3.2.55 \(\int x^2 \sqrt {a+b \text {ArcSin}(c+d x)} \, dx\) [155]

Optimal. Leaf size=535 \[ \frac {c^2 (c+d x) \sqrt {a+b \text {ArcSin}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \text {ArcSin}(c+d x)}}{3 d^3}+\frac {c \sqrt {a+b \text {ArcSin}(c+d x)} \cos (2 \text {ArcSin}(c+d x))}{2 d^3}-\frac {\sqrt {b} c \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{4 d^3}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}-\frac {\sqrt {b} c^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{d^3}+\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{12 d^3}+\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 d^3}+\frac {\sqrt {b} c^2 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{d^3}-\frac {\sqrt {b} c \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{4 d^3}-\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{12 d^3} \]

[Out]

1/72*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*6^(1/2)*Pi^(1/2)/d^3-1/72
*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(3*a/b)*b^(1/2)*6^(1/2)*Pi^(1/2)/d^3-1/8*cos(
a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/d^3-1/2*c^2*cos(a/b
)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/d^3+1/8*FresnelC(2^(1/
2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)/d^3+1/2*c^2*FresnelC(2^(1/2)/
Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)/d^3-1/4*c*cos(2*a/b)*FresnelC(2*
(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*b^(1/2)*Pi^(1/2)/d^3-1/4*c*FresnelS(2*(a+b*arcsin(d*x+c))^(1/2)/b^
(1/2)/Pi^(1/2))*sin(2*a/b)*b^(1/2)*Pi^(1/2)/d^3+c^2*(d*x+c)*(a+b*arcsin(d*x+c))^(1/2)/d^3+1/3*(d*x+c)^3*(a+b*a
rcsin(d*x+c))^(1/2)/d^3+1/2*c*cos(2*arcsin(d*x+c))*(a+b*arcsin(d*x+c))^(1/2)/d^3

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Rubi [A]
time = 1.72, antiderivative size = 535, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4889, 4831, 6873, 6874, 3467, 3434, 3433, 3432, 3466, 3435, 3524, 3438} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c^2 \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{d^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{d^3}+\frac {c^2 (c+d x) \sqrt {a+b \text {ArcSin}(c+d x)}}{d^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{12 d^3}-\frac {\sqrt {\pi } \sqrt {b} c \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{4 d^3}-\frac {\sqrt {\pi } \sqrt {b} c \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{4 d^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{12 d^3}+\frac {(c+d x)^3 \sqrt {a+b \text {ArcSin}(c+d x)}}{3 d^3}+\frac {c \cos (2 \text {ArcSin}(c+d x)) \sqrt {a+b \text {ArcSin}(c+d x)}}{2 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[a + b*ArcSin[c + d*x]],x]

[Out]

(c^2*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/d^3 + ((c + d*x)^3*Sqrt[a + b*ArcSin[c + d*x]])/(3*d^3) + (c*Sqrt[
a + b*ArcSin[c + d*x]]*Cos[2*ArcSin[c + d*x]])/(2*d^3) - (Sqrt[b]*c*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a +
b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(4*d^3) - (Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a +
b*ArcSin[c + d*x]])/Sqrt[b]])/(4*d^3) - (Sqrt[b]*c^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcS
in[c + d*x]])/Sqrt[b]])/d^3 + (Sqrt[b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]
])/Sqrt[b]])/(12*d^3) + (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b
])/(4*d^3) + (Sqrt[b]*c^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/d^3
- (Sqrt[b]*c*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(4*d^3) - (Sq
rt[b]*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(12*d^3)

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 3438

Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(a +
b*Sin[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 1] && IGtQ[n, 1]

