3.251 \(\int (c e+d e x)^2 (a+b \sin ^{-1}(c+d x))^{5/2} \, dx\)

Optimal. Leaf size=427 \[ -\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^2 \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}+\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} e^2 \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{144 d}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^2 \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}-\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} e^2 \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{144 d}-\frac{5 b^2 e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{36 d}-\frac{5 b^2 e^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{6 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac{5 b e^2 \sqrt{1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{5 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d} \]

[Out]

(-5*b^2*e^2*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/(6*d) - (5*b^2*e^2*(c + d*x)^3*Sqrt[a + b*ArcSin[c + d*x]])
/(36*d) + (5*b*e^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(9*d) + (5*b*e^2*(c + d*x)^2*Sqrt[1 -
(c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(18*d) + (e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^(5/2))/(3*d) + (
15*b^(5/2)*e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(16*d) - (5*b^(
5/2)*e^2*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(144*d) - (15*b^(
5/2)*e^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(16*d) + (5*b^(5/2)*e
^2*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(144*d)

________________________________________________________________________________________

Rubi [A]  time = 1.32544, antiderivative size = 427, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {4805, 12, 4629, 4707, 4677, 4619, 4723, 3306, 3305, 3351, 3304, 3352, 3312} \[ -\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^2 \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}+\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} e^2 \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{144 d}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^2 \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}-\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} e^2 \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{144 d}-\frac{5 b^2 e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{36 d}-\frac{5 b^2 e^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{6 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac{5 b e^2 \sqrt{1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{5 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*b^2*e^2*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/(6*d) - (5*b^2*e^2*(c + d*x)^3*Sqrt[a + b*ArcSin[c + d*x]])
/(36*d) + (5*b*e^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(9*d) + (5*b*e^2*(c + d*x)^2*Sqrt[1 -
(c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(18*d) + (e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^(5/2))/(3*d) + (
15*b^(5/2)*e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(16*d) - (5*b^(
5/2)*e^2*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(144*d) - (15*b^(
5/2)*e^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(16*d) + (5*b^(5/2)*e
^2*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(144*d)

Rule 4805

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 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 4723

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

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 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps

\begin{align*} \int (c e+d e x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int e^2 x^2 \left (a+b \sin ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac{\left (5 b e^2\right ) \operatorname{Subst}\left (\int \frac{x^3 \left (a+b \sin ^{-1}(x)\right )^{3/2}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{6 d}\\ &=\frac{5 b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac{\left (5 b e^2\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \sin ^{-1}(x)\right )^{3/2}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{9 d}-\frac{\left (5 b^2 e^2\right ) \operatorname{Subst}\left (\int x^2 \sqrt{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{12 d}\\ &=-\frac{5 b^2 e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{36 d}+\frac{5 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac{5 b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac{\left (5 b^2 e^2\right ) \operatorname{Subst}\left (\int \sqrt{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{6 d}+\frac{\left (5 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{1-x^2} \sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{72 d}\\ &=-\frac{5 b^2 e^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{6 d}-\frac{5 b^2 e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{36 d}+\frac{5 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac{5 b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac{\left (5 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{\sin ^3(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{72 d}+\frac{\left (5 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2} \sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{12 d}\\ &=-\frac{5 b^2 e^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{6 d}-\frac{5 b^2 e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{36 d}+\frac{5 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac{5 b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac{\left (5 b^3 e^2\right ) \operatorname{Subst}\left (\int \left (\frac{3 \sin (x)}{4 \sqrt{a+b x}}-\frac{\sin (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{72 d}+\frac{\left (5 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{12 d}\\ &=-\frac{5 b^2 e^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{6 d}-\frac{5 b^2 e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{36 d}+\frac{5 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac{5 b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac{\left (5 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{288 d}+\frac{\left (5 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{96 d}+\frac{\left (5 b^3 e^2 \cos \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+d x)\right )}{12 d}-\frac{\left (5 b^3 e^2 \sin \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+d x)\right )}{12 d}\\ &=-\frac{5 b^2 e^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{6 d}-\frac{5 b^2 e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{36 d}+\frac{5 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac{5 b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac{\left (5 b^2 e^2 \cos \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+d x)}\right )}{6 d}+\frac{\left (5 b^3 e^2 \cos \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+d x)\right )}{96 d}-\frac{\left (5 b^3 e^2 \cos \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+d x)\right )}{288 d}-\frac{\left (5 b^2 e^2 \sin \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+d x)}\right )}{6 d}-\frac{\left (5 b^3 e^2 \sin \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+d x)\right )}{96 d}+\frac{\left (5 b^3 e^2 \sin \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+d x)\right )}{288 d}\\ &=-\frac{5 b^2 e^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{6 d}-\frac{5 b^2 e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{36 d}+\frac{5 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac{5 b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac{5 b^{5/2} e^2 \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{6 d}-\frac{5 b^{5/2} e^2 \sqrt{\frac{\pi }{2}} C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{6 d}+\frac{\left (5 b^2 e^2 \cos \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+d x)}\right )}{48 d}-\frac{\left (5 b^2 e^2 \cos \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+d x)}\right )}{144 d}-\frac{\left (5 b^2 e^2 \sin \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+d x)}\right )}{48 d}+\frac{\left (5 b^2 e^2 \sin \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+d x)}\right )}{144 d}\\ &=-\frac{5 b^2 e^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{6 d}-\frac{5 b^2 e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{36 d}+\frac{5 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac{5 b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac{15 b^{5/2} e^2 \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}-\frac{5 b^{5/2} e^2 \sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{144 d}-\frac{15 b^{5/2} e^2 \sqrt{\frac{\pi }{2}} C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{16 d}+\frac{5 b^{5/2} e^2 \sqrt{\frac{\pi }{6}} C\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{3 a}{b}\right )}{144 d}\\ \end{align*}

