3.158 \(\int x (a+b \sin ^{-1}(c+d x))^{3/2} \, dx\)

Optimal. Leaf size=343 \[ \frac{3 \sqrt{\pi } b^{3/2} \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{32 d^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} c \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 d^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} c \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 d^2}-\frac{3 \sqrt{\pi } b^{3/2} \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{32 d^2}+\frac{3 b \sin \left (2 \sin ^{-1}(c+d x)\right ) \sqrt{a+b \sin ^{-1}(c+d x)}}{16 d^2}-\frac{3 b c \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d^2}-\frac{c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d^2}-\frac{\cos \left (2 \sin ^{-1}(c+d x)\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d^2} \]

[Out]

(-3*b*c*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(2*d^2) - (c*(c + d*x)*(a + b*ArcSin[c + d*x])^(3/2
))/d^2 - ((a + b*ArcSin[c + d*x])^(3/2)*Cos[2*ArcSin[c + d*x]])/(4*d^2) + (3*b^(3/2)*c*Sqrt[Pi/2]*Cos[a/b]*Fre
snelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*d^2) - (3*b^(3/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(
2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(32*d^2) + (3*b^(3/2)*c*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sq
rt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(2*d^2) + (3*b^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c +
d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(32*d^2) + (3*b*Sqrt[a + b*ArcSin[c + d*x]]*Sin[2*ArcSin[c + d*x]])/(
16*d^2)

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Rubi [A]  time = 1.04347, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 16, 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, 3354, 3352, 3351, 3353} \[ \frac{3 \sqrt{\pi } b^{3/2} \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{32 d^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} c \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 d^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} c \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 d^2}-\frac{3 \sqrt{\pi } b^{3/2} \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{32 d^2}+\frac{3 b \sin \left (2 \sin ^{-1}(c+d x)\right ) \sqrt{a+b \sin ^{-1}(c+d x)}}{16 d^2}-\frac{3 b c \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d^2}-\frac{c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d^2}-\frac{\cos \left (2 \sin ^{-1}(c+d x)\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSin[c + d*x])^(3/2),x]

[Out]

(-3*b*c*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(2*d^2) - (c*(c + d*x)*(a + b*ArcSin[c + d*x])^(3/2
))/d^2 - ((a + b*ArcSin[c + d*x])^(3/2)*Cos[2*ArcSin[c + d*x]])/(4*d^2) + (3*b^(3/2)*c*Sqrt[Pi/2]*Cos[a/b]*Fre
snelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*d^2) - (3*b^(3/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(
2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(32*d^2) + (3*b^(3/2)*c*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sq
rt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(2*d^2) + (3*b^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c +
d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(32*d^2) + (3*b*Sqrt[a + b*ArcSin[c + d*x]]*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 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]

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 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]

Rubi steps

\begin{align*} \int x \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{c}{d}+\frac{x}{d}\right ) \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^{3/2} \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^4 \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^4 \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^4 \cos \left (\frac{a}{b}-\frac{x^2}{b}\right )+\frac{1}{2} x^4 \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^4 \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^4 \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 )^{3/2}}{d^2}-\frac{\left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \cos \left (2 \sin ^{-1}(c+d x)\right )}{4 d^2}+\frac{3 \operatorname{Subst}\left (\int x^2 \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{(3 c) \operatorname{Subst}\left (\int x^2 \sin \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{d^2}\\ &=-\frac{3 b c \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d^2}-\frac{c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d^2}-\frac{\left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \cos \left (2 \sin ^{-1}(c+d x)\right )}{4 d^2}+\frac{3 b \sqrt{a+b \sin ^{-1}(c+d x)} \sin \left (2 \sin ^{-1}(c+d x)\right )}{16 d^2}+\frac{(3 b) \operatorname{Subst}\left (\int \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{(3 b c) \operatorname{Subst}\left (\int \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{3 b c \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d^2}-\frac{c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d^2}-\frac{\left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \cos \left (2 \sin ^{-1}(c+d x)\right )}{4 d^2}+\frac{3 b \sqrt{a+b \sin ^{-1}(c+d x)} \sin \left (2 \sin ^{-1}(c+d x)\right )}{16 d^2}+\frac{\left (3 b 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 )}{2 d^2}-\frac{\left (3 b \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 )}{16 d^2}+\frac{\left (3 b 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 )}{2 d^2}+\frac{\left (3 b \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 )}{16 d^2}\\ &=-\frac{3 b c \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d^2}-\frac{c (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d^2}-\frac{\left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \cos \left (2 \sin ^{-1}(c+d x)\right )}{4 d^2}+\frac{3 b^{3/2} c \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 d^2}-\frac{3 b^{3/2} \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 )}{32 d^2}+\frac{3 b^{3/2} c \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{2 d^2}+\frac{3 b^{3/2} \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 )}{32 d^2}+\frac{3 b \sqrt{a+b \sin ^{-1}(c+d x)} \sin \left (2 \sin ^{-1}(c+d x)\right )}{16 d^2}\\ \end{align*}

