3.3.22 \(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {ArcSin}(c x))^2}{x} \, dx\) [222]

Optimal. Leaf size=545 \[ -\frac {22}{9} b^2 d \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}} \]

[Out]

1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2-22/9*b^2*d*(-c^2*d*x^2+d)^(1/2)-2/27*b^2*d*(-c^2*x^2+1)*(-c^2*d*x
^2+d)^(1/2)+d*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)-2*a*b*c*d*x*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2*b
^2*c*d*x*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2/3*b*c*d*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2
)/(-c^2*x^2+1)^(1/2)+2/9*b*c^3*d*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2*d*(a+b*arcsin
(c*x))^2*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+2*I*b*d*(a+b*arcsin(c*x))*p
olylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2*I*b*d*(a+b*arcsin(c*x))*polylog(
2,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2*b^2*d*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2
))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+2*b^2*d*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-
c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.40, antiderivative size = 545, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {4787, 4783, 4803, 4268, 2611, 2320, 6724, 4715, 267, 4739, 455, 45} \begin {gather*} \frac {2 i b d \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{\sqrt {1-c^2 x^2}}-\frac {2 i b d \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2+d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2-\frac {2 d \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{\sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{9 \sqrt {1-c^2 x^2}}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c d x \text {ArcSin}(c x) \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {22}{9} b^2 d \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-22*b^2*d*Sqrt[d - c^2*d*x^2])/9 - (2*a*b*c*d*x*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] - (2*b^2*d*(1 - c^2*x^
2)*Sqrt[d - c^2*d*x^2])/27 - (2*b^2*c*d*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (2*b*c*d*x*Sqrt
[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[
c*x]))/(9*Sqrt[1 - c^2*x^2]) + d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + ((d - c^2*d*x^2)^(3/2)*(a + b*Arc
Sin[c*x])^2)/3 - (2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]
+ ((2*I)*b*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - ((2*I
)*b*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (2*b^2*d*Sqrt
[d - c^2*d*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (2*b^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, E^
(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 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 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 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 4715

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4739

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 4783

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

Rule 4787

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*((a + b*ArcSin[c*x])^n/(f*(m + 2*p + 1))), x] + (Dist[2*d*(p/(m + 2*p + 1)), Int[(
f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 2*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]) /; FreeQ[{a, b
, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -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*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx &=\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+d \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (1-\frac {c^2 x^2}{3}\right )}{\sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c d \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {c^2 x}{3}}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {2}{3 \sqrt {1-c^2 x}}+\frac {1}{3} \sqrt {1-c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {22}{9} b^2 d \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {22}{9} b^2 d \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {22}{9} b^2 d \sqrt {d-c^2 d x^2}-\frac {2 a b c d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {2 b^2 c d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.60, size = 576, normalized size = 1.06 \begin {gather*} -\frac {1}{3} a^2 d \left (-4+c^2 x^2\right ) \sqrt {d-c^2 d x^2}+a^2 d^{3/2} \log (c x)-a^2 d^{3/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {2 a b d \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {1-c^2 x^2} \text {ArcSin}(c x)+\text {ArcSin}(c x) \log \left (1-e^{i \text {ArcSin}(c x)}\right )-\text {ArcSin}(c x) \log \left (1+e^{i \text {ArcSin}(c x)}\right )+i \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-i \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 d \sqrt {d-c^2 d x^2} \left (2 \sqrt {1-c^2 x^2}+2 c x \text {ArcSin}(c x)-\sqrt {1-c^2 x^2} \text {ArcSin}(c x)^2-\text {ArcSin}(c x)^2 \left (\log \left (1-e^{i \text {ArcSin}(c x)}\right )-\log \left (1+e^{i \text {ArcSin}(c x)}\right )\right )-2 i \text {ArcSin}(c x) \left (\text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )\right )+2 \left (\text {PolyLog}\left (3,-e^{i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (3,e^{i \text {ArcSin}(c x)}\right )\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {a b d \sqrt {d-c^2 d x^2} \left (9 c x-3 \text {ArcSin}(c x) \left (3 \sqrt {1-c^2 x^2}+\cos (3 \text {ArcSin}(c x))\right )+\sin (3 \text {ArcSin}(c x))\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b^2 d \sqrt {d-c^2 d x^2} \left (27 \sqrt {1-c^2 x^2} \left (-2+\text {ArcSin}(c x)^2\right )+\left (-2+9 \text {ArcSin}(c x)^2\right ) \cos (3 \text {ArcSin}(c x))-6 \text {ArcSin}(c x) (9 c x+\sin (3 \text {ArcSin}(c x)))\right )}{108 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(a^2*d*(-4 + c^2*x^2)*Sqrt[d - c^2*d*x^2]) + a^2*d^(3/2)*Log[c*x] - a^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d -
c^2*d*x^2]] + (2*a*b*d*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]*Log[1 - E^(I*
ArcSin[c*x])] - ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + I*PolyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[2, E^(I*
ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - (b^2*d*Sqrt[d - c^2*d*x^2]*(2*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x] - Sqrt
[1 - c^2*x^2]*ArcSin[c*x]^2 - ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) - (2*I)*
ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 2*(PolyLog[3, -E^(I*ArcSin[c*x]
)] - PolyLog[3, E^(I*ArcSin[c*x])])))/Sqrt[1 - c^2*x^2] - (a*b*d*Sqrt[d - c^2*d*x^2]*(9*c*x - 3*ArcSin[c*x]*(3
*Sqrt[1 - c^2*x^2] + Cos[3*ArcSin[c*x]]) + Sin[3*ArcSin[c*x]]))/(18*Sqrt[1 - c^2*x^2]) + (b^2*d*Sqrt[d - c^2*d
*x^2]*(27*Sqrt[1 - c^2*x^2]*(-2 + ArcSin[c*x]^2) + (-2 + 9*ArcSin[c*x]^2)*Cos[3*ArcSin[c*x]] - 6*ArcSin[c*x]*(
9*c*x + Sin[3*ArcSin[c*x]])))/(108*Sqrt[1 - c^2*x^2])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1275 vs. \(2 (529 ) = 1058\).
time = 0.30, size = 1276, normalized size = 2.34