Mathematica [C]  time = 0.258587, size = 249, normalized size = 0.58 \[ \frac{b^3 e^2 e^{-\frac{3 i a}{b}} \left (-81 e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{7}{2},-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-81 e^{\frac{4 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{7}{2},\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\sqrt{3} \left (\sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{7}{2},-\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{6 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{7}{2},\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{648 d \sqrt{a+b \sin ^{-1}(c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b^3*e^2*(-81*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((-I)*(a + b*ArcSin[c + d*x]))
/b] - 81*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, (I*(a + b*ArcSin[c + d*x]))/b] + Sqrt[
3]*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((6*I)*a)/b)*Sq
rt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b])))/(648*d*E^(((3*I)*a)/b)*Sqrt
[a + b*ArcSin[c + d*x]])

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Maple [B]  time = 0.145, size = 873, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^2*(a+b*arcsin(d*x+c))^(5/2),x)

[Out]

1/864/d*e^2/(a+b*arcsin(d*x+c))^(1/2)*(-5*3^(1/2)*(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*2^(1/2)*cos(3
*a/b)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3+5*3^(1/2)*(1/b)^(1/2)*(a+
b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*2^(1/2)*sin(3*a/b)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d
*x+c))^(1/2)/b)*b^3+405*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1
/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3-405*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*si
n(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3+216*arcsin(d*x+c)^3*sin((a+b*arc
sin(d*x+c))/b-a/b)*b^3-72*arcsin(d*x+c)^3*sin(3*(a+b*arcsin(d*x+c))/b-3*a/b)*b^3+648*arcsin(d*x+c)^2*sin((a+b*
arcsin(d*x+c))/b-a/b)*a*b^2+540*arcsin(d*x+c)^2*cos((a+b*arcsin(d*x+c))/b-a/b)*b^3-216*arcsin(d*x+c)^2*sin(3*(
a+b*arcsin(d*x+c))/b-3*a/b)*a*b^2-60*arcsin(d*x+c)^2*cos(3*(a+b*arcsin(d*x+c))/b-3*a/b)*b^3+648*arcsin(d*x+c)*
sin((a+b*arcsin(d*x+c))/b-a/b)*a^2*b-810*arcsin(d*x+c)*sin((a+b*arcsin(d*x+c))/b-a/b)*b^3+1080*arcsin(d*x+c)*c
os((a+b*arcsin(d*x+c))/b-a/b)*a*b^2-216*arcsin(d*x+c)*sin(3*(a+b*arcsin(d*x+c))/b-3*a/b)*a^2*b+30*arcsin(d*x+c
)*sin(3*(a+b*arcsin(d*x+c))/b-3*a/b)*b^3-120*arcsin(d*x+c)*cos(3*(a+b*arcsin(d*x+c))/b-3*a/b)*a*b^2+216*sin((a
+b*arcsin(d*x+c))/b-a/b)*a^3-810*sin((a+b*arcsin(d*x+c))/b-a/b)*a*b^2+540*cos((a+b*arcsin(d*x+c))/b-a/b)*a^2*b
-72*sin(3*(a+b*arcsin(d*x+c))/b-3*a/b)*a^3+30*sin(3*(a+b*arcsin(d*x+c))/b-3*a/b)*a*b^2-60*cos(3*(a+b*arcsin(d*
x+c))/b-3*a/b)*a^2*b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{2}{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 4.17627, size = 3895, normalized size = 9.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(2)*sqrt(pi)*a^2*b^3*i*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt
