\(\int \frac {(a+b \arcsin (c x))^2}{(f+g x) (d-c^2 d x^2)^{3/2}} \, dx\) [78]

Optimal result

Integrand size = 33, antiderivative size = 1137 \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-i e^{-i \arcsin (c x)}\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{-i \arcsin (c x)}\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {2 b g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 b g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 g^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}} \]

[Out]

-1/2*I*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/d/(c*f-g)/(-c^2*d*x^2+d)^(1/2)+1/2*I*(a+b*arcsin(c*x))^2*(-c^2*x
^2+1)^(1/2)/d/(c*f+g)/(-c^2*d*x^2+d)^(1/2)-1/2*(a+b*arcsin(c*x))^2*cot(1/4*Pi+1/2*arcsin(c*x))*(-c^2*x^2+1)^(1
/2)/d/(c*f-g)/(-c^2*d*x^2+d)^(1/2)+2*b*(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)
/d/(c*f+g)/(-c^2*d*x^2+d)^(1/2)+2*b*(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)/d/
(c*f-g)/(-c^2*d*x^2+d)^(1/2)+I*g^2*(a+b*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*x^2+1)^(1/2)/d/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)-I*g^2*(a+b*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*x^2+1)^(1/2)/d/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)
+2*I*b^2*polylog(2,I/(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/d/(c*f+g)/(-c^2*d*x^2+d)^(1/2)-2*I*b^2*pol
ylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/d/(c*f-g)/(-c^2*d*x^2+d)^(1/2)+2*b*g^2*(a+b*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*x^2+1)^(1/2)/d/(c^2*f^2-g^2)^(3/2
)/(-c^2*d*x^2+d)^(1/2)-2*b*g^2*(a+b*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*x^2+1)^(1/2)/d/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)+2*I*b^2*g^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*x^2+1)^(1/2)/d/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)-2*I*b^2*g
^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*x^2+1)^(1/2)/d/(c^2*f^2-g^2)^(3/2
)/(-c^2*d*x^2+d)^(1/2)+1/2*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)*tan(1/4*Pi+1/2*arcsin(c*x))/d/(c*f+g)/(-c^2*
d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 1137, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {4861, 4859, 4857, 3399, 4269, 3798, 2221, 2317, 2438, 3404, 2296, 2611, 2320, 6724} \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {2 i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{-i \arcsin (c x)}\right ) b^2}{d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {2 i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) b^2}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {2 i g^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) b^2}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 i g^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) b^2}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-i e^{-i \arcsin (c x)}\right ) b}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right ) b}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {2 g^2 \sqrt {1-c^2 x^2} (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 ) b}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 g^2 \sqrt {1-c^2 x^2} (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 ) b}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \tan \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

((-1/2*I)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(d*(c*f - g)*Sqrt[d - c^2*d*x^2]) + ((I/2)*Sqrt[1 - c^2*x^2
]*(a + b*ArcSin[c*x])^2)/(d*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Cot[Pi/4
+ ArcSin[c*x]/2])/(2*d*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I/
E^(I*ArcSin[c*x])])/(d*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I*E
^(I*ArcSin[c*x])])/(d*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (I*g^2*Sqrt[1 - c^2*x^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])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (I*g^2*S
qrt[1 - c^2*x^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])])/(d*(c^2*f
^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + ((2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I/E^(I*ArcSin[c*x])])/(d*(c*f +
g)*Sqrt[d - c^2*d*x^2]) - ((2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(d*(c*f - g)*Sqrt[d -
c^2*d*x^2]) + (2*b*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c
^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (2*b*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]
)*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2
]) + ((2*I)*b^2*g^2*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d*(c^2
*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - ((2*I)*b^2*g^2*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(
c*f + Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[
c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2])/(2*d*(c*f + g)*Sqrt[d - c^2*d*x^2])

Rule 2221

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]
- Dist[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 2296

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]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2438

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 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 3404

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4857

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol]
:> Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(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] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 4859

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] /; Free
Q[{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]

