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

Optimal. Leaf size=440 \[ \frac {\sqrt {2 \pi } c^2 \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d^3}+\frac {\sqrt {2 \pi } c^2 \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d^3}+\frac {\sqrt {\pi } c \sin \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{\sqrt {b} d^3}+\frac {\sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {\sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {\sqrt {\frac {\pi }{2}} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {\sqrt {\frac {\pi }{6}} \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {\sqrt {\pi } c \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{\sqrt {b} d^3} \]

[Out]

-1/12*cos(3*a/b)*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/d^3/b^(1/2)-1/1
2*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(3*a/b)*6^(1/2)*Pi^(1/2)/d^3/b^(1/2)+1/4*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^3/b^(1/2)+1/4*FresnelS(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^3/b^(1/2)-c*cos(2*a/b)*FresnelS
(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/d^3/b^(1/2)+c*FresnelC(2*(a+b*arcsin(d*x+c))^(1/2)/b^(
1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/d^3/b^(1/2)+c^2*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^3/b^(1/2)+c^2*FresnelS(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^3/b^(1/2)

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Rubi [A]  time = 1.02, antiderivative size = 440, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4805, 4747, 6741, 6742, 3354, 3352, 3351, 3353, 4574} \[ \frac {\sqrt {2 \pi } c^2 \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d^3}+\frac {\sqrt {2 \pi } c^2 \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d^3}+\frac {\sqrt {\pi } c \sin \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{\sqrt {b} d^3}+\frac {\sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {\sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {\sqrt {\frac {\pi }{2}} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {\sqrt {\frac {\pi }{6}} \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {\sqrt {\pi } c \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{\sqrt {b} d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*Sqrt[b]*d^3) + (c^2*Sqrt[2
*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(Sqrt[b]*d^3) - (Sqrt[Pi/6]*Cos[(3*a
)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*Sqrt[b]*d^3) - (c*Sqrt[Pi]*Cos[(2*a)/b]*Fr
esnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(Sqrt[b]*d^3) + (Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*S
qrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(2*Sqrt[b]*d^3) + (c^2*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a +
b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*d^3) + (c*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(
Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(Sqrt[b]*d^3) - (Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])
/Sqrt[b]]*Sin[(3*a)/b])/(2*Sqrt[b]*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 4574

Int[Cos[w_]^(q_.)*Sin[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sin[v]^p*Cos[w]^q, x], x] /; IGtQ[p, 0] &&
IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w],
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 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 \frac {x^2}{\sqrt {a+b \sin ^{-1}(c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\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 \frac {\cos (x) \left (-\frac {c}{d}+\frac {\sin (x)}{d}\right )^2}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \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 \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 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right )+c \sin \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )+\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 \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 \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 \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 {2 \operatorname {Subst}\left (\int \left (-\frac {1}{4} \cos \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right )+\frac {1}{4} \cos \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 {\left (2 c^2 \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 )}{b d^3}-\frac {\left (2 c \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 )}{b d^3}+\frac {\left (2 c^2 \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 )}{b d^3}+\frac {\left (2 c \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 )}{b d^3}\\ &=\frac {c^2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d^3}-\frac {c \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 )}{\sqrt {b} d^3}+\frac {c^2 \sqrt {2 \pi } S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{\sqrt {b} d^3}+\frac {c \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 )}{\sqrt {b} d^3}-\frac {\operatorname {Subst}\left (\int \cos \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{2 b d^3}+\frac {\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 b d^3}\\ &=\frac {c^2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d^3}-\frac {c \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 )}{\sqrt {b} d^3}+\frac {c^2 \sqrt {2 \pi } S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{\sqrt {b} d^3}+\frac {c \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 )}{\sqrt {b} d^3}+\frac {\cos \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 )}{2 b d^3}-\frac {\cos \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 )}{2 b d^3}+\frac {\sin \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 )}{2 b d^3}-\frac {\sin \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 )}{2 b d^3}\\ &=\frac {\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 \sqrt {b} d^3}+\frac {c^2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d^3}-\frac {\sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {c \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 )}{\sqrt {b} d^3}+\frac {\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 \sqrt {b} d^3}+\frac {c^2 \sqrt {2 \pi } S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{\sqrt {b} d^3}+\frac {c \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 )}{\sqrt {b} d^3}-\frac {\sqrt {\frac {\pi }{6}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{2 \sqrt {b} d^3}\\ \end {align*}