Mathematica [C]  time = 7.39647, size = 635, normalized size = 1.85 \[ -\frac{a b c e^{-\frac{i a}{b}} \left (\sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{2 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{b c \left (-\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left (2 a \sin \left (\frac{a}{b}\right )+3 b \cos \left (\frac{a}{b}\right )\right ) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}\right )+\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left (2 a \cos \left (\frac{a}{b}\right )-3 b \sin \left (\frac{a}{b}\right )\right ) S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right )+2 \left (3 \sqrt{1-(c+d x)^2}+2 (c+d x) \sin ^{-1}(c+d x)\right ) \sqrt{a+b \sin ^{-1}(c+d x)}\right )}{4 d^2}+\frac{a \left (\sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right )+\sqrt{\pi } \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right )-2 \sqrt{\frac{1}{b}} \cos \left (2 \sin ^{-1}(c+d x)\right ) \sqrt{a+b \sin ^{-1}(c+d x)}\right )}{8 \sqrt{\frac{1}{b}} d^2}+\frac{b \left (\sqrt{\pi } \sqrt{\frac{1}{b}} \left (3 b \sin \left (\frac{2 a}{b}\right )-4 a \cos \left (\frac{2 a}{b}\right )\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right )+\sqrt{\pi } \left (-\sqrt{\frac{1}{b}}\right ) \left (4 a \sin \left (\frac{2 a}{b}\right )+3 b \cos \left (\frac{2 a}{b}\right )\right ) S\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right )+2 \left (3 \sin \left (2 \sin ^{-1}(c+d x)\right )-4 \sin ^{-1}(c+d x) \cos \left (2 \sin ^{-1}(c+d x)\right )\right ) \sqrt{a+b \sin ^{-1}(c+d x)}\right )}{32 d^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*(a + b*ArcSin[c + d*x])^(3/2),x]

[Out]

-(a*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]]) - (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*Cos[a/
b] - 3*b*Sin[a/b])))/(4*d^2) + (a*(-2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]]*Cos[2*ArcSin[c + d*x]] + Sqrt[P
i]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]] + Sqrt[Pi]*FresnelS[(2*Sqrt[b^
(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b]))/(8*Sqrt[b^(-1)]*d^2) + (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]*FresnelC[(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
]])))/(32*d^2)

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Maple [B]  time = 0.133, size = 577, normalized size = 1.7 \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))^(3/2),x)

[Out]

-1/32/d^2*(-24*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^2*c-24*2^(1/2)*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(a/b)*F
resnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2*c+3*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x
+c))^(1/2)*cos(2*a/b)*FresnelS(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2-3*(1/b)^(1/2)*Pi^(1/2)*
(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/b)*FresnelC(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2+32*arcsi
n(d*x+c)^2*sin((a+b*arcsin(d*x+c))/b-a/b)*b^2*c+8*arcsin(d*x+c)^2*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*b^2+64*ar
csin(d*x+c)*sin((a+b*arcsin(d*x+c))/b-a/b)*a*b*c+48*arcsin(d*x+c)*cos((a+b*arcsin(d*x+c))/b-a/b)*b^2*c+16*arcs
in(d*x+c)*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*a*b-6*arcsin(d*x+c)*sin(2*(a+b*arcsin(d*x+c))/b-2*a/b)*b^2+32*sin
((a+b*arcsin(d*x+c))/b-a/b)*a^2*c+48*cos((a+b*arcsin(d*x+c))/b-a/b)*a*b*c+8*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)
*a^2-6*sin(2*(a+b*arcsin(d*x+c))/b-2*a/b)*a*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{3}{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))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*arcsin(d*x + c) + a)^(3/2)*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))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \operatorname{asin}{\left (c + d x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*asin(d*x+c))**(3/2),x)

[Out]

Integral(x*(a + b*asin(c + d*x))**(3/2), x)

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Giac [B]  time = 2.31657, size = 1856, normalized size = 5.41 \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))^(3/2),x, algorithm="giac")

[Out]

1/4*sqrt(2)*sqrt(pi)*a*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) + 1/4*sqrt(2)*s
qrt(pi)*a*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) - 3/8*sqrt(2)*sqrt(pi)*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(ab
s(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d^2) - 1/4*sqrt(2)*sqrt(pi)*a*b^2*c*i*erf(-1/2*sqr
t(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) + 3/8*sqrt(2)*sqrt(pi)*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) - 1/4*sqrt(2)*sqrt(pi)*a*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*s
qrt(abs(b)))*d^2) + 3/64*sqrt(pi)*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) + 3/64*sqrt(pi)*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) + 1/
2*sqrt(b*arcsin(d*x + c) + a)*b*c*i*arcsin(d*x + c)*e^(i*arcsin(d*x + c))/d^2 - 1/2*sqrt(b*arcsin(d*x + c) + a
)*b*c*i*arcsin(d*x + c)*e^(-i*arcsin(d*x + c))/d^2 + 1/16*sqrt(pi)*a*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) - 1/16*sqrt(pi)
*a*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) + 1/2*sqrt(b*arcsin(d*x + c) + a)*a*c*i*e^(i*arcsin(d*x + c))/d^2 - 1/2*sqrt(b*ar
csin(d*x + c) + a)*a*c*i*e^(-i*arcsin(d*x + c))/d^2 - 1/16*sqrt(pi)*a*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) + 1/16*sqrt(pi)*
a*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) - 3/32*sqrt(b*arcsin(d*x + c) + a)*b*i*e^(2*i*arcsin(d*x + c))/d^2 - 1/8*sqrt(b*arcs
in(d*x + c) + a)*b*arcsin(d*x + c)*e^(2*i*arcsin(d*x + c))/d^2 - 3/4*sqrt(b*arcsin(d*x + c) + a)*b*c*e^(i*arcs
in(d*x + c))/d^2 - 3/4*sqrt(b*arcsin(d*x + c) + a)*b*c*e^(-i*arcsin(d*x + c))/d^2 + 3/32*sqrt(b*arcsin(d*x + c
) + a)*b*i*e^(-2*i*arcsin(d*x + c))/d^2 - 1/8*sqrt(b*arcsin(d*x + c) + 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*e^(2*i*arcsin(d*x + c))/d^2 - 1/8*sqrt(b*arcsin(d*x + c) + a)*a
*e^(-2*i*arcsin(d*x + c))/d^2