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

Optimal. Leaf size=406 \[ \frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} c \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^2}-\frac{15 \sqrt{\pi } b^{5/2} \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{128 d^2}-\frac{15 \sqrt{\pi } b^{5/2} \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{128 d^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} c \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^2}+\frac{15 b^2 c (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d^2}+\frac{15 b^2 \cos \left (2 \sin ^{-1}(c+d x)\right ) \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d^2}-\frac{5 b c \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d^2}+\frac{5 b \sin \left (2 \sin ^{-1}(c+d x)\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{16 d^2}-\frac{c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{d^2}-\frac{\cos \left (2 \sin ^{-1}(c+d x)\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d^2} \]

[Out]

(15*b^2*c*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/(4*d^2) - (5*b*c*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]
)^(3/2))/(2*d^2) - (c*(c + d*x)*(a + b*ArcSin[c + d*x])^(5/2))/d^2 + (15*b^2*Sqrt[a + b*ArcSin[c + d*x]]*Cos[2
*ArcSin[c + d*x]])/(64*d^2) - ((a + b*ArcSin[c + d*x])^(5/2)*Cos[2*ArcSin[c + d*x]])/(4*d^2) - (15*b^(5/2)*Sqr
t[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(128*d^2) - (15*b^(5/2)*c*Sqr
t[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*d^2) + (15*b^(5/2)*c*Sqrt[Pi/2
]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(4*d^2) - (15*b^(5/2)*Sqrt[Pi]*FresnelS
[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(128*d^2) + (5*b*(a + b*ArcSin[c + d*x])^(3
/2)*Sin[2*ArcSin[c + d*x]])/(16*d^2)

________________________________________________________________________________________

Rubi [A]  time = 1.17029, antiderivative size = 406, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4805, 4747, 6741, 6742, 3386, 3385, 3353, 3352, 3351, 3354} \[ \frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} c \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^2}-\frac{15 \sqrt{\pi } b^{5/2} \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{128 d^2}-\frac{15 \sqrt{\pi } b^{5/2} \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{128 d^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} c \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^2}+\frac{15 b^2 c (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d^2}+\frac{15 b^2 \cos \left (2 \sin ^{-1}(c+d x)\right ) \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d^2}-\frac{5 b c \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d^2}+\frac{5 b \sin \left (2 \sin ^{-1}(c+d x)\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{16 d^2}-\frac{c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{d^2}-\frac{\cos \left (2 \sin ^{-1}(c+d x)\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*b^2*c*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/(4*d^2) - (5*b*c*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]
)^(3/2))/(2*d^2) - (c*(c + d*x)*(a + b*ArcSin[c + d*x])^(5/2))/d^2 + (15*b^2*Sqrt[a + b*ArcSin[c + d*x]]*Cos[2
*ArcSin[c + d*x]])/(64*d^2) - ((a + b*ArcSin[c + d*x])^(5/2)*Cos[2*ArcSin[c + d*x]])/(4*d^2) - (15*b^(5/2)*Sqr
t[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(128*d^2) - (15*b^(5/2)*c*Sqr
t[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*d^2) + (15*b^(5/2)*c*Sqrt[Pi/2
]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(4*d^2) - (15*b^(5/2)*Sqrt[Pi]*FresnelS
[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(128*d^2) + (5*b*(a + b*ArcSin[c + d*x])^(3
/2)*Sin[2*ArcSin[c + d*x]])/(16*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 4747

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 3386

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Sin[c + d*
x^n])/(d*n), x] - Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x
] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3385

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> -Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Cos[c + d
*x^n])/(d*n), x] + Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e},
x] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3353

Int[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Sin[c], Int[Cos[d*(e + f*x)^2], x], x] + Dist[
Cos[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, 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 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 3354

Int[Cos[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Cos[c], Int[Cos[d*(e + f*x)^2], x], x] - Dist[
Sin[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]

