3.2.68 \(\int \frac {x}{(a+b \text {ArcSin}(c+d x))^{5/2}} \, dx\) [168]

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

[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 = 0.62, 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 = {4889, 4829, 4717, 4807, 4719, 3387, 3386, 3432, 3385, 3433, 4729, 4731, 4491, 12, 4737} \begin {gather*} \frac {8 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(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 \text {ArcSin}(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 \text {ArcSin}(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 \text {ArcSin}(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 \text {ArcSin}(c+d x)}}-\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \text {ArcSin}(c+d x)}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \text {ArcSin}(c+d x)}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{3 b d^2 (a+b \text {ArcSin}(c+d x))^{3/2}}+\frac {2 c \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \text {ArcSin}(c+d x))^{3/2}} \end {gather*}

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 3385

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 3386

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 3387

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 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 4491

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 4717

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_), 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 4729

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 4731

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

Rule 4737

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

Rule 4807

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

Rule 4829

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 4889

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 {\text {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 {\text {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 {\text {Subst}\left (\int \frac {x}{\left (a+b \sin ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d^2}-\frac {c \text {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 \text {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 \text {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) \text {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 \text {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) \text {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 \text {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) \text {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 \text {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 ) \text {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 ) \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Result contains complex when optimal does not.
time = 1.79, size = 392, normalized size = 1.02 \begin {gather*} \frac {-4 a \cos (2 \text {ArcSin}(c+d x))-4 b \text {ArcSin}(c+d x) \cos (2 \text {ArcSin}(c+d x))-8 \sqrt {\frac {1}{b}} \sqrt {\pi } (a+b \text {ArcSin}(c+d x))^{3/2} \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi }}\right )+2 b c e^{-\frac {i a}{b}} \left (-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )+c e^{-i \text {ArcSin}(c+d x)} \left (-2 i a+b+2 i a e^{2 i \text {ArcSin}(c+d x)}+b e^{2 i \text {ArcSin}(c+d x)}+2 i b \left (-1+e^{2 i \text {ArcSin}(c+d x)}\right ) \text {ArcSin}(c+d x)+2 b e^{\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \left (\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )+8 \sqrt {\frac {1}{b}} \sqrt {\pi } (a+b \text {ArcSin}(c+d x))^{3/2} \text {FresnelC}\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )-b \sin (2 \text {ArcSin}(c+d x))}{3 b^2 d^2 (a+b \text {ArcSin}(c+d x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[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] Leaf count of result is larger than twice the leaf count of optimal. \(737\) vs. \(2(312)=624\).
time = 0.49, size = 738, normalized size = 1.92

method	result	size
default	\(-\frac {-4 \arcsin \left (d x +c \right ) \sqrt {\pi }\, \sqrt {2}\, \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 ) \sqrt {-\frac {1}{b}}\, b c +4 \arcsin \left (d x +c \right ) \sqrt {\pi }\, \sqrt {2}\, \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 ) \sqrt {-\frac {1}{b}}\, b c -4 \arcsin \left (d x +c \right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b -4 \arcsin \left (d x +c \right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b -4 \sqrt {\pi }\, \sqrt {2}\, \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 ) \sqrt {-\frac {1}{b}}\, a c +4 \sqrt {\pi }\, \sqrt {2}\, \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 ) \sqrt {-\frac {1}{b}}\, a c -4 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) a -4 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\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 \left (a +b \arcsin \left (d x +c \right )\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 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b +4 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\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}}}\)	\(738\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/d^2/b^2*(-4*arcsin(d*x+c)*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)*(-1/b)^(1/2)*b*c+4*arcsin(d*x+c)*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)*(-1/b)^(1/2)*b*c-4*arcsin(
d*x+c)*2^(1/2)*Pi^(1/2)*(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(
1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b-4*arcsin(d*x+c)*2^(1/2)*Pi^(1/2)*(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*si
n(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b-4*Pi^(1/2)*2^(1/2)*(a+b*arcsi
n(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)*(-1/b)^(1/2)*a*c+
4*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)*(-1/b)^(1/2)*a*c-4*2^(1/2)*Pi^(1/2)*(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*FresnelS(2*2
^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*a-4*2^(1/2)*Pi^(1/2)*(-2/b)^(1/2)*(a+b*arcsin(d*x+c)
)^(1/2)*sin(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*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*arcsi
n(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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