\(\int \frac {a+b \arcsin (c x)}{(f+g x) (d-c^2 d x^2)^{5/2}} \, dx\) [57]

Optimal result

Integrand size = 31, antiderivative size = 1300 \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {(c f-2 g) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{4 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{12 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {i g^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {i g^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{6 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {b (c f+2 g) \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{2 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-2 g) \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{2 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{6 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {b g^4 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {b g^4 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{12 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {(c f+2 g) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{4 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \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 )}{24 d^2 (c f+g) \sqrt {d-c^2 d x^2}} \]

[Out]

-1/4*(c*f-2*g)*(a+b*arcsin(c*x))*cot(1/4*Pi+1/2*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/d^2/(c*f-g)^2/(-c^2*d*x^2+d)^(
1/2)-1/12*(a+b*arcsin(c*x))*cot(1/4*Pi+1/2*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/d^2/(c*f-g)/(-c^2*d*x^2+d)^(1/2)-1/
24*b*csc(1/4*Pi+1/2*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/d^2/(c*f-g)/(-c^2*d*x^2+d)^(1/2)-1/24*(a+b*arcsin(c*x))*
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)/d^2/(c*f-g)/(-c^2*d*x^2+d)^(1/2)+
1/6*b*ln(cos(1/4*Pi+1/2*arcsin(c*x)))*(-c^2*x^2+1)^(1/2)/d^2/(c*f+g)/(-c^2*d*x^2+d)^(1/2)+1/2*b*(c*f+2*g)*ln(c
os(1/4*Pi+1/2*arcsin(c*x)))*(-c^2*x^2+1)^(1/2)/d^2/(c*f+g)^2/(-c^2*d*x^2+d)^(1/2)+1/2*b*(c*f-2*g)*ln(sin(1/4*P
i+1/2*arcsin(c*x)))*(-c^2*x^2+1)^(1/2)/d^2/(c*f-g)^2/(-c^2*d*x^2+d)^(1/2)+1/6*b*ln(sin(1/4*Pi+1/2*arcsin(c*x))
)*(-c^2*x^2+1)^(1/2)/d^2/(c*f-g)/(-c^2*d*x^2+d)^(1/2)-I*g^4*(a+b*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*x^2+1)^(1/2)/d^2/(c^2*f^2-g^2)^(5/2)/(-c^2*d*x^2+d)^(1/2)+I*g^4*(a+b*arcs
in(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*x^2+1)^(1/2)/d^2/(c^2*f^2-g^2)^(
5/2)/(-c^2*d*x^2+d)^(1/2)-b*g^4*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^2/(c^2*f^2-g^2)^(5/2)/(-c^2*d*x^2+d)^(1/2)+b*g^4*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^2/(c^2*f^2-g^2)^(5/2)/(-c^2*d*x^2+d)^(1/2)-1/24*b*sec(1/4*Pi+1/2*arcsin
(c*x))^2*(-c^2*x^2+1)^(1/2)/d^2/(c*f+g)/(-c^2*d*x^2+d)^(1/2)+1/12*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)*tan(1/4
*Pi+1/2*arcsin(c*x))/d^2/(c*f+g)/(-c^2*d*x^2+d)^(1/2)+1/4*(c*f+2*g)*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)*tan(1
/4*Pi+1/2*arcsin(c*x))/d^2/(c*f+g)^2/(-c^2*d*x^2+d)^(1/2)+1/24*(a+b*arcsin(c*x))*sec(1/4*Pi+1/2*arcsin(c*x))^2
*(-c^2*x^2+1)^(1/2)*tan(1/4*Pi+1/2*arcsin(c*x))/d^2/(c*f+g)/(-c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 1300, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {4861, 4859, 4857, 3399, 4270, 4269, 3556, 3404, 2296, 2221, 2317, 2438} \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {i \sqrt {1-c^2 x^2} (a+b \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}{d^2 (c f-g)^2 (c f+g)^2 \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)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) g^4}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^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}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^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}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \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 )}{24 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \csc ^2\left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{24 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \sec ^2\left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{24 d^2 (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{12 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {(c f-2 g) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{4 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f+2 g) \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )\right )}{2 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )\right )}{6 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )\right )}{6 d^2 (c f-g) \sqrt {d-c^2 d x^2}}+\frac {b (c f-2 g) \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )\right )}{2 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \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 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {(c f+2 g) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{4 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )}{12 d^2 (c f+g) \sqrt {d-c^2 d x^2}} \]

