\(\int \frac {(a+b \arcsin (c x))^2}{x (d-c^2 d x^2)^{5/2}} \, dx\) [260]

Optimal result

Integrand size = 29, antiderivative size = 577 \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(a+b \arcsin (c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

1/3*(a+b*arcsin(c*x))^2/d/(-c^2*d*x^2+d)^(3/2)+1/3*b^2/d^2/(-c^2*d*x^2+d)^(1/2)+(a+b*arcsin(c*x))^2/d^2/(-c^2*
d*x^2+d)^(1/2)-1/3*b*c*x*(a+b*arcsin(c*x))/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+14/3*I*b*(a+b*arcsin(c*
x))*arctan(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-2*(a+b*arcsin(c*x))^2*arctanh
(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)+2*I*b*(a+b*arcsin(c*x))*polylog(2,-I*c*
x-(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-7/3*I*b^2*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(
1/2)))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)+7/3*I*b^2*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2
+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-2*I*b*(a+b*arcsin(c*x))*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1
/2)/d^2/(-c^2*d*x^2+d)^(1/2)-2*b^2*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^
(1/2)+2*b^2*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {4793, 4803, 4268, 2611, 2320, 6724, 4749, 4266, 2317, 2438, 4747, 267} \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {14 i b \sqrt {1-c^2 x^2} \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {7 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

b^2/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) +
(a + b*ArcSin[c*x])^2/(3*d*(d - c^2*d*x^2)^(3/2)) + (a + b*ArcSin[c*x])^2/(d^2*Sqrt[d - c^2*d*x^2]) + (((14*I)
/3)*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (2*Sqrt[1 -
c^2*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) + ((2*I)*b*Sqrt[1 - c^2*
x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (((7*I)/3)*b^2*Sqrt[1 - c
^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) + (((7*I)/3)*b^2*Sqrt[1 - c^2*x^2]*PolyL
og[2, I*E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - ((2*I)*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog
[2, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/(
d^2*Sqrt[d - c^2*d*x^2]) + (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2])

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

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 4266

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

Rule 4268

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

Rule 4747

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(p
+ 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a +
b*ArcSin[c*x])^n, x], x] + Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[x*(1 - c^2*x^2)^(
p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] &
& LtQ[p, -1] && NeQ[p, -3/2]

Rule 4749

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/(c*d), Subst[Int[(a +
b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 4793

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(p
+ 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*c*(n/(2*f*(p + 1)))*Simp[(d + e*
x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; Fre
eQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] ||
IntegerQ[p] || EqQ[n, 1])

Rule 4803

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(1/c^(m
+ 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; Free
Q[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]

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 {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx}{d}-\frac {\left (2 b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c x (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(a+b \arcsin (c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx}{d^2}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{1-c^2 x^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(a+b \arcsin (c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\arcsin (c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \sec (x) \, dx,x,\arcsin (c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \sec (x) \, dx,x,\arcsin (c x))}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(a+b \arcsin (c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (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 )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (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 )}{3 d^2 \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^2 \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^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(a+b \arcsin (c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (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 )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (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 )}{3 d^2 \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^2 \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^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(a+b \arcsin (c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(a+b \arcsin (c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 8.74 (sec) , antiderivative size = 935, normalized size of antiderivative = 1.62 \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (\frac {a^2}{3 d^3 \left (-1+c^2 x^2\right )^2}-\frac {a^2}{d^3 \left (-1+c^2 x^2\right )}\right )+\frac {a^2 \log (c x)}{d^{5/2}}-\frac {a^2 \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{d^{5/2}}+\frac {b^2 \left (1-c^2 x^2\right )^{3/2} \left (4-\frac {(-2+\arcsin (c x)) \arcsin (c x)}{-1+c x}+14 \arcsin (c x)^2+12 \arcsin (c x)^2 \left (\log \left (1-e^{i \arcsin (c x)}\right )-\log \left (1+e^{i \arcsin (c x)}\right )\right )-28 \left (\arcsin (c x) \left (\log \left (1-i e^{i \arcsin (c x)}\right )-\log \left (1+i e^{i \arcsin (c x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )\right )+24 i \arcsin (c x) \left (\operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )+24 \left (-\operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+\operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )+\frac {2 \arcsin (c x)^2 \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^3}+\frac {2 \left (2+7 \arcsin (c x)^2\right ) \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )}-\frac {2 \arcsin (c x)^2 \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^3}+\frac {\arcsin (c x) (2+\arcsin (c x))}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^2}-\frac {2 \left (2+7 \arcsin (c x)^2\right ) \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )}\right )}{12 d \left (d \left (1-c^2 x^2\right )\right )^{3/2}}+\frac {a b \left (20 \arcsin (c x)+12 \arcsin (c x) \cos (2 \arcsin (c x))+18 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+6 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1-e^{i \arcsin (c x)}\right )-18 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-6 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1+e^{i \arcsin (c x)}\right )+21 \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+7 \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-21 \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-7 \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+24 i \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-24 i \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-2 \sin (2 \arcsin (c x))\right )}{12 d \left (d \left (1-c^2 x^2\right )\right )^{3/2}} \]

[In]