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 3524

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[x^(m - n +
1)*(Sin[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sin[a + b*x^n]^
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -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^2 \sqrt {a+b \sin ^{-1}(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \left (-\frac {c}{d}+\frac {x}{d}\right )^2 \sqrt {a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \sqrt {a+b x} \cos (x) \left (-\frac {c}{d}+\frac {\sin (x)}{d}\right )^2 \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac {2 \text {Subst}\left (\int x^2 \cos \left (\frac {a-x^2}{b}\right ) \left (c+\sin \left (\frac {a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac {2 \text {Subst}\left (\int x^2 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right ) \left (c+\sin \left (\frac {a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac {2 \text {Subst}\left (\int \left (c^2 x^2 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right )+c x^2 \sin \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )+x^2 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac {2 \text {Subst}\left (\int x^2 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{b d^3}+\frac {(2 c) \text {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 )}{b d^3}+\frac {\left (2 c^2\right ) \text {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 )}{b d^3}\\ &=\frac {c^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d^3}+\frac {c \sqrt {a+b \sin ^{-1}(c+d x)} \cos \left (2 \sin ^{-1}(c+d x)\right )}{2 d^3}+\frac {\text {Subst}\left (\int \sin ^3\left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{3 d^3}-\frac {c \text {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 )}{2 d^3}+\frac {c^2 \text {Subst}\left (\int \sin \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{d^3}\\ &=\frac {c^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d^3}+\frac {c \sqrt {a+b \sin ^{-1}(c+d x)} \cos \left (2 \sin ^{-1}(c+d x)\right )}{2 d^3}+\frac {\text {Subst}\left (\int \left (-\frac {1}{4} \sin \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right )+\frac {3}{4} \sin \left (\frac {a}{b}-\frac {x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{3 d^3}-\frac {\left (c^2 \cos \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 )}{d^3}-\frac {\left (c \cos \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 )}{2 d^3}+\frac {\left (c^2 \sin \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 )}{d^3}-\frac {\left (c \sin \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 )}{2 d^3}\\ &=\frac {c^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d^3}+\frac {c \sqrt {a+b \sin ^{-1}(c+d x)} \cos \left (2 \sin ^{-1}(c+d x)\right )}{2 d^3}-\frac {\sqrt {b} c \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 )}{4 d^3}-\frac {\sqrt {b} c^2 \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 )}{d^3}+\frac {\sqrt {b} c^2 \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 )}{d^3}-\frac {\sqrt {b} c \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 )}{4 d^3}-\frac {\text {Subst}\left (\int \sin \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{12 d^3}+\frac {\text {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^3}\\ &=\frac {c^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d^3}+\frac {c \sqrt {a+b \sin ^{-1}(c+d x)} \cos \left (2 \sin ^{-1}(c+d x)\right )}{2 d^3}-\frac {\sqrt {b} c \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 )}{4 d^3}-\frac {\sqrt {b} c^2 \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 )}{d^3}+\frac {\sqrt {b} c^2 \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 )}{d^3}-\frac {\sqrt {b} c \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 )}{4 d^3}-\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{4 d^3}+\frac {\cos \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{12 d^3}+\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{4 d^3}-\frac {\sin \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{12 d^3}\\ &=\frac {c^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d^3}+\frac {c \sqrt {a+b \sin ^{-1}(c+d x)} \cos \left (2 \sin ^{-1}(c+d x)\right )}{2 d^3}-\frac {\sqrt {b} c \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 )}{4 d^3}-\frac {\sqrt {b} \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^3}-\frac {\sqrt {b} c^2 \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 )}{d^3}+\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{12 d^3}+\frac {\sqrt {b} \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^3}+\frac {\sqrt {b} c^2 \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 )}{d^3}-\frac {\sqrt {b} c \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 )}{4 d^3}-\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{12 d^3}\\ \end {align*}

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Mathematica [A]
time = 1.09, size = 473, normalized size = 0.88 \begin {gather*} \frac {18 \sqrt {\frac {1}{b}} (c+d x) \sqrt {a+b \text {ArcSin}(c+d x)}+72 \sqrt {\frac {1}{b}} c^2 (c+d x) \sqrt {a+b \text {ArcSin}(c+d x)}+36 \sqrt {\frac {1}{b}} c \sqrt {a+b \text {ArcSin}(c+d x)} \cos (2 \text {ArcSin}(c+d x))-18 c \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 )-9 \left (1+4 c^2\right ) \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}\right )+\sqrt {6 \pi } \cos \left (\frac {3 a}{b}\right ) S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}\right )+9 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}\right ) \sin \left (\frac {a}{b}\right )+36 c^2 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}\right ) \sin \left (\frac {a}{b}\right )-18 c \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 )-\sqrt {6 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}\right ) \sin \left (\frac {3 a}{b}\right )-6 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}(c+d x)} \sin (3 \text {ArcSin}(c+d x))}{72 \sqrt {\frac {1}{b}} d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[a + b*ArcSin[c + d*x]],x]