[In]

Int[(a + b*ArcSin[c*x])/((f + g*x)*(d - c^2*d*x^2)^(5/2)),x]

[Out]

-1/4*((c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + ArcSin[c*x]/2])/(d^2*(c*f - g)^2*Sqrt[d - c
^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + ArcSin[c*x]/2])/(12*d^2*(c*f - g)*Sqrt[d - c^2*
d*x^2]) - (b*Sqrt[1 - c^2*x^2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(24*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 -
c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + ArcSin[c*x]/2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(24*d^2*(c*f - g)*Sqrt[d
- c^2*d*x^2]) - (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f
^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (I*g^4*Sqrt[1 - c^2*x^2]*
(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2
*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + ArcSin[c*x]/2]])/(6*d^2*(c*f +
g)*Sqrt[d - c^2*d*x^2]) + (b*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + ArcSin[c*x]/2]])/(2*d^2*(c*f + g)^2
*Sqrt[d - c^2*d*x^2]) + (b*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*Log[Sin[Pi/4 + ArcSin[c*x]/2]])/(2*d^2*(c*f - g)^2*Sq
rt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[Sin[Pi/4 + ArcSin[c*x]/2]])/(6*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]
) - (b*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2
*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (b*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x
])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (b
*Sqrt[1 - c^2*x^2]*Sec[Pi/4 + ArcSin[c*x]/2]^2)/(24*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a
+ b*ArcSin[c*x])*Tan[Pi/4 + ArcSin[c*x]/2])/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + ((c*f + 2*g)*Sqrt[1 - c^
2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + ArcSin[c*x]/2])/(4*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*
x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2])/(24*d^2*(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 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 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 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

