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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 2.23346, antiderivative size = 535, normalized size of antiderivative = 1., 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 4805

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rule 4747

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[I
nt[(a + b*x)^n*Cos[x]*(c*d + e*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0
]

Rule 6741

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

Rule 6742

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

Rule 3386

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Sin[c + d*
x^n])/(d*n), x] - Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x
] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 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 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3385

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> -Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Cos[c + d
*x^n])/(d*n), x] + Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e},
x] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3354

Int[Cos[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Cos[c], Int[Cos[d*(e + f*x)^2], x], x] - Dist[
Sin[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]

Rule 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 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]

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*}

Mathematica [A]  time = 1.67363, size = 473, normalized size = 0.88 \[ \frac{36 \sqrt{2 \pi } c^2 \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}\right )-9 \sqrt{2 \pi } \left (4 c^2+1\right ) \cos \left (\frac{a}{b}\right ) S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right )+72 \sqrt{\frac{1}{b}} c^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}+9 \sqrt{2 \pi } \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}\right )-\sqrt{6 \pi } \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}\right )-18 \sqrt{\pi } c \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right )-18 \sqrt{\pi } c \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right )+\sqrt{6 \pi } \cos \left (\frac{3 a}{b}\right ) S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right )-6 \sqrt{\frac{1}{b}} \sin \left (3 \sin ^{-1}(c+d x)\right ) \sqrt{a+b \sin ^{-1}(c+d x)}+18 \sqrt{\frac{1}{b}} (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}+36 \sqrt{\frac{1}{b}} c \cos \left (2 \sin ^{-1}(c+d x)\right ) \sqrt{a+b \sin ^{-1}(c+d x)}}{72 \sqrt{\frac{1}{b}} d^3} \]

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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Maple [A]  time = 0.169, size = 748, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

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*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(1/b)^(1/2)*Pi^(1/2)
*2^(1/2)*(a+b*arcsin(d*x+c))^(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)*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*b*c^2+3^(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)*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*
b-3^(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)*(1/b)^(1/2)*Pi
^(1/2)*2^(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)*cos(a/b)*F
resnelS(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-18*cos(2*a/b)*Fre
snelC(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*b*c-1
8*sin(2*a/b)*FresnelS(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x
+c))^(1/2)*b*c+72*arcsin(d*x+c)*sin((a+b*arcsin(d*x+c))/b-a/b)*b*c^2+36*arcsin(d*x+c)*cos(2*(a+b*arcsin(d*x+c)
)/b-2*a/b)*b*c+72*sin((a+b*arcsin(d*x+c))/b-a/b)*a*c^2+18*arcsin(d*x+c)*sin((a+b*arcsin(d*x+c))/b-a/b)*b-6*arc
sin(d*x+c)*sin(3*(a+b*arcsin(d*x+c))/b-3*a/b)*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)/(a+b*arcsin(d*x+c))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \arcsin \left (d x + c\right ) + a} x^{2}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

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 [B]  time = 2.39308, size = 1235, normalized size = 2.31 \begin{align*} \text{result too large to display} \end{align*}

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^2*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^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d^3) + 1/4*sqrt(2)*sqr
t(pi)*b^2*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^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*d^3) + 1/16*sqrt(2)*sqrt(pi)*b^2*i*er
f(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b
))/b)*e^(a*i/b)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d^3) + 1/16*sqrt(2)*sqrt(pi)*b^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^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*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 - 1/24*sqrt(pi)*b^(3/2)*i*erf(-1/2*sqrt(6)*sq
rt(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)/((sq
rt(6)*b^2*i/abs(b) + sqrt(6)*b)*d^3) + 1/8*sqrt(pi)*b^(3/2)*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^2*i/abs(b) + b)*d^3) - 1/8*sqrt(pi)*b^(3/2)*c*erf(sqr
t(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) - 1/24*sqrt(pi)*b^(3/2)*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^2*i/abs(b) - sqrt(6)*b)*d^3) + 1/24*sqrt(b*arcsi
n(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