[Out]

(18*Sqrt[b^(-1)]*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]] + 72*Sqrt[b^(-1)]*c^2*(c + d*x)*Sqrt[a + b*ArcSin[c + d
*x]] + 36*Sqrt[b^(-1)]*c*Sqrt[a + b*ArcSin[c + d*x]]*Cos[2*ArcSin[c + d*x]] - 18*c*Sqrt[Pi]*Cos[(2*a)/b]*Fresn
elC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]] - 9*(1 + 4*c^2)*Sqrt[2*Pi]*Cos[a/b]*FresnelS[Sqrt[b
^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]] + Sqrt[6*Pi]*Cos[(3*a)/b]*FresnelS[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt
[a + b*ArcSin[c + d*x]]] + 9*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*Sin[a/b]
+ 36*c^2*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*Sin[a/b] - 18*c*Sqrt[Pi]*Fr
esnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b] - Sqrt[6*Pi]*FresnelC[Sqrt[b^(-1)]*
Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*Sin[(3*a)/b] - 6*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]]*Sin[3*ArcSin
[c + d*x]])/(72*Sqrt[b^(-1)]*d^3)

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Maple [A]
time = 1.19, size = 783, normalized size = 1.46

method	result	size
default	\(\frac {36 \cos \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 {a +b \arcsin \left (d x +c \right )}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b \,c^{2}+36 \sin \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 {a +b \arcsin \left (d x +c \right )}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b \,c^{2}-9 \cos \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 {a +b \arcsin \left (d x +c \right )}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b c +9 \sin \left (\frac {2 a}{b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, \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 c +9 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \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 +9 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \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 -\cos \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b -\sin \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b -72 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) \arcsin \left (d x +c \right ) b \,c^{2}-72 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,c^{2}+36 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) \arcsin \left (d x +c \right ) b c -18 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) \arcsin \left (d x +c \right ) b +6 \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) \arcsin \left (d x +c \right ) b +36 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a c -18 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a +6 \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) a}{72 d^{3} \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(783\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arcsin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/72/d^3/(a+b*arcsin(d*x+c))^(1/2)*(36*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/
2)/b)*(a+b*arcsin(d*x+c))^(1/2)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*b*c^2+36*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/
b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(a+b*arcsin(d*x+c))^(1/2)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*b*c^2-9*cos(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+b*arcsin(d*x+c))^(1/2)*2^(1/2)*Pi^
(1/2)*(-2/b)^(1/2)*b*c+9*sin(2*a/b)*(a+b*arcsin(d*x+c))^(1/2)*2^(1/2)*Pi^(1/2)*(-2/b)^(1/2)*FresnelS(2*2^(1/2)
/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b*c+9*2^(1/2)*Pi^(1/2)*(-1/b)^(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+9*2^(1/2)*Pi^(1/2)*(-1/b)^(
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-
cos(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(a+b*arcsin(d*x+c))^(1/2)*2^(
1/2)*Pi^(1/2)*(-3/b)^(1/2)*b-sin(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*
(a+b*arcsin(d*x+c))^(1/2)*2^(1/2)*Pi^(1/2)*(-3/b)^(1/2)*b-72*sin(-(a+b*arcsin(d*x+c))/b+a/b)*arcsin(d*x+c)*b*c
^2-72*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a*c^2+36*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*arcsin(d*x+c)*b*c-18*sin(-(
a+b*arcsin(d*x+c))/b+a/b)*arcsin(d*x+c)*b+6*sin(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*arcsin(d*x+c)*b+36*cos(-2*(a+b
*arcsin(d*x+c))/b+2*a/b)*a*c-18*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a+6*sin(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a)

