3.155 \(\int x^2 \sqrt {a+b \sin ^{-1}(c+d x)} \, dx\)

Optimal. Leaf size=535 \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c^2 \sin \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{d^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{d^3}+\frac {c^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{12 d^3}-\frac {\sqrt {\pi } \sqrt {b} c \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{4 d^3}-\frac {\sqrt {\pi } \sqrt {b} c \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{4 d^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{12 d^3}+\frac {(c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d^3}+\frac {c \cos \left (2 \sin ^{-1}(c+d x)\right ) \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d^3} \]

[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 = 2.23, 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 = {4805, 4747, 6741, 6742, 3386, 3353, 3352, 3351, 3385, 3354, 3443, 3357} \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c^2 \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{d^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{d^3}+\frac {c^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{12 d^3}-\frac {\sqrt {\pi } \sqrt {b} c \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{4 d^3}-\frac {\sqrt {\pi } \sqrt {b} c \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{4 d^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{12 d^3}+\frac {(c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{3 d^3}+\frac {c \cos \left (2 \sin ^{-1}(c+d x)\right ) \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d^3} \]

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 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 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 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 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 3357

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 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 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 3443

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 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 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 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]]

Rubi steps

\begin {align*} \int x^2 \sqrt {a+b \sin ^{-1}(c+d x)} \, dx &=\frac {\operatorname {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 {\operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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) \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 )}{b d^3}+\frac {\left (2 c^2\right ) \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 )}{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 {\operatorname {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 \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 )}{2 d^3}+\frac {c^2 \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 )}{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 {\operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 {\operatorname {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 {\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^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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 1.93, size = 473, normalized size = 0.88 \[ \frac {36 \sqrt {2 \pi } c^2 \sin \left (\frac {a}{b}\right ) C\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )-9 \sqrt {2 \pi } \left (4 c^2+1\right ) \cos \left (\frac {a}{b}\right ) S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )+72 \sqrt {\frac {1}{b}} c^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}+9 \sqrt {2 \pi } \sin \left (\frac {a}{b}\right ) C\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )-\sqrt {6 \pi } \sin \left (\frac {3 a}{b}\right ) C\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )-18 \sqrt {\pi } c \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi }}\right )-18 \sqrt {\pi } c \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi }}\right )+\sqrt {6 \pi } \cos \left (\frac {3 a}{b}\right ) S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )-6 \sqrt {\frac {1}{b}} \sin \left (3 \sin ^{-1}(c+d x)\right ) \sqrt {a+b \sin ^{-1}(c+d x)}+18 \sqrt {\frac {1}{b}} (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}+36 \sqrt {\frac {1}{b}} c \cos \left (2 \sin ^{-1}(c+d x)\right ) \sqrt {a+b \sin ^{-1}(c+d x)}}{72 \sqrt {\frac {1}{b}} d^3} \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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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giac [B]  time = 6.21, size = 2330, normalized size = 4.36 \[ \text {result too large to display} \]

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]

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

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maple [A]  time = 0.60, size = 748, normalized size = 1.40 \[ \frac {36 \sqrt {\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sin \left (\frac {a}{b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, b \,c^{2}-36 \sqrt {\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \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 )}\, b \,c^{2}+\sqrt {\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \cos \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {3}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, b -\sqrt {\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sin \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {3}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, b +9 \sqrt {2}\, \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sin \left (\frac {a}{b}\right ) b -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 -18 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sin \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {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 )}\, b c -18 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \cos \left (\frac {2 a}{b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, b c +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}+36 \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) \arcsin \left (d x +c \right ) b c +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 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a c -6 \sin \left (\frac {3 a +3 b \arcsin \left (d x +c \right )}{b}-\frac {3 a}{b}\right ) \arcsin \left (d x +c \right ) b +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 a +3 b \arcsin \left (d x +c \right )}{b}-\frac {3 a}{b}\right ) a +18 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a}{72 d^{3} \sqrt {a +b \arcsin \left (d x +c \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \arcsin \left (d x + c\right ) + a} x^{2}\,{d x} \]

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

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx \]

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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