Rubi steps

\begin{align*} \int x \left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{c}{d}+\frac{x}{d}\right ) \left (a+b \sin ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^{5/2} \cos (x) \left (-\frac{c}{d}+\frac{\sin (x)}{d}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=-\frac{2 \operatorname{Subst}\left (\int x^6 \cos \left (\frac{a-x^2}{b}\right ) \left (c+\sin \left (\frac{a-x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac{2 \operatorname{Subst}\left (\int x^6 \cos \left (\frac{a}{b}-\frac{x^2}{b}\right ) \left (c+\sin \left (\frac{a-x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac{2 \operatorname{Subst}\left (\int \left (c x^6 \cos \left (\frac{a}{b}-\frac{x^2}{b}\right )+\frac{1}{2} x^6 \sin \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac{\operatorname{Subst}\left (\int x^6 \sin \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^2}-\frac{(2 c) \operatorname{Subst}\left (\int x^6 \cos \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac{c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{d^2}-\frac{\left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \cos \left (2 \sin ^{-1}(c+d x)\right )}{4 d^2}+\frac{5 \operatorname{Subst}\left (\int x^4 \cos \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{4 d^2}-\frac{(5 c) \operatorname{Subst}\left (\int x^4 \sin \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{d^2}\\ &=-\frac{5 b c \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac{c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{d^2}-\frac{\left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \cos \left (2 \sin ^{-1}(c+d x)\right )}{4 d^2}+\frac{5 b \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \sin \left (2 \sin ^{-1}(c+d x)\right )}{16 d^2}+\frac{(15 b) \operatorname{Subst}\left (\int x^2 \sin \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{16 d^2}+\frac{(15 b c) \operatorname{Subst}\left (\int x^2 \cos \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{2 d^2}\\ &=\frac{15 b^2 c (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d^2}-\frac{5 b c \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac{c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac{15 b^2 \sqrt{a+b \sin ^{-1}(c+d x)} \cos \left (2 \sin ^{-1}(c+d x)\right )}{64 d^2}-\frac{\left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \cos \left (2 \sin ^{-1}(c+d x)\right )}{4 d^2}+\frac{5 b \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \sin \left (2 \sin ^{-1}(c+d x)\right )}{16 d^2}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{64 d^2}+\frac{\left (15 b^2 c\right ) \operatorname{Subst}\left (\int \sin \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{4 d^2}\\ &=\frac{15 b^2 c (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d^2}-\frac{5 b c \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac{c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac{15 b^2 \sqrt{a+b \sin ^{-1}(c+d x)} \cos \left (2 \sin ^{-1}(c+d x)\right )}{64 d^2}-\frac{\left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \cos \left (2 \sin ^{-1}(c+d x)\right )}{4 d^2}+\frac{5 b \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \sin \left (2 \sin ^{-1}(c+d x)\right )}{16 d^2}-\frac{\left (15 b^2 c \cos \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 )}{4 d^2}-\frac{\left (15 b^2 \cos \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 )}{64 d^2}+\frac{\left (15 b^2 c \sin \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 )}{4 d^2}-\frac{\left (15 b^2 \sin \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 )}{64 d^2}\\ &=\frac{15 b^2 c (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d^2}-\frac{5 b c \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac{c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac{15 b^2 \sqrt{a+b \sin ^{-1}(c+d x)} \cos \left (2 \sin ^{-1}(c+d x)\right )}{64 d^2}-\frac{\left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \cos \left (2 \sin ^{-1}(c+d x)\right )}{4 d^2}-\frac{15 b^{5/2} \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{128 d^2}-\frac{15 b^{5/2} c \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^2}+\frac{15 b^{5/2} c \sqrt{\frac{\pi }{2}} C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{4 d^2}-\frac{15 b^{5/2} \sqrt{\pi } S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{128 d^2}+\frac{5 b \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \sin \left (2 \sin ^{-1}(c+d x)\right )}{16 d^2}\\ \end{align*}

