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

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

[Out]

-8/3*cos(2*a/b)*FresnelS(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(5/2)/d^2+8/3*FresnelC(2*(a+
b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/b^(5/2)/d^2+4/3*c*cos(a/b)*FresnelC(2^(1/2)/Pi^(1
/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(5/2)/d^2+4/3*c*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsi
n(d*x+c))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2)/b^(5/2)/d^2+2/3*c*(1-(d*x+c)^2)^(1/2)/b/d^2/(a+b*arcsin(d*x
+c))^(3/2)-2/3*(d*x+c)*(1-(d*x+c)^2)^(1/2)/b/d^2/(a+b*arcsin(d*x+c))^(3/2)-4/3/b^2/d^2/(a+b*arcsin(d*x+c))^(1/
2)-4/3*c*(d*x+c)/b^2/d^2/(a+b*arcsin(d*x+c))^(1/2)+8/3*(d*x+c)^2/b^2/d^2/(a+b*arcsin(d*x+c))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.90, antiderivative size = 384, normalized size of antiderivative = 1.00, 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 12

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

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 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 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]

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*}

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Mathematica [C]  time = 3.13, size = 392, normalized size = 1.02 \[ \frac {8 \sqrt {\pi } \sqrt {\frac {1}{b}} \sin \left (\frac {2 a}{b}\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} C\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi }}\right )-8 \sqrt {\pi } \sqrt {\frac {1}{b}} \cos \left (\frac {2 a}{b}\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi }}\right )+2 b c e^{-\frac {i a}{b}} \left (-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+c e^{-i \sin ^{-1}(c+d x)} \left (2 b e^{\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+2 i a e^{2 i \sin ^{-1}(c+d x)}-2 i a+b e^{2 i \sin ^{-1}(c+d x)}+2 i b \left (-1+e^{2 i \sin ^{-1}(c+d x)}\right ) \sin ^{-1}(c+d x)+b\right )-4 a \cos \left (2 \sin ^{-1}(c+d x)\right )-b \sin \left (2 \sin ^{-1}(c+d x)\right )-4 b \sin ^{-1}(c+d x) \cos \left (2 \sin ^{-1}(c+d x)\right )}{3 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}} \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]

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)

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maple [B]  time = 0.40, size = 681, normalized size = 1.77 \[ -\frac {-4 \sqrt {2}\, \arcsin \left (d x +c \right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b c -4 \sqrt {2}\, \arcsin \left (d x +c \right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b c -4 \sqrt {2}\, \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) a c -4 \sqrt {2}\, \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) a c +8 \arcsin \left (d x +c \right ) \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, b -8 \arcsin \left (d x +c \right ) \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, b +8 \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, a -8 \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, a +4 \arcsin \left (d x +c \right ) \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) b c +4 \arcsin \left (d x +c \right ) \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) b -2 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) b c +4 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a c +\sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) b +4 \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a}{3 d^{2} b^{2} \left (a +b \arcsin \left (d x +c \right )\right )^{\frac {3}{2}}} \]

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*(-4*2^(1/2)*arcsin(d*x+c)*(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*2^(1/2)*arcsin(d*x+c)*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcs
in(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)*FresnelS(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)/(a+b*arcsin(d*x+c))^(3/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*asin(c + d*x))^(5/2),x)

[Out]

int(x/(a + b*asin(c + d*x))^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]

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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