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

Optimal. Leaf size=361 \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^2 \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} e^2 \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{24 d}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^2 \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} e^2 \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{24 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{3 d}+\frac{b e^2 \sqrt{1-(c+d x)^2} (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{6 d}+\frac{b e^2 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.03263, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 24, 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, 4623, 3306, 3305, 3351, 3304, 3352, 4635, 4406} \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^2 \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} e^2 \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{24 d}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^2 \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} e^2 \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{24 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{3 d}+\frac{b e^2 \sqrt{1-(c+d x)^2} (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{6 d}+\frac{b e^2 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e^2*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(3*d) + (b*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*Sqr
t[a + b*ArcSin[c + d*x]])/(6*d) + (e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^(3/2))/(3*d) - (3*b^(3/2)*e^2*Sqrt[
Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(8*d) + (b^(3/2)*e^2*Sqrt[Pi/6]*Cos
[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(24*d) - (3*b^(3/2)*e^2*Sqrt[Pi/2]*Fresn
elS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(8*d) + (b^(3/2)*e^2*Sqrt[Pi/6]*FresnelS[(Sqrt
[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(24*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 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 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 (c e+d e x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int e^2 x^2 \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{x^3 \sqrt{a+b \sin ^{-1}(x)}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \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 )^{3/2}}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{a+b \sin ^{-1}(x)}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{3 d}-\frac{\left (b^2 e^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{12 d}\\ &=\frac{b e^2 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}+\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \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 )^{3/2}}{3 d}-\frac{\left (b^2 e^2\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{12 d}-\frac{\left (b^2 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{6 d}\\ &=\frac{b e^2 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}+\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \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 )^{3/2}}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{6 d}-\frac{\left (b^2 e^2\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+d x)\right )}{12 d}\\ &=\frac{b e^2 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}+\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \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 )^{3/2}}{3 d}-\frac{\left (b^2 e^2\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{48 d}+\frac{\left (b^2 e^2\right ) \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{48 d}-\frac{\left (b e^2 \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+d x)\right )}{6 d}-\frac{\left (b e^2 \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+d x)\right )}{6 d}\\ &=\frac{b e^2 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}+\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \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 )^{3/2}}{3 d}-\frac{\left (b e^2 \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+d x)}\right )}{3 d}-\frac{\left (b^2 e^2 \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+d x)\right )}{48 d}+\frac{\left (b^2 e^2 \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+d x)\right )}{48 d}-\frac{\left (b e^2 \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+d x)}\right )}{3 d}-\frac{\left (b^2 e^2 \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+d x)\right )}{48 d}+\frac{\left (b^2 e^2 \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+d x)\right )}{48 d}\\ &=\frac{b e^2 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}+\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \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 )^{3/2}}{3 d}-\frac{b^{3/2} e^2 \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 d}-\frac{b^{3/2} e^2 \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{3 d}-\frac{\left (b e^2 \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+d x)}\right )}{24 d}+\frac{\left (b e^2 \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+d x)}\right )}{24 d}-\frac{\left (b e^2 \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+d x)}\right )}{24 d}+\frac{\left (b e^2 \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+d x)}\right )}{24 d}\\ &=\frac{b e^2 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}+\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \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 )^{3/2}}{3 d}-\frac{3 b^{3/2} e^2 \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}+\frac{b^{3/2} e^2 \sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) C\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{24 d}-\frac{3 b^{3/2} e^2 \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{8 d}+\frac{b^{3/2} e^2 \sqrt{\frac{\pi }{6}} S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{3 a}{b}\right )}{24 d}\\ \end{align*}

Mathematica [C]  time = 0.272283, size = 268, normalized size = 0.74 \[ \frac{b e^2 e^{-\frac{3 i a}{b}} \sqrt{a+b \sin ^{-1}(c+d x)} \left (27 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{5}{2},-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+27 e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{5}{2},\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-\sqrt{3} \left (\sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{5}{2},-\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{5}{2},\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{216 d \sqrt{\frac{\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*e^2*Sqrt[a + b*ArcSin[c + d*x]]*(27*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((-I)*(a
+ b*ArcSin[c + d*x]))/b] + 27*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, (I*(a + b*Arc
Sin[c + d*x]))/b] - Sqrt[3]*(Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b
] + E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b])))/(2
16*d*E^(((3*I)*a)/b)*Sqrt[(a + b*ArcSin[c + d*x])^2/b^2])

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Maple [B]  time = 0.122, size = 593, normalized size = 1.6 \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))^(3/2),x)

[Out]

1/144/d*e^2/(a+b*arcsin(d*x+c))^(1/2)*(3^(1/2)*(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(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)*Pi^(1/2)*2^(1/2)*b^2+3^(1/2)*(1/b)^(1/2)*(a+b*arc
sin(d*x+c))^(1/2)*sin(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)*Pi^(1/
2)*2^(1/2)*b^2-27*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(1
/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2-27*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(a/b)*
FresnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2+36*arcsin(d*x+c)^2*sin((a+b*arcsin(d*x+
c))/b-a/b)*b^2-12*arcsin(d*x+c)^2*sin(3*(a+b*arcsin(d*x+c))/b-3*a/b)*b^2+72*arcsin(d*x+c)*sin((a+b*arcsin(d*x+
c))/b-a/b)*a*b+54*arcsin(d*x+c)*cos((a+b*arcsin(d*x+c))/b-a/b)*b^2-24*arcsin(d*x+c)*sin(3*(a+b*arcsin(d*x+c))/
b-3*a/b)*a*b-6*arcsin(d*x+c)*cos(3*(a+b*arcsin(d*x+c))/b-3*a/b)*b^2+36*sin((a+b*arcsin(d*x+c))/b-a/b)*a^2+54*c
os((a+b*arcsin(d*x+c))/b-a/b)*a*b-12*sin(3*(a+b*arcsin(d*x+c))/b-3*a/b)*a^2-6*cos(3*(a+b*arcsin(d*x+c))/b-3*a/
b)*a*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{3}{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))^(3/2),x, algorithm="maxima")

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

[Out]

e**2*(Integral(a*c**2*sqrt(a + b*asin(c + d*x)), x) + Integral(a*d**2*x**2*sqrt(a + b*asin(c + d*x)), x) + Int
egral(b*c**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x), x) + Integral(2*a*c*d*x*sqrt(a + b*asin(c + d*x)), x) +
Integral(b*d**2*x**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x), x) + Integral(2*b*c*d*x*sqrt(a + b*asin(c + d*x)
)*asin(c + d*x), x))

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Giac [B]  time = 2.92723, size = 2016, normalized size = 5.58 \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))^(3/2),x, algorithm="giac")

[Out]

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