(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b + 2)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d) - 1/8*sqrt(2
)*sqrt(pi)*a^2*b^3*i*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*
x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b + 2)/((b^3*i/sqrt(abs(b)) - b^2*sqrt(abs(b)))*d) + 1/12*sqrt(pi)*a^2*b^(
5/2)*i*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)
/sqrt(b))*e^(3*a*i/b + 2)/((sqrt(6)*b^3*i/abs(b) + sqrt(6)*b^2)*d) + 3/16*sqrt(2)*sqrt(pi)*a*b^4*erf(-1/2*sqrt
(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*
i/b + 2)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d) + 1/8*sqrt(2)*sqrt(pi)*a^2*b^2*i*erf(-1/2*sqrt(2)*sqrt(b*
arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b + 2)/((
b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d) - 15/64*sqrt(2)*sqrt(pi)*b^4*i*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c)
+ a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b + 2)/((b^2*i/sqrt(abs(
b)) + b*sqrt(abs(b)))*d) - 3/16*sqrt(2)*sqrt(pi)*a*b^4*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(
b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b + 2)/((b^3*i/sqrt(abs(b)) - b^2*sqrt(a
bs(b)))*d) + 1/8*sqrt(2)*sqrt(pi)*a^2*b^2*i*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*s
qrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b + 2)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*d) -
15/64*sqrt(2)*sqrt(pi)*b^4*i*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*a
rcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b + 2)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*d) + 1/12*sqrt(pi)*a
^2*b^(5/2)*i*erf(1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arcsin(d*x + c)
+ a)/sqrt(b))*e^(-3*a*i/b + 2)/((sqrt(6)*b^3*i/abs(b) - sqrt(6)*b^2)*d) + 1/24*sqrt(b*arcsin(d*x + c) + a)*b^
2*i*arcsin(d*x + c)^2*e^(3*i*arcsin(d*x + c) + 2)/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*b^2*i*arcsin(d*x + c)^2*
e^(i*arcsin(d*x + c) + 2)/d + 1/8*sqrt(b*arcsin(d*x + c) + a)*b^2*i*arcsin(d*x + c)^2*e^(-i*arcsin(d*x + c) +
2)/d - 1/24*sqrt(b*arcsin(d*x + c) + a)*b^2*i*arcsin(d*x + c)^2*e^(-3*i*arcsin(d*x + c) + 2)/d - 1/24*sqrt(pi)
*a^2*b^2*i*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arcsin(d*x + c)
+ a)/sqrt(b))*e^(3*a*i/b + 2)/((sqrt(6)*b^(5/2)*i/abs(b) + sqrt(6)*b^(3/2))*d) - 1/24*sqrt(pi)*a^2*b^2*i*erf(1
/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(
-3*a*i/b + 2)/((sqrt(6)*b^(5/2)*i/abs(b) - sqrt(6)*b^(3/2))*d) - 1/24*sqrt(pi)*a*b^(7/2)*erf(-1/2*sqrt(6)*sqrt
(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(3*a*i/b + 2)/((
sqrt(6)*b^3*i/abs(b) + sqrt(6)*b^2)*d) - 1/24*sqrt(pi)*a^2*b^(3/2)*i*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) +
a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(3*a*i/b + 2)/((sqrt(6)*b^2*i/abs(b)