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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^2*(a+b*arcsin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*arcsin(d*x + c) + a)*x^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^2*(a+b*arcsin(d*x+c))^(1/2),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^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*asin(d*x+c))**(1/2),x)

[Out]

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

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Giac [C] Result contains complex when optimal does not.
time = 1.30, size = 2255, normalized size = 4.21 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arcsin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(2)*sqrt(pi)*a*b^2*c^2*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^3) + 1/4*I*sqrt(2)
*sqrt(pi)*b^3*c^2*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^3) + 1/2*sqrt(2)*sqrt(pi)*a*b^2
*c^2*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^3) - 1/4*I*sqrt(2)*sqrt(pi)*b^3*c^2*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^3) - sqrt(pi)*a*b*c^2*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^3) - sqrt(pi)*a*b*c^2*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^3) - 1/2*I*sqrt(pi)*a*b^(3/2)*c*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqr
t(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^2 + I*b^3/abs(b))*d^3) + 1/8*sqrt(pi)*b^(5/2)*c*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^3) + 1/8*sqrt(2)*sqrt(pi)*a*b^2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sq
rt(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^3) + 1/
16*I*sqrt(2)*sqrt(pi)*b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arc
sin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d^3) + 1/8*sqrt(2)*sqrt(p
i)*a*b^2*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^3) - 1/16*I*sqrt(2)*sqrt(pi)*b^3*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^3) + 1/2*I*sqrt(pi)*a*b^(3/2)*c*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^3) + 1
/8*sqrt(pi)*b^(5/2)*c*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^3) - 1/2*I*sqrt(pi)*a*b*c*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*s
qrt(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^3) - 1/4*sqrt(pi)*a*b^
(3/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)
/abs(b))*e^(3*I*a/b)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*d^3) - 1/24*I*sqrt(pi)*b^(5/2)*erf(-1/2*sqrt(6)*sqr
t(b*arcsin(d*x + c) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sqr
t(6)*b^2 + I*sqrt(6)*b^3/abs(b))*d^3) + 1/2*I*sqrt(pi)*a*sqrt(b)*c*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^3) - 1/4*sqrt(pi)*a*b^(3/2)*er
f(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))
*e^(-3*I*a/b)/((sqrt(6)*b^2 - I*sqrt(6)*b^3/abs(b))*d^3) + 1/24*I*sqrt(pi)*b^(5/2)*erf(-1/2*sqrt(6)*sqrt(b*arc
sin(d*x + c) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/((sqrt(6)*b
^2 - I*sqrt(6)*b^3/abs(b))*d^3) - 1/2*I*sqrt(b*arcsin(d*x + c) + a)*c^2*e^(I*arcsin(d*x + c))/d^3 + 1/2*I*sqrt
(b*arcsin(d*x + c) + a)*c^2*e^(-I*arcsin(d*x + c))/d^3 + 1/4*sqrt(pi)*a*b*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x +
c) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*b^(3/2) + I
*sqrt(6)*b^(5/2)/abs(b))*d^3) - 1/4*sqrt(pi)*a*b*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*sq
rt(abs(b)))*d^3) - 1/4*sqrt(pi)*a*b*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*s
qrt(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^3) + 1/4*sqrt(pi)*a*b*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x
+ c) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/((sqrt(6)*b^(3/2) - I*sqrt(6)*b^(5/2)/abs(b))*d^3) + 1/4*sqrt(b*arcsin
(d*x + c) + a)*c*e^(2*I*arcsin(d*x + c))/d^3 + 1/4*sqrt(b*arcsin(d*x + c) + a)*c*e^(-2*I*arcsin(d*x + c))/d^3
+ 1/24*I*sqrt(b*arcsin(d*x + c) + a)*e^(3*I*arcsin(d*x + c))/d^3 - 1/8*I*sqrt(b*arcsin(d*x + c) + a)*e^(I*arcs
in(d*x + c))/d^3 + 1/8*I*sqrt(b*arcsin(d*x + c)...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*asin(c + d*x))^(1/2),x)

[Out]

int(x^2*(a + b*asin(c + d*x))^(1/2), x)

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