Mathematica [C]  time = 9.42034, size = 1083, normalized size = 2.67 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a^2*b*c*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + E^(((2*I)*a)/
b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(2*d^2*E^((I*a)/b)*Sqrt[a +
b*ArcSin[c + d*x]]) - (a*b*c*(2*Sqrt[a + b*ArcSin[c + d*x]]*(3*Sqrt[1 - (c + d*x)^2] + 2*(c + d*x)*ArcSin[c +
d*x]) - Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(3*b*Cos[a/b] +
2*a*Sin[a/b]) + Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(2*a*Co
s[a/b] - 3*b*Sin[a/b])))/(2*d^2) - (c*((2*Sqrt[a + b*ArcSin[c + d*x]]*(-2*Sqrt[1 - (c + d*x)^2]*(a - 5*b*ArcSi
n[c + d*x]) + b*(c + d*x)*(-15 + 4*ArcSin[c + d*x]^2)))/Sqrt[b^(-1)] + Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2
/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*((-4*a^2 + 15*b^2)*Cos[a/b] + 12*a*b*Sin[a/b]) + Sqrt[2*Pi]*FresnelC[Sqrt[b^
(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(12*a*b*Cos[a/b] + (4*a^2 - 15*b^2)*Sin[a/b])))/(8*Sqrt[b^(-1)]*
d^2) + (a^2*(-2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]]*Cos[2*ArcSin[c + d*x]] + Sqrt[Pi]*Cos[(2*a)/b]*Fresne
lC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]] + Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSi
n[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b]))/(8*Sqrt[b^(-1)]*d^2) + (a*b*(-(Sqrt[b^(-1)]*Sqrt[Pi]*FresnelS[(2*Sqrt[b^
(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*(3*b*Cos[(2*a)/b] + 4*a*Sin[(2*a)/b])) + Sqrt[b^(-1)]*Sqrt[Pi]*Fr
esnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*(-4*a*Cos[(2*a)/b] + 3*b*Sin[(2*a)/b]) + 2*Sqrt[
a + b*ArcSin[c + d*x]]*(-4*ArcSin[c + d*x]*Cos[2*ArcSin[c + d*x]] + 3*Sin[2*ArcSin[c + d*x]])))/(16*d^2) + (Sq
rt[Pi]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*((16*a^2 - 15*b^2)*Cos[(2*a)/b] - 24*a*
b*Sin[(2*a)/b]) - Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*(-24*a*b*Cos[(2*a)/
b] + (-16*a^2 + 15*b^2)*Sin[(2*a)/b]) - (2*Sqrt[a + b*ArcSin[c + d*x]]*(b*(-15 + 16*ArcSin[c + d*x]^2)*Cos[2*A
rcSin[c + d*x]] + 4*(a - 5*b*ArcSin[c + d*x])*Sin[2*ArcSin[c + d*x]]))/Sqrt[b^(-1)])/(128*Sqrt[b^(-1)]*d^2)

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Maple [B]  time = 0.157, size = 859, 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(x*(a+b*arcsin(d*x+c))^(5/2),x)

[Out]

-1/128/d^2*(240*2^(1/2)*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(1/b
)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3*c-240*2^(1/2)*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(a/b)
*FresnelC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3*c+128*arcsin(d*x+c)^3*sin((a+b*arcsin(
d*x+c))/b-a/b)*b^3*c+15*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*FresnelC(2/Pi^(1/2)/(1/b)^(1
/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3+15*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/b)*FresnelS(2/P
i^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3+32*arcsin(d*x+c)^3*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*b^3
+384*arcsin(d*x+c)^2*sin((a+b*arcsin(d*x+c))/b-a/b)*a*b^2*c+320*arcsin(d*x+c)^2*cos((a+b*arcsin(d*x+c))/b-a/b)
*b^3*c+96*arcsin(d*x+c)^2*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*a*b^2-40*arcsin(d*x+c)^2*sin(2*(a+b*arcsin(d*x+c)
)/b-2*a/b)*b^3+384*arcsin(d*x+c)*sin((a+b*arcsin(d*x+c))/b-a/b)*a^2*b*c-480*arcsin(d*x+c)*sin((a+b*arcsin(d*x+
c))/b-a/b)*b^3*c+640*arcsin(d*x+c)*cos((a+b*arcsin(d*x+c))/b-a/b)*a*b^2*c+96*arcsin(d*x+c)*cos(2*(a+b*arcsin(d
*x+c))/b-2*a/b)*a^2*b-30*arcsin(d*x+c)*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*b^3-80*arcsin(d*x+c)*sin(2*(a+b*arcs
in(d*x+c))/b-2*a/b)*a*b^2+128*sin((a+b*arcsin(d*x+c))/b-a/b)*a^3*c-480*sin((a+b*arcsin(d*x+c))/b-a/b)*a*b^2*c+
320*cos((a+b*arcsin(d*x+c))/b-a/b)*a^2*b*c+32*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*a^3-30*cos(2*(a+b*arcsin(d*x+
c))/b-2*a/b)*a*b^2-40*sin(2*(a+b*arcsin(d*x+c))/b-2*a/b)*a^2*b)/(a+b*arcsin(d*x+c))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} x\,{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((b*arcsin(d*x + c) + a)^(5/2)*x, 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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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]

