3.691 \(\int (d+e x^2) (a+b \sin ^{-1}(c x))^{3/2} \, dx\)

Optimal. Leaf size=482 \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} e \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} e \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} d \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{2 c}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} d \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{2 c}+\frac{3 b d \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{2 c}+\frac{b e x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{6 c}+\frac{b e \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{3 c^3}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2} \]

[Out]

(3*b*d*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(2*c) + (b*e*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(3*c
^3) + (b*e*x^2*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(6*c) + d*x*(a + b*ArcSin[c*x])^(3/2) + (e*x^3*(a +
b*ArcSin[c*x])^(3/2))/3 - (3*b^(3/2)*d*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[
b]])/(2*c) - (3*b^(3/2)*e*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*c^3)
+ (b^(3/2)*e*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(24*c^3) - (3*b^(
3/2)*d*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c) - (3*b^(3/2)*e*Sqrt[P
i/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(8*c^3) + (b^(3/2)*e*Sqrt[Pi/6]*FresnelS
[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(24*c^3)

________________________________________________________________________________________

Rubi [A]  time = 1.42438, antiderivative size = 482, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 13, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.65, Rules used = {4667, 4619, 4677, 4623, 3306, 3305, 3351, 3304, 3352, 4629, 4707, 4635, 4406} \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} e \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} e \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} d \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{2 c}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} d \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{2 c}+\frac{3 b d \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{2 c}+\frac{b e x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{6 c}+\frac{b e \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{3 c^3}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*d*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(2*c) + (b*e*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(3*c
^3) + (b*e*x^2*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(6*c) + d*x*(a + b*ArcSin[c*x])^(3/2) + (e*x^3*(a +
b*ArcSin[c*x])^(3/2))/3 - (3*b^(3/2)*d*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[
b]])/(2*c) - (3*b^(3/2)*e*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*c^3)
+ (b^(3/2)*e*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(24*c^3) - (3*b^(
3/2)*d*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c) - (3*b^(3/2)*e*Sqrt[P
i/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(8*c^3) + (b^(3/2)*e*Sqrt[Pi/6]*FresnelS
[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(24*c^3)

Rule 4667

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[n, 0])

Rule 4619

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 4677

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

Rule 4623

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

Rule 3306

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(f*x^2)/d], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4629

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

Rule 4707

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

Rule 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^{3/2} \, dx &=\int \left (d \left (a+b \sin ^{-1}(c x)\right )^{3/2}+e x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}\right ) \, dx\\ &=d \int \left (a+b \sin ^{-1}(c x)\right )^{3/2} \, dx+e \int x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2} \, dx\\ &=d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{1}{2} (3 b c d) \int \frac{x \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{2} (b c e) \int \frac{x^3 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{3 b d \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{2 c}+\frac{b e x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{1}{4} \left (3 b^2 d\right ) \int \frac{1}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx-\frac{1}{12} \left (b^2 e\right ) \int \frac{x^2}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx-\frac{(b e) \int \frac{x \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{1-c^2 x^2}} \, dx}{3 c}\\ &=\frac{3 b d \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{2 c}+\frac{b e \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{3 c^3}+\frac{b e x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{(3 b d) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{4 c}-\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}-\frac{\left (b^2 e\right ) \int \frac{1}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx}{6 c^2}\\ &=\frac{3 b d \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{2 c}+\frac{b e \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{3 c^3}+\frac{b e x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{6 c^3}-\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 \sqrt{a+b x}}-\frac{\cos (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}-\frac{\left (3 b d \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{4 c}-\frac{\left (3 b d \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{4 c}\\ &=\frac{3 b d \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{2 c}+\frac{b e \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{3 c^3}+\frac{b e x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}+\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}-\frac{\left (3 b d \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{2 c}-\frac{\left (b e \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{6 c^3}-\frac{\left (3 b d \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{2 c}-\frac{\left (b e \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{6 c^3}\\ &=\frac{3 b d \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{2 c}+\frac{b e \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{3 c^3}+\frac{b e x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{3 b^{3/2} d \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{2 c}-\frac{3 b^{3/2} d \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{2 c}-\frac{\left (b e \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{3 c^3}-\frac{\left (b^2 e \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}+\frac{\left (b^2 e \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}-\frac{\left (b e \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{3 c^3}-\frac{\left (b^2 e \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}+\frac{\left (b^2 e \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{48 c^3}\\ &=\frac{3 b d \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{2 c}+\frac{b e \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{3 c^3}+\frac{b e x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{3 b^{3/2} d \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{2 c}-\frac{b^{3/2} e \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{3 c^3}-\frac{3 b^{3/2} d \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{2 c}-\frac{b^{3/2} e \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{3 c^3}-\frac{\left (b e \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{24 c^3}+\frac{\left (b e \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{24 c^3}-\frac{\left (b e \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{24 c^3}+\frac{\left (b e \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{24 c^3}\\ &=\frac{3 b d \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{2 c}+\frac{b e \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{3 c^3}+\frac{b e x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{6 c}+d x \left (a+b \sin ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{3 b^{3/2} d \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{2 c}-\frac{3 b^{3/2} e \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}+\frac{b^{3/2} e \sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) C\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}-\frac{3 b^{3/2} d \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{2 c}-\frac{3 b^{3/2} e \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{8 c^3}+\frac{b^{3/2} e \sqrt{\frac{\pi }{6}} S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{3 a}{b}\right )}{24 c^3}\\ \end{align*}

