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

Optimal. Leaf size=342 \[ -\frac{\sqrt{2 \pi } e^2 \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}+\frac{\sqrt{6 \pi } e^2 \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{b^{5/2} d}-\frac{\sqrt{2 \pi } e^2 \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}+\frac{\sqrt{6 \pi } e^2 \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{b^{5/2} d}+\frac{4 e^2 (c+d x)^3}{b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{8 e^2 (c+d x)}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{2 e^2 \sqrt{1-(c+d x)^2} (c+d x)^2}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}} \]

[Out]

(-2*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^(3/2)) - (8*e^2*(c + d*x))/(3*b^2*d*
Sqrt[a + b*ArcSin[c + d*x]]) + (4*e^2*(c + d*x)^3)/(b^2*d*Sqrt[a + b*ArcSin[c + d*x]]) - (e^2*Sqrt[2*Pi]*Cos[a
/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) + (e^2*Sqrt[6*Pi]*Cos[(3*a)/b]*F
resnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(b^(5/2)*d) - (e^2*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*
Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*d) + (e^2*Sqrt[6*Pi]*FresnelS[(Sqrt[6/Pi]*Sqrt[a +
b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(5/2)*d)

________________________________________________________________________________________

Rubi [A]  time = 1.05408, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {4805, 12, 4633, 4719, 4635, 4406, 3306, 3305, 3351, 3304, 3352, 4623} \[ -\frac{\sqrt{2 \pi } e^2 \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}+\frac{\sqrt{6 \pi } e^2 \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{b^{5/2} d}-\frac{\sqrt{2 \pi } e^2 \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}+\frac{\sqrt{6 \pi } e^2 \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{b^{5/2} d}+\frac{4 e^2 (c+d x)^3}{b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{8 e^2 (c+d x)}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{2 e^2 \sqrt{1-(c+d x)^2} (c+d x)^2}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^(3/2)) - (8*e^2*(c + d*x))/(3*b^2*d*
Sqrt[a + b*ArcSin[c + d*x]]) + (4*e^2*(c + d*x)^3)/(b^2*d*Sqrt[a + b*ArcSin[c + d*x]]) - (e^2*Sqrt[2*Pi]*Cos[a
/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) + (e^2*Sqrt[6*Pi]*Cos[(3*a)/b]*F
resnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(b^(5/2)*d) - (e^2*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*
Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*d) + (e^2*Sqrt[6*Pi]*FresnelS[(Sqrt[6/Pi]*Sqrt[a +
b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(5/2)*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 4633

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

Rule 4719

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

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]

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 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]

Rubi steps

\begin{align*} \int \frac{(c e+d e x)^2}{\left (a+b \sin ^{-1}(c+d x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{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 \frac{x^2}{\left (a+b \sin ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac{\left (4 e^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2} \left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d}-\frac{\left (2 e^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{1-x^2} \left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{b d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{8 e^2 (c+d x)}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{4 e^2 (c+d x)^3}{b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{\left (8 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{3 b^2 d}-\frac{\left (12 e^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{b^2 d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{8 e^2 (c+d x)}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{4 e^2 (c+d x)^3}{b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{\left (8 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 )}{3 b^3 d}-\frac{\left (12 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 )}{b^2 d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{8 e^2 (c+d x)}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{4 e^2 (c+d x)^3}{b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{\left (12 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 )}{b^2 d}+\frac{\left (8 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 )}{3 b^3 d}+\frac{\left (8 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 )}{3 b^3 d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{8 e^2 (c+d x)}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{4 e^2 (c+d x)^3}{b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{b^2 d}+\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{b^2 d}+\frac{\left (16 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 b^3 d}+\frac{\left (16 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 b^3 d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{8 e^2 (c+d x)}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{4 e^2 (c+d x)^3}{b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{8 e^2 \sqrt{2 \pi } \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 b^{5/2} d}+\frac{8 e^2 \sqrt{2 \pi } 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 b^{5/2} d}-\frac{\left (3 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 )}{b^2 d}+\frac{\left (3 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 )}{b^2 d}-\frac{\left (3 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 )}{b^2 d}+\frac{\left (3 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 )}{b^2 d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{8 e^2 (c+d x)}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{4 e^2 (c+d x)^3}{b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{8 e^2 \sqrt{2 \pi } \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 b^{5/2} d}+\frac{8 e^2 \sqrt{2 \pi } 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 b^{5/2} d}-\frac{\left (6 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 )}{b^3 d}+\frac{\left (6 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 )}{b^3 d}-\frac{\left (6 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 )}{b^3 d}+\frac{\left (6 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 )}{b^3 d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{8 e^2 (c+d x)}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{4 e^2 (c+d x)^3}{b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{e^2 \sqrt{2 \pi } \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 b^{5/2} d}+\frac{e^2 \sqrt{6 \pi } \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 )}{b^{5/2} d}-\frac{e^2 \sqrt{2 \pi } 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 b^{5/2} d}+\frac{e^2 \sqrt{6 \pi } 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 )}{b^{5/2} d}\\ \end{align*}

