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

Optimal. Leaf size=384 \[ \frac{8 \sqrt{\pi } \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{3 b^{5/2} d^2}+\frac{4 \sqrt{2 \pi } c \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d^2}+\frac{4 \sqrt{2 \pi } c \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d^2}-\frac{8 \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{3 b^{5/2} d^2}+\frac{8 (c+d x)^2}{3 b^2 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{4 c (c+d x)}{3 b^2 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{4}{3 b^2 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{2 \sqrt{1-(c+d x)^2} (c+d x)}{3 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac{2 c \sqrt{1-(c+d x)^2}}{3 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}} \]

[Out]

(2*c*Sqrt[1 - (c + d*x)^2])/(3*b*d^2*(a + b*ArcSin[c + d*x])^(3/2)) - (2*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(3*b
*d^2*(a + b*ArcSin[c + d*x])^(3/2)) - 4/(3*b^2*d^2*Sqrt[a + b*ArcSin[c + d*x]]) - (4*c*(c + d*x))/(3*b^2*d^2*S
qrt[a + b*ArcSin[c + d*x]]) + (8*(c + d*x)^2)/(3*b^2*d^2*Sqrt[a + b*ArcSin[c + d*x]]) + (4*c*Sqrt[2*Pi]*Cos[a/
b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d^2) - (8*Sqrt[Pi]*Cos[(2*a)/b]*Fres
nelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(3*b^(5/2)*d^2) + (4*c*Sqrt[2*Pi]*FresnelS[(Sqrt[2/P
i]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*d^2) + (8*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin
[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(3*b^(5/2)*d^2)

________________________________________________________________________________________

Rubi [A]  time = 0.89594, antiderivative size = 384, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 15, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.938, Rules used = {4805, 4745, 4621, 4719, 4623, 3306, 3305, 3351, 3304, 3352, 4633, 4635, 4406, 12, 4641} \[ \frac{8 \sqrt{\pi } \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{3 b^{5/2} d^2}+\frac{4 \sqrt{2 \pi } c \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d^2}+\frac{4 \sqrt{2 \pi } c \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d^2}-\frac{8 \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{3 b^{5/2} d^2}+\frac{8 (c+d x)^2}{3 b^2 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{4 c (c+d x)}{3 b^2 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{4}{3 b^2 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{2 \sqrt{1-(c+d x)^2} (c+d x)}{3 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac{2 c \sqrt{1-(c+d x)^2}}{3 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c*Sqrt[1 - (c + d*x)^2])/(3*b*d^2*(a + b*ArcSin[c + d*x])^(3/2)) - (2*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(3*b
*d^2*(a + b*ArcSin[c + d*x])^(3/2)) - 4/(3*b^2*d^2*Sqrt[a + b*ArcSin[c + d*x]]) - (4*c*(c + d*x))/(3*b^2*d^2*S
qrt[a + b*ArcSin[c + d*x]]) + (8*(c + d*x)^2)/(3*b^2*d^2*Sqrt[a + b*ArcSin[c + d*x]]) + (4*c*Sqrt[2*Pi]*Cos[a/
b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d^2) - (8*Sqrt[Pi]*Cos[(2*a)/b]*Fres
nelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(3*b^(5/2)*d^2) + (4*c*Sqrt[2*Pi]*FresnelS[(Sqrt[2/P
i]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*d^2) + (8*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin
[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(3*b^(5/2)*d^2)

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 4745

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

Rule 4621

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

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 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 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 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 12

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

Rule 4641

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

Rubi steps

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

Mathematica [C]  time = 2.70806, size = 392, normalized size = 1.02 \[ \frac{2 b c e^{-\frac{i a}{b}} \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+c e^{-i \sin ^{-1}(c+d x)} \left (2 b e^{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+2 i a e^{2 i \sin ^{-1}(c+d x)}-2 i a+b e^{2 i \sin ^{-1}(c+d x)}+2 i b \left (-1+e^{2 i \sin ^{-1}(c+d x)}\right ) \sin ^{-1}(c+d x)+b\right )+8 \sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left (\frac{2 a}{b}\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \text{FresnelC}\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right )-8 \sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left (\frac{2 a}{b}\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} S\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right )-4 a \cos \left (2 \sin ^{-1}(c+d x)\right )-b \sin \left (2 \sin ^{-1}(c+d x)\right )-4 b \sin ^{-1}(c+d x) \cos \left (2 \sin ^{-1}(c+d x)\right )}{3 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-4*a*Cos[2*ArcSin[c + d*x]] - 4*b*ArcSin[c + d*x]*Cos[2*ArcSin[c + d*x]] - 8*Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*Arc
Sin[c + d*x])^(3/2)*Cos[(2*a)/b]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]] + (2*b*c*(((-
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) + (c*((-2*I)*a
+ b + (2*I)*a*E^((2*I)*ArcSin[c + d*x]) + b*E^((2*I)*ArcSin[c + d*x]) + (2*I)*b*(-1 + E^((2*I)*ArcSin[c + d*x]
))*ArcSin[c + d*x] + 2*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]) + 8*Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcSin[c + d*x])^(3/2)*F
resnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b] - b*Sin[2*ArcSin[c + d*x]])/(3*b^2
*d^2*(a + b*ArcSin[c + d*x])^(3/2))

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Maple [B]  time = 0.133, size = 681, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/d^2/b^2/(a+b*arcsin(d*x+c))^(3/2)*(-4*arcsin(d*x+c)*2^(1/2)*(1/b)^(1/2)*Pi^(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*c-4*arcsin(d*x+c)*2^(1/2)*(1/b
)^(1/2)*Pi^(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*c-4*2^(1/2)*(1/b)^(1/2)*Pi^(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)*a*c-4*2^(1/2)*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(a/b)*Fres
nelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*a*c+8*arcsin(d*x+c)*Pi^(1/2)*(1/b)^(1/2)*cos(2*
a/b)*FresnelS(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(a+b*arcsin(d*x+c))^(1/2)*b-8*arcsin(d*x+c)*
Pi^(1/2)*(1/b)^(1/2)*sin(2*a/b)*FresnelC(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(a+b*arcsin(d*x+c
))^(1/2)*b+8*Pi^(1/2)*(1/b)^(1/2)*cos(2*a/b)*FresnelS(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(a+b
*arcsin(d*x+c))^(1/2)*a-8*Pi^(1/2)*(1/b)^(1/2)*sin(2*a/b)*FresnelC(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^
(1/2)/b)*(a+b*arcsin(d*x+c))^(1/2)*a+4*arcsin(d*x+c)*sin((a+b*arcsin(d*x+c))/b-a/b)*b*c+4*arcsin(d*x+c)*cos(2*
(a+b*arcsin(d*x+c))/b-2*a/b)*b-2*cos((a+b*arcsin(d*x+c))/b-a/b)*b*c+4*sin((a+b*arcsin(d*x+c))/b-a/b)*a*c+sin(2
*(a+b*arcsin(d*x+c))/b-2*a/b)*b+4*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\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(x/(a+b*arcsin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(x/(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(x/(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*} \int \frac{x}{\left (a + b \operatorname{asin}{\left (c + d x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\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(x/(a+b*arcsin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

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