3.2.58 \(\int x (a+b \text {ArcSin}(c+d x))^{3/2} \, dx\) [158]

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

[Out]

-c*(d*x+c)*(a+b*arcsin(d*x+c))^(3/2)/d^2-1/4*(a+b*arcsin(d*x+c))^(3/2)*cos(2*arcsin(d*x+c))/d^2+3/4*b^(3/2)*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)/d^2+3/4*b^(3/2)*c*Fresn
elS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2)/d^2-3/32*b^(3/2)*cos(2*a/b)*
FresnelS(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/d^2+3/32*b^(3/2)*FresnelC(2*(a+b*arcsin(d*x+c)
)^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/d^2+3/16*b*sin(2*arcsin(d*x+c))*(a+b*arcsin(d*x+c))^(1/2)/d^2-3/
2*b*c*(1-(d*x+c)^2)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/d^2

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Rubi [A]
time = 0.77, antiderivative size = 343, normalized size of antiderivative = 1.00, 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 = {4889, 4831, 6873, 6874, 3467, 3466, 3435, 3433, 3432, 3434} \begin {gather*} \frac {3 \sqrt {\pi } b^{3/2} \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 )}{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 \text {ArcSin}(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 \text {ArcSin}(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 \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 d^2}-\frac {3 b c \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{2 d^2}-\frac {c (c+d x) (a+b \text {ArcSin}(c+d x))^{3/2}}{d^2}+\frac {3 b \sin (2 \text {ArcSin}(c+d x)) \sqrt {a+b \text {ArcSin}(c+d x)}}{16 d^2}-\frac {\cos (2 \text {ArcSin}(c+d x)) (a+b \text {ArcSin}(c+d x))^{3/2}}{4 d^2} \end {gather*}

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 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 3434

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 3435

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 3466

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 3467

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 4831

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 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]

Rule 6873

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

Rule 6874

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

Rubi steps

\begin {align*} \int x \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac {\text {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 {\text {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 \text {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 \text {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 \text {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 {\text {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) \text {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 \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 6.92, size = 635, normalized size = 1.85 \begin {gather*} -\frac {a b c e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {3}{2},-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {3}{2},\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )}{2 d^2 \sqrt {a+b \text {ArcSin}(c+d x)}}-\frac {b c \left (2 \sqrt {a+b \text {ArcSin}(c+d x)} \left (3 \sqrt {1-(c+d x)^2}+2 (c+d x) \text {ArcSin}(c+d x)\right )-\sqrt {\frac {1}{b}} \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )+\sqrt {\frac {1}{b}} \sqrt {2 \pi } S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )\right )}{4 d^2}+\frac {a \left (-2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}(c+d x)} \cos (2 \text {ArcSin}(c+d x))+\sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi }}\right )+\sqrt {\pi } S\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 )\right )}{8 \sqrt {\frac {1}{b}} d^2}+\frac {b \left (-\sqrt {\frac {1}{b}} \sqrt {\pi } S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi }}\right ) \left (3 b \cos \left (\frac {2 a}{b}\right )+4 a \sin \left (\frac {2 a}{b}\right )\right )+\sqrt {\frac {1}{b}} \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi }}\right ) \left (-4 a \cos \left (\frac {2 a}{b}\right )+3 b \sin \left (\frac {2 a}{b}\right )\right )+2 \sqrt {a+b \text {ArcSin}(c+d x)} (-4 \text {ArcSin}(c+d x) \cos (2 \text {ArcSin}(c+d x))+3 \sin (2 \text {ArcSin}(c+d x)))\right )}{32 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(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]))/(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
[Pi]*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[P
i]*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])) + Sq
rt[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*S
in[(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] Leaf count of result is larger than twice the leaf count of optimal. \(604\) vs. \(2(275)=550\).
time = 0.49, size = 605, normalized size = 1.76

method	result	size
default	\(\frac {48 \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 ) \sqrt {2}\, \sqrt {-\frac {1}{b}}\, b^{2} c -48 \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 ) \sqrt {2}\, \sqrt {-\frac {1}{b}}\, b^{2} c +3 \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \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 ) \sqrt {2}\, b^{2}+3 \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \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 ) \sqrt {2}\, b^{2}+64 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2} c -16 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}+128 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b c -96 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2} c -32 \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 ) a b -12 \arcsin \left (d x +c \right ) \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}+64 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2} c -96 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b c -16 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2}-12 \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a b}{64 d^{2} \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(605\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arcsin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/64/d^2*(48*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)*2^(1/2)*(-1/b)^(1/2)*b^2*c-48*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)*2^(1/2)*(-1/b)^(1/2)*b^2*c+3*(-2/b)^(1/2)*Pi^(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)*2^(1/2)*b^2+3*(
-2/b)^(1/2)*Pi^(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)*2^(1/2)*b^2+64*arcsin(d*x+c)^2*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b^2*c-16*arcsin(d*x+c)^2*cos(
-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^2+128*arcsin(d*x+c)*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a*b*c-96*arcsin(d*x+c)*c
os(-(a+b*arcsin(d*x+c))/b+a/b)*b^2*c-32*arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b-12*arcsin(d*x+c)
*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^2+64*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a^2*c-96*cos(-(a+b*arcsin(d*x+c))/
b+a/b)*a*b*c-16*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^2-12*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b)/(a+b*arcsi
n(d*x+c))^(1/2)