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

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*(a^2/(3*d^3*(-1 + c^2*x^2)^2) - a^2/(d^3*(-1 + c^2*x^2))) + (a^2*Log[c*x])/d^(5/2) -
(a^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/d^(5/2) + (b^2*(1 - c^2*x^2)^(3/2)*(4 - ((-2 + ArcSin[c*x])*
ArcSin[c*x])/(-1 + c*x) + 14*ArcSin[c*x]^2 + 12*ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSi
n[c*x])]) - 28*(ArcSin[c*x]*(Log[1 - I*E^(I*ArcSin[c*x])] - Log[1 + I*E^(I*ArcSin[c*x])]) + I*(PolyLog[2, (-I)
*E^(I*ArcSin[c*x])] - PolyLog[2, I*E^(I*ArcSin[c*x])])) + (24*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] -
PolyLog[2, E^(I*ArcSin[c*x])]) + 24*(-PolyLog[3, -E^(I*ArcSin[c*x])] + PolyLog[3, E^(I*ArcSin[c*x])]) + (2*Ar
cSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (2*(2 + 7*ArcSin[c*x]^2)*Sin[Arc
Sin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - (2*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]
/2] + Sin[ArcSin[c*x]/2])^3 + (ArcSin[c*x]*(2 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (2
*(2 + 7*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(12*d*(d*(1 - c^2*x^2))
^(3/2)) + (a*b*(20*ArcSin[c*x] + 12*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + 18*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 -
E^(I*ArcSin[c*x])] + 6*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 - E^(I*ArcSin[c*x])] - 18*Sqrt[1 - c^2*x^2]*ArcSin
[c*x]*Log[1 + E^(I*ArcSin[c*x])] - 6*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + E^(I*ArcSin[c*x])] + 21*Sqrt[1 - c
^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 7*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSi
n[c*x]/2]] - 21*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 7*Cos[3*ArcSin[c*x]]*Log[Cos[
ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + (24*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^(I*ArcSin[c*x])] - (24*I)*(1 -
c^2*x^2)^(3/2)*PolyLog[2, E^(I*ArcSin[c*x])] - 2*Sin[2*ArcSin[c*x]]))/(12*d*(d*(1 - c^2*x^2))^(3/2))

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 924, normalized size of antiderivative = 1.60

method	result	size
default	\(\frac {a^{2}}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a^{2}}{d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arcsin \left (c x \right )^{2} x^{2} c^{2}+\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +c^{2} x^{2}-4 \arcsin \left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3}}-\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 i \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 i \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-7 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+7 i \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+6 \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-7 \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+7 \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{3 d^{3} \left (c^{2} x^{2}-1\right )}\right )-\frac {i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}+6 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+6 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+14 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}-i x^{3} c^{3}-8 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )+i c x -12 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-12 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-28 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}+6 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+6 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-12 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}+6 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+14 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{3 \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\)	\(924\)
parts	\(\frac {a^{2}}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a^{2}}{d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arcsin \left (c x \right )^{2} x^{2} c^{2}+\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +c^{2} x^{2}-4 \arcsin \left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3}}-\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 i \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 i \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-7 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+7 i \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+6 \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-7 \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+7 \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{3 d^{3} \left (c^{2} x^{2}-1\right )}\right )-\frac {i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}+6 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+6 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+14 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}-i x^{3} c^{3}-8 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )+i c x -12 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-12 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-28 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}+6 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+6 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-12 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}+6 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+14 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{3 \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\)	\(924\)

[In]

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

[Out]

1/3*a^2/d/(-c^2*d*x^2+d)^(3/2)+a^2/d^2/(-c^2*d*x^2+d)^(1/2)-a^2/d^(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2)
)/x)+b^2*(-1/3*(-d*(c^2*x^2-1))^(1/2)*(3*arcsin(c*x)^2*x^2*c^2+(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x*c+c^2*x^2-4*ar
csin(c*x)^2-1)/(c^2*x^2-1)^2/d^3-1/3*I*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(3*I*arcsin(c*x)^2*ln(1+I*c*x
+(-c^2*x^2+1)^(1/2))-3*I*arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-7*I*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+
1)^(1/2)))+7*I*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+6*arcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2
))-6*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+6*I*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))-6*I*polylog(3,I*
c*x+(-c^2*x^2+1)^(1/2))-7*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+7*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))))/d^3/(
c^2*x^2-1))-1/3*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(6*I*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2*c^2+6*
dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4+6*dilog(I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4+14*arctan(I*c*x+(-c^2*x^2+
1)^(1/2))*c^4*x^4-I*x^3*c^3-8*I*(-c^2*x^2+1)^(1/2)*arcsin(c*x)+I*c*x-12*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^2*
x^2-12*dilog(I*c*x+(-c^2*x^2+1)^(1/2))*c^2*x^2-28*arctan(I*c*x+(-c^2*x^2+1)^(1/2))*c^2*x^2+6*I*arcsin(c*x)*ln(
1+I*c*x+(-c^2*x^2+1)^(1/2))*x^4*c^4+6*I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-12*I*arcsin(c*x)*ln(1+I*c*x
+(-c^2*x^2+1)^(1/2))*x^2*c^2+6*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))+6*dilog(I*c*x+(-c^2*x^2+1)^(1/2))+14*arctan(I
*c*x+(-c^2*x^2+1)^(1/2)))/(c^6*x^6-3*c^4*x^4+3*c^2*x^2-1)/d^3

Fricas [F]

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

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-1/3*a^2*(3*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(5/2) - 3/(sqrt(-c^2*d*x^2 + d)*d^2) - 1
/((-c^2*d*x^2 + d)^(3/2)*d)) - sqrt(d)*integrate((b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*arc
tan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^6*d^3*x^7 - 3*c^4*d^3*x^5 + 3*c^2*d^3
*x^3 - d^3*x), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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