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 4270

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{(f+g x) \left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {\sqrt {1-c^2 x^2} \int \left (\frac {c (a+b \arcsin (c x))}{4 (c f+g) (-1+c x)^2 \sqrt {1-c^2 x^2}}-\frac {c (c f+2 g) (a+b \arcsin (c x))}{4 (c f+g)^2 (-1+c x) \sqrt {1-c^2 x^2}}+\frac {c (a+b \arcsin (c x))}{4 (c f-g) (1+c x)^2 \sqrt {1-c^2 x^2}}+\frac {c (c f-2 g) (a+b \arcsin (c x))}{4 (c f-g)^2 (1+c x) \sqrt {1-c^2 x^2}}+\frac {g^4 (a+b \arcsin (c x))}{(-c f+g)^2 (c f+g)^2 (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {\left (c (c f-2 g) \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{(1+c x) \sqrt {1-c^2 x^2}} \, dx}{4 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}+\frac {\left (c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{(1+c x)^2 \sqrt {1-c^2 x^2}} \, dx}{4 d^2 (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\left (g^4 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {\left (c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{(-1+c x)^2 \sqrt {1-c^2 x^2}} \, dx}{4 d^2 (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\left (c (c f+2 g) \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{(-1+c x) \sqrt {1-c^2 x^2}} \, dx}{4 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {\left (c (c f-2 g) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{c+c \sin (x)} \, dx,x,\arcsin (c x)\right )}{4 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{(c+c \sin (x))^2} \, dx,x,\arcsin (c x)\right )}{4 d^2 (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\left (g^4 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{c f+g \sin (x)} \, dx,x,\arcsin (c x)\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{(-c+c \sin (x))^2} \, dx,x,\arcsin (c x)\right )}{4 d^2 (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\left (c (c f+2 g) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{-c+c \sin (x)} \, dx,x,\arcsin (c x)\right )}{4 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {\left ((c f-2 g) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \csc ^2\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{8 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x) \csc ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{16 d^2 (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\left (2 g^4 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\arcsin (c x)\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x) \csc ^4\left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{16 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left ((c f+2 g) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \csc ^2\left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{8 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {(c f-2 g) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{4 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {(c f+2 g) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{4 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \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 )}{24 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (b (c f-2 g) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \cot \left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{4 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x) \csc ^2\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{24 d^2 (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x) \csc ^2\left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{24 d^2 (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+2 g) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \cot \left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{4 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 i g^5 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{2 c f-2 i e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\arcsin (c x)\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 i g^5 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{2 c f-2 i e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\arcsin (c x)\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}} \\ & = -\frac {(c f-2 g) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{4 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{12 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {i g^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {i g^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {b (c f+2 g) \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{2 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-2 g) \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{2 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{12 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {(c f+2 g) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{4 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \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 )}{24 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \cot \left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{12 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \cot \left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{12 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (i b g^4 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \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^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\left (i b g^4 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \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^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}} \\ & = -\frac {(c f-2 g) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{4 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{12 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {i g^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {i g^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{6 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {b (c f+2 g) \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{2 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-2 g) \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{2 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{6 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{12 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {(c f+2 g) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{4 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \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 )}{24 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (b g^4 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\left (b g^4 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}} \\ & = -\frac {(c f-2 g) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{4 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{12 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {i g^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {i g^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{6 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {b (c f+2 g) \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{2 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-2 g) \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{2 d^2 (c f-g)^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{6 d^2 (c f-g) \sqrt {d-c^2 d x^2}}-\frac {b g^4 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {b g^4 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d^2 (c f-g)^2 (c f+g)^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{24 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{12 d^2 (c f+g) \sqrt {d-c^2 d x^2}}+\frac {(c f+2 g) \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{4 d^2 (c f+g)^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \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 )}{24 d^2 (c f+g) \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 12.89 (sec) , antiderivative size = 2078, normalized size of antiderivative = 1.60 \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Result too large to show} \]

[In]

Integrate[(a + b*ArcSin[c*x])/((f + g*x)*(d - c^2*d*x^2)^(5/2)),x]