Mathematica [C]  time = 10.1017, size = 873, normalized size = 1.81 \[ \frac{a b d e^{-\frac{i a}{b}} \left (\sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{2 c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{a b e e^{-\frac{3 i a}{b}} \left (9 e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+9 e^{\frac{4 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt{3} \left (\sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac{6 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )\right )}{72 c^3 \sqrt{a+b \sin ^{-1}(c x)}}+\frac{b d \left (2 \sqrt{a+b \sin ^{-1}(c x)} \left (2 c x \sin ^{-1}(c x)+3 \sqrt{1-c^2 x^2}\right )-\sqrt{\frac{1}{b}} \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}\right ) \left (3 b \cos \left (\frac{a}{b}\right )+2 a \sin \left (\frac{a}{b}\right )\right )+\sqrt{\frac{1}{b}} \sqrt{2 \pi } S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}\right ) \left (2 a \cos \left (\frac{a}{b}\right )-3 b \sin \left (\frac{a}{b}\right )\right )\right )}{4 c}+\frac{b e \left (18 \sqrt{a+b \sin ^{-1}(c x)} \left (2 c x \sin ^{-1}(c x)+3 \sqrt{1-c^2 x^2}\right )-9 \sqrt{\frac{1}{b}} \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}\right ) \left (3 b \cos \left (\frac{a}{b}\right )+2 a \sin \left (\frac{a}{b}\right )\right )+9 \sqrt{\frac{1}{b}} \sqrt{2 \pi } S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}\right ) \left (2 a \cos \left (\frac{a}{b}\right )-3 b \sin \left (\frac{a}{b}\right )\right )+\sqrt{\frac{1}{b}} \sqrt{6 \pi } \text{FresnelC}\left (\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}\right ) \left (b \cos \left (\frac{3 a}{b}\right )+2 a \sin \left (\frac{3 a}{b}\right )\right )+\sqrt{\frac{1}{b}} \sqrt{6 \pi } S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}\right ) \left (b \sin \left (\frac{3 a}{b}\right )-2 a \cos \left (\frac{3 a}{b}\right )\right )-6 \sqrt{a+b \sin ^{-1}(c x)} \left (\cos \left (3 \sin ^{-1}(c x)\right )+2 \sin ^{-1}(c x) \sin \left (3 \sin ^{-1}(c x)\right )\right )\right )}{144 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*b*d*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*
(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b]))/(2*c*E^((I*a)/b)*Sqrt[a + b*ArcSin[c*x]]) + (a
*b*e*(9*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 9*E^(((4
*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*(Sqrt[((-I)*(a + b*A
rcSin[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*G
amma[3/2, ((3*I)*(a + b*ArcSin[c*x]))/b])))/(72*c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]]) + (b*d*(2*Sqrt[a
+ b*ArcSin[c*x]]*(3*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x]) - Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqr
t[2/Pi]*Sqrt[a + b*ArcSin[c*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) + Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]
*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]]]*(2*a*Cos[a/b] - 3*b*Sin[a/b])))/(4*c) + (b*e*(18*Sqrt[a + b*ArcSin[c*x]]*
(3*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x]) - 9*Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a
+ b*ArcSin[c*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) + 9*Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*S
qrt[a + b*ArcSin[c*x]]]*(2*a*Cos[a/b] - 3*b*Sin[a/b]) + Sqrt[b^(-1)]*Sqrt[6*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[6/P
i]*Sqrt[a + b*ArcSin[c*x]]]*(b*Cos[(3*a)/b] + 2*a*Sin[(3*a)/b]) + Sqrt[b^(-1)]*Sqrt[6*Pi]*FresnelS[Sqrt[b^(-1)
]*Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]]]*(-2*a*Cos[(3*a)/b] + b*Sin[(3*a)/b]) - 6*Sqrt[a + b*ArcSin[c*x]]*(Cos[3*
ArcSin[c*x]] + 2*ArcSin[c*x]*Sin[3*ArcSin[c*x]])))/(144*c^3)