method	result	size
default	\(\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-a^{2} d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )+a^{2} \sqrt {-c^{2} d \,x^{2}+d}\, d +\frac {68 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d}{27 \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right )^{2} x^{4} c^{4}}{3 \left (c^{2} x^{2}-1\right )}+\frac {5 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right )^{2} x^{2} c^{2}}{3 \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{4} c^{4}}{27 \left (c^{2} x^{2}-1\right )}-\frac {70 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{2} c^{2}}{27 \left (c^{2} x^{2}-1\right )}-\frac {4 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right )^{2}}{3 \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {2 i b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {2 i b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a^{2}}{3}+\frac {8 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, x c}{3 \left (c^{2} x^{2}-1\right )}-\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}}{9 \left (c^{2} x^{2}-1\right )}-\frac {8 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right )}{3 \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) x^{4} c^{4}}{3 \left (c^{2} x^{2}-1\right )}+\frac {10 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) x^{2} c^{2}}{3 \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}}{9 \left (c^{2} x^{2}-1\right )}+\frac {8 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \sqrt {-c^{2} x^{2}+1}\, x c}{3 \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}\)	\(1276\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x,x,method=_RETURNVERBOSE)

[Out]

-a^2*d^(3/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)+a^2*(-c^2*d*x^2+d)^(1/2)*d+68/27*b^2*(-d*(c^2*x^2-1))^
(1/2)*d/(c^2*x^2-1)+b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*d*arcsin(c*x)^2*ln(1+I*c*x+(-c^2
*x^2+1)^(1/2))-b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*d*arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+
1)^(1/2))-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*arcsin(c*x)^2*x^4*c^4+5/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/
(c^2*x^2-1)*arcsin(c*x)^2*x^2*c^2+2/27*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*x^4*c^4-70/27*b^2*(-d*(c^2*x^2
-1))^(1/2)*d/(c^2*x^2-1)*x^2*c^2+2*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*d*polylog(3,-I*c*
x-(-c^2*x^2+1)^(1/2))-2*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*d*polylog(3,I*c*x+(-c^2*x^2+
1)^(1/2))-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*arcsin(c*x)^2+1/3*(-c^2*d*x^2+d)^(3/2)*a^2+8/3*b^2*(-d*
(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x*c-2/9*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-
1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^3*c^3-8/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*arcsin(c*x)-2*I*b^2*(-d
*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*d*arcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+2*I*b^2*
(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*d*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+2*I*a*
b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)*d*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-2/3*a*b*(-d*(c^2
*x^2-1))^(1/2)*d/(c^2*x^2-1)*arcsin(c*x)*x^4*c^4+10/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*arcsin(c*x)*x^2
*c^2-2/9*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^3*c^3+8/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/
(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x*c+2*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)*d*arcsin(c*x)*l
n(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)*d*arcsin(c*x)*ln(1-I
*c*x-(-c^2*x^2+1)^(1/2))-2*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)*d*polylog(2,-I*c*x-(-c^
2*x^2+1)^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*d^(3/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) - (-c^2*d*x^2 + d)^(3/2) - 3*sqrt(-c^2
*d*x^2 + d)*d)*a^2 - sqrt(d)*integrate(((b^2*c^2*d*x^2 - b^2*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 +
2*(a*b*c^2*d*x^2 - a*b*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/x, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arcsin(c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arcsin(c*
x))*sqrt(-c^2*d*x^2 + d)/x, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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