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*((a*g - a*c^2*f*x)/(3*d^3*(-(c^2*f^2) + g^2)*(-1 + c^2*x^2)^2) + (-3*a*g^3 - 2*a*c^4
*f^3*x + 5*a*c^2*f*g^2*x)/(3*d^3*(-(c^2*f^2) + g^2)^2*(-1 + c^2*x^2))) + (a*g^4*Log[f + g*x])/(d^(5/2)*(-(c*f)
+ g)^2*(c*f + g)^2*Sqrt[-(c^2*f^2) + g^2]) - (a*g^4*Log[d*g + c^2*d*f*x + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt
[-(d*(-1 + c^2*x^2))]])/(d^(5/2)*(-(c*f) + g)^2*(c*f + g)^2*Sqrt[-(c^2*f^2) + g^2]) + (b*((g*(-(c^2*f^2) + 7*g
^2)*(1 - c^2*x^2)^(3/2)*ArcSin[c*x])/(6*(-(c^2*f^2) + g^2)^2*(d*(1 - c^2*x^2))^(3/2)) + ((4*c*f + 7*g)*(1 - c^
2*x^2)^(3/2)*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])/(6*(c*f + g)^2*(d*(1 - c^2*x^2))^(3/2)) + ((4*c*f -
7*g)*(1 - c^2*x^2)^(3/2)*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])/(6*(c*f - g)^2*(d*(1 - c^2*x^2))^(3/2)
) + (g^4*(1 - c^2*x^2)^(3/2)*((Pi*ArcTan[(g + c*f*Tan[ArcSin[c*x]/2])/Sqrt[c^2*f^2 - g^2]])/Sqrt[c^2*f^2 - g^2
] + (2*(Pi/2 - ArcSin[c*x])*ArcTanh[((c*f + g)*Cot[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]] - 2*ArcCos
[-((c*f)/g)]*ArcTanh[((-(c*f) + g)*Tan[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)]
- (2*I)*(ArcTanh[((c*f + g)*Cot[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[(
Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*(Pi/2 - ArcSin[
c*x]))*Sqrt[g]*Sqrt[c*f + c*g*x])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[(Pi/2 - ArcSin[c*x])/
2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log
[(E^((I/2)*(Pi/2 - ArcSin[c*x]))*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*f + c*g*x])] - (ArcCos[-((c*f
)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[1 - ((c*f - I*Sq
rt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))/(g*(c*f + g + Sqrt[-(c^2
*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))] + (-ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[(Pi/2 - A
rcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[1 - ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2)
+ g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))] + I*(
PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))/(
g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^
2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[(
Pi/2 - ArcSin[c*x])/2]))]))/Sqrt[-(c^2*f^2) + g^2]))/((-(c*f) + g)^2*(c*f + g)^2*(d*(1 - c^2*x^2))^(3/2)) + ((
1 - c^2*x^2)^(3/2)*(-1 + ArcSin[c*x]))/(12*(c*f + g)*(d*(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[
c*x]/2])^2) + ((1 - c^2*x^2)^(3/2)*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(6*(c*f + g)*(d*(1 - c^2*x^2))^(3/2)*(Cos[A
rcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3) + ((1 - c^2*x^2)^(3/2)*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(6*(c*f - g)*(d*
(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3) + ((1 - c^2*x^2)^(3/2)*(-1 - ArcSin[c*x]))/(
12*(c*f - g)*(d*(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + ((1 - c^2*x^2)^(3/2)*(4*c*
f*ArcSin[c*x]*Sin[ArcSin[c*x]/2] - 7*g*ArcSin[c*x]*Sin[ArcSin[c*x]/2]))/(6*(c*f - g)^2*(d*(1 - c^2*x^2))^(3/2)
*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) + ((1 - c^2*x^2)^(3/2)*(4*c*f*ArcSin[c*x]*Sin[ArcSin[c*x]/2] + 7*g
*ArcSin[c*x]*Sin[ArcSin[c*x]/2]))/(6*(c*f + g)^2*(d*(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]
/2]))))/d

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 7970 vs. \(2 (1102 ) = 2204\).

Time = 1.50 (sec) , antiderivative size = 7971, normalized size of antiderivative = 6.13

method	result	size
default	\(\text {Expression too large to display}\)	\(7971\)
parts	\(\text {Expression too large to display}\)	\(7971\)

[In]

int((a+b*arcsin(c*x))/(g*x+f)/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

\[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}} \,d x } \]

[In]

integrate((a+b*arcsin(c*x))/(g*x+f)/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)/(c^6*d^3*g*x^7 + c^6*d^3*f*x^6 - 3*c^4*d^3*g*x^5 - 3*c^4*d^
3*f*x^4 + 3*c^2*d^3*g*x^3 + 3*c^2*d^3*f*x^2 - d^3*g*x - d^3*f), x)

Sympy [F]

\[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (f + g x\right )}\, dx \]

[In]

integrate((a+b*asin(c*x))/(g*x+f)/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Integral((a + b*asin(c*x))/((-d*(c*x - 1)*(c*x + 1))**(5/2)*(f + g*x)), x)

Maxima [F]

\[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}} \,d x } \]

[In]

integrate((a+b*arcsin(c*x))/(g*x+f)/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*arcsin(c*x) + a)/((-c^2*d*x^2 + d)^(5/2)*(g*x + f)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arcsin(c*x))/(g*x+f)/(-c^2*d*x^2+d)^(5/2),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. \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{\left (f+g\,x\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]

[In]

int((a + b*asin(c*x))/((f + g*x)*(d - c^2*d*x^2)^(5/2)),x)

[Out]

int((a + b*asin(c*x))/((f + g*x)*(d - c^2*d*x^2)^(5/2)), x)