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Maple [B]  time = 0.154, size = 835, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144/c^3*(-108*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(1/b)^
(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b^2*c^2*d-108*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(a/b)*F
resnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b^2*c^2*d+(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*ar
csin(c*x))^(1/2)*cos(3*a/b)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*3^(1/2)*b
^2*e+(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(3*a/b)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(1/b)^(
1/2)*(a+b*arcsin(c*x))^(1/2)/b)*3^(1/2)*b^2*e-27*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(a/b)
*FresnelC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b^2*e-27*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*a
rcsin(c*x))^(1/2)*sin(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b^2*e+144*arcsin(c
*x)^2*sin((a+b*arcsin(c*x))/b-a/b)*b^2*c^2*d+288*arcsin(c*x)*sin((a+b*arcsin(c*x))/b-a/b)*a*b*c^2*d+216*arcsin
(c*x)*cos((a+b*arcsin(c*x))/b-a/b)*b^2*c^2*d+36*arcsin(c*x)^2*sin((a+b*arcsin(c*x))/b-a/b)*b^2*e-12*arcsin(c*x
)^2*sin(3*(a+b*arcsin(c*x))/b-3*a/b)*b^2*e+144*sin((a+b*arcsin(c*x))/b-a/b)*a^2*c^2*d+216*cos((a+b*arcsin(c*x)
)/b-a/b)*a*b*c^2*d+72*arcsin(c*x)*sin((a+b*arcsin(c*x))/b-a/b)*a*b*e+54*arcsin(c*x)*cos((a+b*arcsin(c*x))/b-a/
b)*b^2*e-24*arcsin(c*x)*sin(3*(a+b*arcsin(c*x))/b-3*a/b)*a*b*e-6*arcsin(c*x)*cos(3*(a+b*arcsin(c*x))/b-3*a/b)*
b^2*e+36*sin((a+b*arcsin(c*x))/b-a/b)*a^2*e+54*cos((a+b*arcsin(c*x))/b-a/b)*a*b*e-12*sin(3*(a+b*arcsin(c*x))/b
-3*a/b)*a^2*e-6*cos(3*(a+b*arcsin(c*x))/b-3*a/b)*a*b*e)/(a+b*arcsin(c*x))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 3.82275, size = 2699, normalized size = 5.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*I*sqrt(2)*sqrt(pi)*a*b^3*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*a
rcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 3/8*sqrt(2)*sqrt(pi)*b
^4*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b)
)/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/4*I*sqrt(2)*sqrt(pi)*a*b^3*d*erf(1/2*I*sqrt(2)*
sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3
/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 3/8*sqrt(2)*sqrt(pi)*b^4*d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sq
rt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(a
bs(b)))*c) + 1/4*I*sqrt(2)*sqrt(pi)*a*b^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt
(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c) - 1/4*I*sqrt(2
)*sqrt(pi)*a*b^2*d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a
)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c) - 1/2*I*sqrt(b*arcsin(c*x) + a)*b*d*ar
csin(c*x)*e^(I*arcsin(c*x))/c + 1/2*I*sqrt(b*arcsin(c*x) + a)*b*d*arcsin(c*x)*e^(-I*arcsin(c*x))/c - 1/16*I*sq
rt(2)*sqrt(pi)*a*b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x)
