1/2)-2*I*(-c^2*x^2+1)^(1/2)*c *x-2*arcsin(c*x)^2*c*x-2*c^2*x^2+4*I*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1 )^(1/2)))*x^2*c^2-8*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))*x*c-I*arcsin( c*x)^2*(-c^2*x^2+1)^(1/2)*x*c-2*I*arcsin(c*x)+arcsin(c*x)^2+4*c*x+4*polylo g(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-2)/d/e^3/c/(c^4*x^4-2*c^3*x^3+2*c*x-1)+ 2/3*a*b*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)*(I*arcsin( c*x)*c^2*x^2-2*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)*x^2*c^2+arcsin(c*x)*(-c^2*x^ 2+1)^(1/2)*c*x-2*I*arcsin(c*x)*c*x+4*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)*x*c-2* arcsin(c*x)*(-c^2*x^2+1)^(1/2)+I*arcsin(c*x)-c*x-2*ln(I*c*x+(-c^2*x^2+1)^( 1/2)-I)+1)/d/e^3/c/(c^4*x^4-2*c^3*x^3+2*c*x-1)
\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {c d x + d} {\left (-c e x + e\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(5/2),x, algorith m="fricas")
Output:
integral(-(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)*sqrt(c*d*x + d)*sq rt(-c*e*x + e)/(c^4*d*e^3*x^4 - 2*c^3*d*e^3*x^3 + 2*c*d*e^3*x - d*e^3), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c x + 1\right )} \left (- e \left (c x - 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((a+b*asin(c*x))**2/(c*d*x+d)**(1/2)/(-c*e*x+e)**(5/2),x)
Output:
Integral((a + b*asin(c*x))**2/(sqrt(d*(c*x + 1))*(-e*(c*x - 1))**(5/2)), x )
Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(5/2),x, algorith m="maxima")
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(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {c d x + d} {\left (-c e x + e\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(5/2),x, algorith m="giac")
Output:
integrate((b*arcsin(c*x) + a)^2/(sqrt(c*d*x + d)*(-c*e*x + e)^(5/2)), x)
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d+c\,d\,x}\,{\left (e-c\,e\,x\right )}^{5/2}} \,d x \] Input:
int((a + b*asin(c*x))^2/((d + c*d*x)^(1/2)*(e - c*e*x)^(5/2)),x)
Output:
int((a + b*asin(c*x))^2/((d + c*d*x)^(1/2)*(e - c*e*x)^(5/2)), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{5/2}} \, dx=\frac {6 \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{2}-2 \sqrt {c x +1}\, \sqrt {-c x +1}\, c x +\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) a b \,c^{2} x -6 \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{2}-2 \sqrt {c x +1}\, \sqrt {-c x +1}\, c x +\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) a b c +3 \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{2}-2 \sqrt {c x +1}\, \sqrt {-c x +1}\, c x +\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) b^{2} c^{2} x -3 \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{2}-2 \sqrt {c x +1}\, \sqrt {-c x +1}\, c x +\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) b^{2} c +\sqrt {c x +1}\, a^{2} c x -2 \sqrt {c x +1}\, a^{2}}{3 \sqrt {e}\, \sqrt {d}\, \sqrt {-c x +1}\, c \,e^{2} \left (c x -1\right )} \] Input:
int((a+b*asin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(5/2),x)
Output:
(6*sqrt( - c*x + 1)*int(asin(c*x)/(sqrt(c*x + 1)*sqrt( - c*x + 1)*c**2*x** 2 - 2*sqrt(c*x + 1)*sqrt( - c*x + 1)*c*x + sqrt(c*x + 1)*sqrt( - c*x + 1)) ,x)*a*b*c**2*x - 6*sqrt( - c*x + 1)*int(asin(c*x)/(sqrt(c*x + 1)*sqrt( - c *x + 1)*c**2*x**2 - 2*sqrt(c*x + 1)*sqrt( - c*x + 1)*c*x + sqrt(c*x + 1)*s qrt( - c*x + 1)),x)*a*b*c + 3*sqrt( - c*x + 1)*int(asin(c*x)**2/(sqrt(c*x + 1)*sqrt( - c*x + 1)*c**2*x**2 - 2*sqrt(c*x + 1)*sqrt( - c*x + 1)*c*x + s qrt(c*x + 1)*sqrt( - c*x + 1)),x)*b**2*c**2*x - 3*sqrt( - c*x + 1)*int(asi n(c*x)**2/(sqrt(c*x + 1)*sqrt( - c*x + 1)*c**2*x**2 - 2*sqrt(c*x + 1)*sqrt ( - c*x + 1)*c*x + sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*b**2*c + sqrt(c*x + 1)*a**2*c*x - 2*sqrt(c*x + 1)*a**2)/(3*sqrt(e)*sqrt(d)*sqrt( - c*x + 1)*c* e**2*(c*x - 1))