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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))^(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 = 0.00, 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))^(3/2),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 x \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 1.03, size = 1987, normalized size = 5.79 \begin {gather*} \text {Too large to display} \end {gather*}

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]

-1/2*sqrt(2)*sqrt(pi)*a^2*b^2*c*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt
(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d^2) - 1/2*I*sqrt(2
)*sqrt(pi)*a*b^3*c*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x
+ c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d^2) - 1/2*sqrt(2)*sqrt(pi)*a^2*
b^2*c*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqr
t(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d^2) + 1/2*I*sqrt(2)*sqrt(pi)*a*b^3*c*erf(1/
2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)
*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d^2) + 1/2*I*sqrt(2)*sqrt(pi)*a*b^2*c*erf(-1/2*I*sqrt(2)
*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/
((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*d^2) - 3/8*sqrt(2)*sqrt(pi)*b^3*c*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x
+ c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b)
) + b*sqrt(abs(b)))*d^2) - 1/2*I*sqrt(2)*sqrt(pi)*a*b^2*c*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(a
bs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs
(b)))*d^2) - 3/8*sqrt(2)*sqrt(pi)*b^3*c*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(
2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*d^2) + sqrt(
pi)*a^2*b*c*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) +
a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*d^2) + sqrt(pi)*a^2*b*c*e
rf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b
))/b)*e^(-I*a/b)/((-I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*d^2) + 1/4*I*sqrt(pi)*a^2*b^(3/2)*erf
(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^2 + I*b^
3/abs(b))*d^2) - 1/8*sqrt(pi)*a*b^(5/2)*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) +
a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^2 + I*b^3/abs(b))*d^2) - 1/4*I*sqrt(pi)*a^2*b^(3/2)*erf(-sqrt(b*arcsin(d*x
+ c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^2 - I*b^3/abs(b))*d^2) - 1/
8*sqrt(pi)*a*b^(5/2)*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*
e^(-2*I*a/b)/((b^2 - I*b^3/abs(b))*d^2) + 1/2*I*sqrt(b*arcsin(d*x + c) + a)*b*c*arcsin(d*x + c)*e^(I*arcsin(d*
x + c))/d^2 - 1/2*I*sqrt(b*arcsin(d*x + c) + a)*b*c*arcsin(d*x + c)*e^(-I*arcsin(d*x + c))/d^2 + 1/8*sqrt(pi)*
a*b^2*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b
^(3/2) + I*b^(5/2)/abs(b))*d^2) + 1/4*I*sqrt(pi)*a^2*b*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arc
sin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^(3/2) - I*b^(5/2)/abs(b))*d^2) + 1/8*sqrt(pi)*a*b^2*erf(-sq
rt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^(3/2) - I*b
^(5/2)/abs(b))*d^2) - 1/4*I*sqrt(pi)*a^2*sqrt(b)*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*
x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*d^2) + 3/64*I*sqrt(pi)*b^(5/2)*erf(-sqrt(b*arcsin(
d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*d^2) - 3
/64*I*sqrt(pi)*b^(5/2)*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b)
)*e^(-2*I*a/b)/((b - I*b^2/abs(b))*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 + 1/2*I*sqrt(b*arcsin(d*x + c) + a)*a*c*e^(I*arcsin(d*x + c))/d^2 - 3/4*sqrt(b*arcsin(d*x + c) + a)*
b*c*e^(I*arcsin(d*x + c))/d^2 - 1/2*I*sqrt(b*arcsin(d*x + c) + a)*a*c*e^(-I*arcsin(d*x + c))/d^2 - 3/4*sqrt(b*
arcsin(d*x + c) + a)*b*c*e^(-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 - 3/32*I*sqrt(b*arcsin(
d*x + c) + a)*b*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 +
3/32*I*sqrt(b*arcsin(d*x + c) + a)*b*e^(-2*I*arcsin(d*x + c))/d^2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*asin(c + d*x))^(3/2),x)

[Out]

int(x*(a + b*asin(c + d*x))^(3/2), x)

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