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Mathematica [A]  time = 1.17, size = 335, normalized size = 0.76 \[ \frac {\sqrt {\pi } \sqrt {\frac {1}{b}} \left (3 \sqrt {2} \left (4 c^2+1\right ) \cos \left (\frac {a}{b}\right ) C\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )+12 \sqrt {2} c^2 \sin \left (\frac {a}{b}\right ) S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )+12 c \sin \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi }}\right )-\sqrt {6} \cos \left (\frac {3 a}{b}\right ) C\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )+3 \sqrt {2} \sin \left (\frac {a}{b}\right ) S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )-\sqrt {6} \sin \left (\frac {3 a}{b}\right ) S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )-12 c \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi }}\right )\right )}{12 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b^(-1)]*Sqrt[Pi]*(3*Sqrt[2]*(1 + 4*c^2)*Cos[a/b]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c +
d*x]]] - Sqrt[6]*Cos[(3*a)/b]*FresnelC[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]]] - 12*c*Cos[(2*a)/b
]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]] + 3*Sqrt[2]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]
*Sqrt[a + b*ArcSin[c + d*x]]]*Sin[a/b] + 12*Sqrt[2]*c^2*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c +
d*x]]]*Sin[a/b] + 12*c*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b] - Sqrt[6]
*FresnelS[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*Sin[(3*a)/b]))/(12*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 [A]  time = 2.20, size = 665, normalized size = 1.51 \[ -\frac {\sqrt {\pi } c^{2} \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {a i}{b}\right )}}{{\left (\frac {\sqrt {2} b i}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )} d^{3}} + \frac {\sqrt {\pi } c^{2} \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {a i}{b}\right )}}{{\left (\frac {\sqrt {2} b i}{\sqrt {{\left | b \right |}}} - \sqrt {2} \sqrt {{\left | b \right |}}\right )} d^{3}} + \frac {\sqrt {\pi } c i \operatorname {erf}\left (\frac {\sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b} i}{{\left | b \right |}} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}}\right ) e^{\left (-\frac {2 \, a i}{b}\right )}}{2 \, {\left (\frac {b^{\frac {3}{2}} i}{{\left | b \right |}} - \sqrt {b}\right )} d^{3}} + \frac {\sqrt {\pi } c i \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b} i}{{\left | b \right |}} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}}\right ) e^{\left (\frac {2 \, a i}{b}\right )}}{2 \, \sqrt {b} d^{3} {\left (\frac {b i}{{\left | b \right |}} + 1\right )}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {6} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b} i}{2 \, {\left | b \right |}} - \frac {\sqrt {6} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {b}}\right ) e^{\left (\frac {3 \, a i}{b}\right )}}{4 \, {\left (\frac {\sqrt {6} b^{\frac {3}{2}} i}{{\left | b \right |}} + \sqrt {6} \sqrt {b}\right )} d^{3}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {a i}{b}\right )}}{4 \, {\left (\frac {\sqrt {2} b i}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )} d^{3}} + \frac {\sqrt {\pi } \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {a i}{b}\right )}}{4 \, {\left (\frac {\sqrt {2} b i}{\sqrt {{\left | b \right |}}} - \sqrt {2} \sqrt {{\left | b \right |}}\right )} d^{3}} - \frac {\sqrt {\pi } \operatorname {erf}\left (\frac {\sqrt {6} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b} i}{2 \, {\left | b \right |}} - \frac {\sqrt {6} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {b}}\right ) e^{\left (-\frac {3 \, a i}{b}\right )}}{4 \, {\left (\frac {\sqrt {6} b^{\frac {3}{2}} i}{{\left | b \right |}} - \sqrt {6} \sqrt {b}\right )} d^{3}} \]

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]

-sqrt(pi)*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*i/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b)))*d^3) + sqrt(pi)*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*i/sqrt(abs(b)) - sqrt(2)*sqrt(abs(b)))*d^3) + 1/2*sqrt(pi)*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^(3/2)*i/abs(b) - sqrt(b))*d^3
) + 1/2*sqrt(pi)*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)/(sqrt(b)*d^3*(b*i/abs(b) + 1)) + 1/4*sqrt(pi)*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^(3/2)*i/abs(b) + sqrt(6)*
sqrt(b))*d^3) - 1/4*sqrt(pi)*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*i/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b)))*d^3) + 1/4*s
qrt(pi)*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*s
qrt(abs(b))/b)*e^(-a*i/b)/((sqrt(2)*b*i/sqrt(abs(b)) - sqrt(2)*sqrt(abs(b)))*d^3) - 1/4*sqrt(pi)*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^(3/2)*i/abs(b) - sqrt(6)*sqrt(b))*d^3)

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maple [A]  time = 0.50, size = 339, normalized size = 0.77 \[ -\frac {\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \left (-12 \sqrt {2}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) c^{2}-12 \sqrt {2}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) c^{2}+\sqrt {3}\, \sqrt {2}\, \cos \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )+\sqrt {3}\, \sqrt {2}\, \sin \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )-3 \sqrt {2}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )-3 \sqrt {2}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )+12 \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) c -12 \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) c \right )}{12 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12/d^3*Pi^(1/2)*(1/b)^(1/2)*(-12*2^(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)*c^2-12*2^(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)*c^2+3^(1/
2)*2^(1/2)*cos(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)*2^(1/
2)*sin(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*2^(1/2)*cos(a/b)*Fr
esnelC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)-3*2^(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)+12*cos(2*a/b)*FresnelS(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/
2)/b)*c-12*sin(2*a/b)*FresnelC(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*c)

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

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(x^2/sqrt(b*arcsin(d*x + c) + a), x)

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

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 \[ \int \frac {x^{2}}{\sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}}\, dx \]

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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