Rule 4861

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

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c^2 x^2} \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = \frac {\sqrt {1-c^2 x^2} \int \left (-\frac {c (a+b \arcsin (c x))^2}{2 (c f+g) (-1+c x) \sqrt {1-c^2 x^2}}+\frac {c (a+b \arcsin (c x))^2}{2 (c f-g) (1+c x) \sqrt {1-c^2 x^2}}+\frac {g^2 (a+b \arcsin (c x))^2}{(-c f+g) (c f+g) (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = \frac {\left (c \sqrt {1-c^2 x^2}\right ) \int \frac {(a+b \arcsin (c x))^2}{(1+c x) \sqrt {1-c^2 x^2}} \, dx}{2 d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (c \sqrt {1-c^2 x^2}\right ) \int \frac {(a+b \arcsin (c x))^2}{(-1+c x) \sqrt {1-c^2 x^2}} \, dx}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{d (-c f+g) (c f+g) \sqrt {d-c^2 d x^2}} \\ & = \frac {\left (c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {(a+b x)^2}{c+c \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {(a+b x)^2}{-c+c \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {(a+b x)^2}{c f+g \sin (x)} \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \sqrt {d-c^2 d x^2}} \\ & = \frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{4 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{4 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (2 g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)^2}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \sqrt {d-c^2 d x^2}} \\ & = -\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \cot \left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \cot \left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\left (2 i g^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)^2}{2 c f-2 i e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 i g^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)^2}{2 c f-2 i e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \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))^2}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{1-i e^{i x}} \, dx,x,\arcsin (c x)\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{-i x} (a+b x)}{1-i e^{-i x}} \, dx,x,\arcsin (c x)\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-\frac {2 i e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-\frac {2 i e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \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))^2}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-i e^{-i \arcsin (c x)}\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 b g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 b g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{-i x}\right ) \, dx,x,\arcsin (c x)\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,\frac {2 i e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,\frac {2 i e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \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))^2}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-i e^{-i \arcsin (c x)}\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 b g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 b g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{-i \arcsin (c x)}\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i g x}{c f-\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i g x}{c f+\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d (-c f+g) (c f+g) \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))^2}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-i e^{-i \arcsin (c x)}\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{-i \arcsin (c x)}\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {2 b g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 b g^2 \sqrt {1-c^2 x^2} (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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 g^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 3.81 (sec) , antiderivative size = 597, normalized size of antiderivative = 0.53 \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {1-c^2 x^2} \left (\frac {-\left ((a+b \arcsin (c x)) \left (-i a+a \cot \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )+b \arcsin (c x) \left (-i+\cot \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-4 b \log \left (1+i e^{-i \arcsin (c x)}\right )\right )\right )+4 i b^2 \operatorname {PolyLog}\left (2,-i e^{-i \arcsin (c x)}\right )}{c f-g}+\frac {2 i g^2 \left ((a+b \arcsin (c x))^2 \log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-(a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{(c f-g) (c f+g) \sqrt {c^2 f^2-g^2}}+\frac {-4 i b^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+(a+b \arcsin (c x)) \left (-i a+4 b \log \left (1+i e^{i \arcsin (c x)}\right )+a \tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )+b \arcsin (c x) \left (-i+\tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )}{c f+g}\right )}{2 d \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

(Sqrt[1 - c^2*x^2]*((-((a + b*ArcSin[c*x])*((-I)*a + a*Cot[(Pi + 2*ArcSin[c*x])/4] + b*ArcSin[c*x]*(-I + Cot[(
Pi + 2*ArcSin[c*x])/4]) - 4*b*Log[1 + I/E^(I*ArcSin[c*x])])) + (4*I)*b^2*PolyLog[2, (-I)/E^(I*ArcSin[c*x])])/(
c*f - g) + ((2*I)*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*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])]))/((c*f - g)*(c*f + g)*S
qrt[c^2*f^2 - g^2]) + ((-4*I)*b^2*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (a + b*ArcSin[c*x])*((-I)*a + 4*b*Log[1
+ I*E^(I*ArcSin[c*x])] + a*Tan[(Pi + 2*ArcSin[c*x])/4] + b*ArcSin[c*x]*(-I + Tan[(Pi + 2*ArcSin[c*x])/4])))/(
c*f + g)))/(2*d*Sqrt[d - c^2*d*x^2])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/(c^4*d^2*g*x^5 + c^4*d^2*f*x^4 - 2
*c^2*d^2*g*x^3 - 2*c^2*d^2*f*x^2 + d^2*g*x + d^2*f), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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