+ a)*sqrt(abs(b))/b)*e^(I*a/b + 1)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 3/32*sqrt(2)*sqrt(pi)*b^4*e
rf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e
^(I*a/b + 1)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 1/16*I*sqrt(2)*sqrt(pi)*a*b^3*erf(1/2*I*sqrt(2)*s
qrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b + 1)/((-I*
b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 3/32*sqrt(2)*sqrt(pi)*b^4*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a
)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b + 1)/((-I*b^3/sqrt(abs(b)) + b^
2*sqrt(abs(b)))*c^3) - 1/2*I*sqrt(b*arcsin(c*x) + a)*a*d*e^(I*arcsin(c*x))/c + 3/4*sqrt(b*arcsin(c*x) + a)*b*d
*e^(I*arcsin(c*x))/c + 1/2*I*sqrt(b*arcsin(c*x) + a)*a*d*e^(-I*arcsin(c*x))/c + 3/4*sqrt(b*arcsin(c*x) + a)*b*
d*e^(-I*arcsin(c*x))/c + 1/24*I*sqrt(pi)*a*b^(5/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sq
rt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b + 1)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c^3) - 1/4
8*sqrt(pi)*b^(7/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sq
rt(b)/abs(b))*e^(3*I*a/b + 1)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c^3) + 1/16*I*sqrt(2)*sqrt(pi)*a*b^2*erf(-
1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*
a/b + 1)/((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c^3) - 1/16*I*sqrt(2)*sqrt(pi)*a*b^2*erf(1/2*I*sqrt(2)*sqrt(b*
arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b + 1)/((-I*b^2/sq
rt(abs(b)) + b*sqrt(abs(b)))*c^3) - 1/24*I*sqrt(pi)*a*b^(5/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b)
+ 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b + 1)/((sqrt(6)*b^2 - I*sqrt(6)*b^3/abs(b)
)*c^3) - 1/48*sqrt(pi)*b^(7/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(
c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b + 1)/((sqrt(6)*b^2 - I*sqrt(6)*b^3/abs(b))*c^3) - 1/24*I*sqrt(pi)*a*b^(3
/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e
^(3*I*a/b + 1)/((sqrt(6)*b + I*sqrt(6)*b^2/abs(b))*c^3) + 1/24*I*sqrt(pi)*a*b^(3/2)*erf(-1/2*sqrt(6)*sqrt(b*ar
csin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b + 1)/((sqrt(6)*b -
I*sqrt(6)*b^2/abs(b))*c^3) + 1/24*I*sqrt(b*arcsin(c*x) + a)*b*arcsin(c*x)*e^(3*I*arcsin(c*x) + 1)/c^3 - 1/8*I*
sqrt(b*arcsin(c*x) + a)*b*arcsin(c*x)*e^(I*arcsin(c*x) + 1)/c^3 + 1/8*I*sqrt(b*arcsin(c*x) + a)*b*arcsin(c*x)*
e^(-I*arcsin(c*x) + 1)/c^3 - 1/24*I*sqrt(b*arcsin(c*x) + a)*b*arcsin(c*x)*e^(-3*I*arcsin(c*x) + 1)/c^3 + 1/24*
I*sqrt(b*arcsin(c*x) + a)*a*e^(3*I*arcsin(c*x) + 1)/c^3 - 1/48*sqrt(b*arcsin(c*x) + a)*b*e^(3*I*arcsin(c*x) +
1)/c^3 - 1/8*I*sqrt(b*arcsin(c*x) + a)*a*e^(I*arcsin(c*x) + 1)/c^3 + 3/16*sqrt(b*arcsin(c*x) + a)*b*e^(I*arcsi
n(c*x) + 1)/c^3 + 1/8*I*sqrt(b*arcsin(c*x) + a)*a*e^(-I*arcsin(c*x) + 1)/c^3 + 3/16*sqrt(b*arcsin(c*x) + a)*b*
e^(-I*arcsin(c*x) + 1)/c^3 - 1/24*I*sqrt(b*arcsin(c*x) + a)*a*e^(-3*I*arcsin(c*x) + 1)/c^3 - 1/48*sqrt(b*arcsi
n(c*x) + a)*b*e^(-3*I*arcsin(c*x) + 1)/c^3