Timed out

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Giac [B]  time = 3.46289, size = 3578, normalized size = 8.81 \begin{align*} \text{result too large to display} \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]

1/2*sqrt(2)*sqrt(pi)*a^2*b^3*c*i*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqr
t(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d^2) + 1/2*sqrt(2)
*sqrt(pi)*a^2*b^3*c*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)/((b^3*i/sqrt(abs(b)) - b^2*sqrt(abs(b)))*d^2) - 3/4*sqrt(2)*sqrt(pi)*a
*b^4*c*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)*s
qrt(abs(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d^2) - 1/2*sqrt(2)*sqrt(pi)*a^2*b^2*c*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)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d^2) + 15/16*sqrt(2)*sqrt(pi)*b^4*c*i*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)/(
(b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d^2) + 3/4*sqrt(2)*sqrt(pi)*a*b^4*c*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)/((b^3*i/sqrt(abs(b
)) - b^2*sqrt(abs(b)))*d^2) - 1/2*sqrt(2)*sqrt(pi)*a^2*b^2*c*i*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/s
qrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^2*i/sqrt(abs(b)) - b*sqrt
(abs(b)))*d^2) + 15/16*sqrt(2)*sqrt(pi)*b^4*c*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)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*d^2)
+ 1/2*sqrt(b*arcsin(d*x + c) + a)*b^2*c*i*arcsin(d*x + c)^2*e^(i*arcsin(d*x + c))/d^2 - 1/2*sqrt(b*arcsin(d*x
+ c) + a)*b^2*c*i*arcsin(d*x + c)^2*e^(-i*arcsin(d*x + c))/d^2 + 3/32*sqrt(pi)*a*b^(7/2)*i*erf(-sqrt(b*arcsin(
d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b)/((b^3*i/abs(b) + b^2)*d^2) +
3/4*sqrt(2)*sqrt(pi)*a*b^3*c*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)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d^2) - 3/4*sqrt(2)*sqrt
(pi)*a*b^3*c*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)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*d^2) + 3/32*sqrt(pi)*a*b^(7/2)*i*erf(sqr
t(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b)/((b^3*i/abs(b) -
b^2)*d^2) + sqrt(b*arcsin(d*x + c) + a)*a*b*c*i*arcsin(d*x + c)*e^(i*arcsin(d*x + c))/d^2 - sqrt(b*arcsin(d*x
+ c) + a)*a*b*c*i*arcsin(d*x + c)*e^(-i*arcsin(d*x + c))/d^2 + 1/8*sqrt(pi)*a^2*b^(5/2)*erf(-sqrt(b*arcsin(d*
x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b)/((b^3*i/abs(b) + b^2)*d^2) - 3
/32*sqrt(pi)*a*b^(5/2)*i*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(
b))*e^(2*a*i/b)/((b^2*i/abs(b) + b)*d^2) - 1/8*sqrt(pi)*a^2*b^(5/2)*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/
abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b)/((b^3*i/abs(b) - b^2)*d^2) - 3/32*sqrt(pi)*a*b^(5/2
)*i*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b)/((b^2
*i/abs(b) - b)*d^2) - 5/32*sqrt(b*arcsin(d*x + c) + a)*b^2*i*arcsin(d*x + c)*e^(2*i*arcsin(d*x + c))/d^2 - 1/8
*sqrt(b*arcsin(d*x + c) + a)*b^2*arcsin(d*x + c)^2*e^(2*i*arcsin(d*x + c))/d^2 + 1/2*sqrt(b*arcsin(d*x + c) +
a)*a^2*c*i*e^(i*arcsin(d*x + c))/d^2 - 15/8*sqrt(b*arcsin(d*x + c) + a)*b^2*c*i*e^(i*arcsin(d*x + c))/d^2 - 5/
4*sqrt(b*arcsin(d*x + c) + a)*b^2*c*arcsin(d*x + c)*e^(i*arcsin(d*x + c))/d^2 - 1/2*sqrt(b*arcsin(d*x + c) + a
)*a^2*c*i*e^(-i*arcsin(d*x + c))/d^2 + 15/8*sqrt(b*arcsin(d*x + c) + a)*b^2*c*i*e^(-i*arcsin(d*x + c))/d^2 - 5
/4*sqrt(b*arcsin(d*x + c) + a)*b^2*c*arcsin(d*x + c)*e^(-i*arcsin(d*x + c))/d^2 + 5/32*sqrt(b*arcsin(d*x + c)
+ a)*b^2*i*arcsin(d*x + c)*e^(-2*i*arcsin(d*x + c))/d^2 - 1/8*sqrt(b*arcsin(d*x + c) + a)*b^2*arcsin(d*x + c)^
2*e^(-2*i*arcsin(d*x + c))/d^2 - 1/16*sqrt(pi)*a^2*b^2*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqr
t(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b)/((b^(5/2)*i/abs(b) + b^(3/2))*d^2) + 1/16*sqrt(pi)*a^2*b^2*erf(s
qrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b)/((b^(5/2)*i/ab
s(b) - b^(3/2))*d^2) - 1/16*sqrt(pi)*a^2*b^(3/2)*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*ar
csin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b)/((b^2*i/abs(b) + b)*d^2) + 15/256*sqrt(pi)*b^(7/2)*erf(-sqrt(b*arcsin(
d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b)/((b^2*i/abs(b) + b)*d^2) + 1
/16*sqrt(pi)*a^2*b^(3/2)*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b
))*e^(-2*a*i/b)/((b^2*i/abs(b) - b)*d^2) - 15/256*sqrt(pi)*b^(7/2)*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/a
bs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b)/((b^2*i/abs(b) - b)*d^2) - 5/32*sqrt(b*arcsin(d*x +
c) + a)*a*b*i*e^(2*i*arcsin(d*x + c))/d^2 - 1/4*sqrt(b*arcsin(d*x + c) + a)*a*b*arcsin(d*x + c)*e^(2*i*arcsin(
d*x + c))/d^2 - 5/4*sqrt(b*arcsin(d*x + c) + a)*a*b*c*e^(i*arcsin(d*x + c))/d^2 - 5/4*sqrt(b*arcsin(d*x + c) +
a)*a*b*c*e^(-i*arcsin(d*x + c))/d^2 + 5/32*sqrt(b*arcsin(d*x + c) + a)*a*b*i*e^(-2*i*arcsin(d*x + c))/d^2 - 1
/4*sqrt(b*arcsin(d*x + c) + a)*a*b*arcsin(d*x + c)*e^(-2*i*arcsin(d*x + c))/d^2 - 1/8*sqrt(b*arcsin(d*x + c) +
a)*a^2*e^(2*i*arcsin(d*x + c))/d^2 + 15/128*sqrt(b*arcsin(d*x + c) + a)*b^2*e^(2*i*arcsin(d*x + c))/d^2 - 1/8
*sqrt(b*arcsin(d*x + c) + a)*a^2*e^(-2*i*arcsin(d*x + c))/d^2 + 15/128*sqrt(b*arcsin(d*x + c) + a)*b^2*e^(-2*i
*arcsin(d*x + c))/d^2