Mathematica [C]  time = 1.84144, size = 411, normalized size = 1.2 \[ \frac{e^2 \left (-2 b e^{-\frac{i a}{b}} \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+6 \sqrt{3} b e^{-\frac{3 i a}{b}} \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+i e^{-i \sin ^{-1}(c+d x)} \left (2 i b e^{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+2 a+2 b \sin ^{-1}(c+d x)+i b\right )+6 \sqrt{3} b e^{\frac{3 i a}{b}} \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{3 i \sin ^{-1}(c+d x)} \left (6 i a+6 i b \sin ^{-1}(c+d x)+b\right )-i e^{i \sin ^{-1}(c+d x)} \left (2 a+2 b \sin ^{-1}(c+d x)-i b\right )-6 i a e^{-3 i \sin ^{-1}(c+d x)}+b e^{-3 i \sin ^{-1}(c+d x)} \left (1-6 i \sin ^{-1}(c+d x)\right )\right )}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(e^2*(((-6*I)*a)/E^((3*I)*ArcSin[c + d*x]) + (b*(1 - (6*I)*ArcSin[c + d*x]))/E^((3*I)*ArcSin[c + d*x]) + E^((3
*I)*ArcSin[c + d*x])*((6*I)*a + b + (6*I)*b*ArcSin[c + d*x]) - I*E^(I*ArcSin[c + d*x])*(2*a - I*b + 2*b*ArcSin
[c + d*x]) - (2*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b])/E^((I
*a)/b) + (I*(2*a + I*b + 2*b*ArcSin[c + d*x] + (2*I)*b*E^((I*(a + b*ArcSin[c + d*x]))/b)*((I*(a + b*ArcSin[c +
d*x]))/b)^(3/2)*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b]))/E^(I*ArcSin[c + d*x]) + (6*Sqrt[3]*b*(((-I)*(a +
b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((3*I)*a)/b) + 6*Sqrt[3]*b*E^(
((3*I)*a)/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b]))/(12*b^2*d*(
a + b*ArcSin[c + d*x])^(3/2))

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Maple [B]  time = 0.121, size = 721, normalized size = 2.1 \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]

1/6/d*e^2/b^2*(6*arcsin(d*x+c)*3^(1/2)*2^(1/2)*Pi^(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)*(1/b)^(1/2)*b+6*arcsin(d*x+c)*3^(1/2)*2^(1/2)*Pi^(1/
2)*(a+b*arcsin(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)*(1/b)^(1/2)*b-2*arcsin(d*x+c)*2^(1/2)*Pi^(1/2)*(1/b)^(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*arcsin(d*x+c)*2^(1/2)*Pi^(1/2)*(1/b)^(1/2)*(a+b*ar
csin(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+6*3^(1/2)*2^(
1/2)*Pi^(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)*(1/b)^(1/2)*a+6*3^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*arcsin(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)*(1/b)^(1/2)*a-2*2^(1/2)*Pi^(1/2)*(1/b)^(1/2)*(a+b*a
rcsin(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)*a-2*2^(1/2)*Pi
^(1/2)*(1/b)^(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)*a+2*arcsin(d*x+c)*sin((a+b*arcsin(d*x+c))/b-a/b)*b-6*arcsin(d*x+c)*sin(3*(a+b*arcsin(d*x+c))/b-3*a/
b)*b-cos((a+b*arcsin(d*x+c))/b-a/b)*b+2*sin((a+b*arcsin(d*x+c))/b-a/b)*a+cos(3*(a+b*arcsin(d*x+c))/b-3*a/b)*b-
6*sin(3*(a+b*arcsin(d*x+c))/b-3*a/b)*a)/(a+b*arcsin(d*x+c))^(3/2)

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

[Out]

e**2*(Integral(c**2/(a**2*sqrt(a + b*asin(c + d*x)) + 2*a*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + b**2*sqr
t(a + b*asin(c + d*x))*asin(c + d*x)**2), x) + Integral(d**2*x**2/(a**2*sqrt(a + b*asin(c + d*x)) + 2*a*b*sqrt
(a + b*asin(c + d*x))*asin(c + d*x) + b**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2), x) + Integral(2*c*d*x/
(a**2*sqrt(a + b*asin(c + d*x)) + 2*a*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + b**2*sqrt(a + b*asin(c + d*x
))*asin(c + d*x)**2), x))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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="giac")

[Out]

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