+ sqrt(6)*b)*d) + 5/288*sqrt(pi)*b^(7/2)*i*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - 1/
2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(3*a*i/b + 2)/((sqrt(6)*b^2*i/abs(b) + sqrt(6)*b)*d) - 3/16*s
qrt(2)*sqrt(pi)*a*b^3*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(
d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b + 2)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d) + 3/16*sqrt(2)*sqrt(pi)*
a*b^3*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqr
t(abs(b))/b)*e^(-a*i/b + 2)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*d) + 1/24*sqrt(pi)*a*b^(7/2)*erf(1/2*sqrt(6
)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-3*a*i/b
+ 2)/((sqrt(6)*b^3*i/abs(b) - sqrt(6)*b^2)*d) - 1/24*sqrt(pi)*a^2*b^(3/2)*i*erf(1/2*sqrt(6)*sqrt(b*arcsin(d*x
+ c) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-3*a*i/b + 2)/((sqrt(6)*b^2*i
/abs(b) - sqrt(6)*b)*d) + 5/288*sqrt(pi)*b^(7/2)*i*erf(1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b
) - 1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-3*a*i/b + 2)/((sqrt(6)*b^2*i/abs(b) - sqrt(6)*b)*d) +
1/12*sqrt(b*arcsin(d*x + c) + a)*a*b*i*arcsin(d*x + c)*e^(3*i*arcsin(d*x + c) + 2)/d - 1/4*sqrt(b*arcsin(d*x
+ c) + a)*a*b*i*arcsin(d*x + c)*e^(i*arcsin(d*x + c) + 2)/d + 1/4*sqrt(b*arcsin(d*x + c) + a)*a*b*i*arcsin(d*x
+ c)*e^(-i*arcsin(d*x + c) + 2)/d - 1/12*sqrt(b*arcsin(d*x + c) + a)*a*b*i*arcsin(d*x + c)*e^(-3*i*arcsin(d*x
+ c) + 2)/d + 1/24*sqrt(pi)*a*b^(5/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - 1/2*sqr
t(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(3*a*i/b + 2)/((sqrt(6)*b^2*i/abs(b) + sqrt(6)*b)*d) - 1/24*sqrt(p
i)*a*b^(5/2)*erf(1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arcsin(d*x + c)
+ a)/sqrt(b))*e^(-3*a*i/b + 2)/((sqrt(6)*b^2*i/abs(b) - sqrt(6)*b)*d) + 1/24*sqrt(b*arcsin(d*x + c) + a)*a^2*
i*e^(3*i*arcsin(d*x + c) + 2)/d - 5/288*sqrt(b*arcsin(d*x + c) + a)*b^2*i*e^(3*i*arcsin(d*x + c) + 2)/d - 5/14
4*sqrt(b*arcsin(d*x + c) + a)*b^2*arcsin(d*x + c)*e^(3*i*arcsin(d*x + c) + 2)/d - 1/8*sqrt(b*arcsin(d*x + c) +
a)*a^2*i*e^(i*arcsin(d*x + c) + 2)/d + 15/32*sqrt(b*arcsin(d*x + c) + a)*b^2*i*e^(i*arcsin(d*x + c) + 2)/d +
5/16*sqrt(b*arcsin(d*x + c) + a)*b^2*arcsin(d*x + c)*e^(i*arcsin(d*x + c) + 2)/d + 1/8*sqrt(b*arcsin(d*x + c)
+ a)*a^2*i*e^(-i*arcsin(d*x + c) + 2)/d - 15/32*sqrt(b*arcsin(d*x + c) + a)*b^2*i*e^(-i*arcsin(d*x + c) + 2)/d
+ 5/16*sqrt(b*arcsin(d*x + c) + a)*b^2*arcsin(d*x + c)*e^(-i*arcsin(d*x + c) + 2)/d - 1/24*sqrt(b*arcsin(d*x
+ c) + a)*a^2*i*e^(-3*i*arcsin(d*x + c) + 2)/d + 5/288*sqrt(b*arcsin(d*x + c) + a)*b^2*i*e^(-3*i*arcsin(d*x +
c) + 2)/d - 5/144*sqrt(b*arcsin(d*x + c) + a)*b^2*arcsin(d*x + c)*e^(-3*i*arcsin(d*x + c) + 2)/d - 5/144*sqrt(
b*arcsin(d*x + c) + a)*a*b*e^(3*i*arcsin(d*x + c) + 2)/d + 5/16*sqrt(b*arcsin(d*x + c) + a)*a*b*e^(i*arcsin(d*
x + c) + 2)/d + 5/16*sqrt(b*arcsin(d*x + c) + a)*a*b*e^(-i*arcsin(d*x + c) + 2)/d - 5/144*sqrt(b*arcsin(d*x +
c) + a)*a*b*e^(-3*i*arcsin(d*x + c) + 2)/d