3.183 \(\int x^2 (a+b \sin ^{-1}(c x))^{5/2} \, dx\)

Optimal. Leaf size=358 \[ -\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}+\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{144 c^3}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}-\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{144 c^3}-\frac{5 b^2 x \sqrt{a+b \sin ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{5/2} \]

[Out]

(-5*b^2*x*Sqrt[a + b*ArcSin[c*x]])/(6*c^2) - (5*b^2*x^3*Sqrt[a + b*ArcSin[c*x]])/36 + (5*b*Sqrt[1 - c^2*x^2]*(
a + b*ArcSin[c*x])^(3/2))/(9*c^3) + (5*b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(3/2))/(18*c) + (x^3*(a + b
*ArcSin[c*x])^(5/2))/3 + (15*b^(5/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]
])/(16*c^3) - (5*b^(5/2)*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(144*
c^3) - (15*b^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(16*c^3) + (5*b
^(5/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(144*c^3)

________________________________________________________________________________________

Rubi [A]  time = 1.40912, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.688, Rules used = {4629, 4707, 4677, 4619, 4723, 3306, 3305, 3351, 3304, 3352, 3312} \[ -\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}+\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{144 c^3}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}-\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{144 c^3}-\frac{5 b^2 x \sqrt{a+b \sin ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcSin[c*x])^(5/2),x]

[Out]

(-5*b^2*x*Sqrt[a + b*ArcSin[c*x]])/(6*c^2) - (5*b^2*x^3*Sqrt[a + b*ArcSin[c*x]])/36 + (5*b*Sqrt[1 - c^2*x^2]*(
a + b*ArcSin[c*x])^(3/2))/(9*c^3) + (5*b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(3/2))/(18*c) + (x^3*(a + b
*ArcSin[c*x])^(5/2))/3 + (15*b^(5/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]
])/(16*c^3) - (5*b^(5/2)*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(144*
c^3) - (15*b^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(16*c^3) + (5*b
^(5/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(144*c^3)

Rule 4629

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcSin[c*x])^n)/(m
+ 1), x] - Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^(
m + 1), Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3306

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(f*x^2)/d], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

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 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

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 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps

\begin{align*} \int x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2} \, dx &=\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{5/2}-\frac{1}{6} (5 b c) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{5/2}-\frac{1}{12} \left (5 b^2\right ) \int x^2 \sqrt{a+b \sin ^{-1}(c x)} \, dx-\frac{(5 b) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{\sqrt{1-c^2 x^2}} \, dx}{9 c}\\ &=-\frac{5}{36} b^2 x^3 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{5/2}-\frac{\left (5 b^2\right ) \int \sqrt{a+b \sin ^{-1}(c x)} \, dx}{6 c^2}+\frac{1}{72} \left (5 b^3 c\right ) \int \frac{x^3}{\sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx\\ &=-\frac{5 b^2 x \sqrt{a+b \sin ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{5/2}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\sin ^3(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{72 c^3}+\frac{\left (5 b^3\right ) \int \frac{x}{\sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{12 c}\\ &=-\frac{5 b^2 x \sqrt{a+b \sin ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{5/2}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \left (\frac{3 \sin (x)}{4 \sqrt{a+b x}}-\frac{\sin (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{72 c^3}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}\\ &=-\frac{5 b^2 x \sqrt{a+b \sin ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{5/2}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{288 c^3}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{96 c^3}+\frac{\left (5 b^3 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}-\frac{\left (5 b^3 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}\\ &=-\frac{5 b^2 x \sqrt{a+b \sin ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{5/2}+\frac{\left (5 b^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 x)}\right )}{6 c^3}+\frac{\left (5 b^3 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{96 c^3}-\frac{\left (5 b^3 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{288 c^3}-\frac{\left (5 b^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 x)}\right )}{6 c^3}-\frac{\left (5 b^3 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{96 c^3}+\frac{\left (5 b^3 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{288 c^3}\\ &=-\frac{5 b^2 x \sqrt{a+b \sin ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{5/2}+\frac{5 b^{5/2} \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{6 c^3}-\frac{5 b^{5/2} \sqrt{\frac{\pi }{2}} C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{6 c^3}+\frac{\left (5 b^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 x)}\right )}{48 c^3}-\frac{\left (5 b^2 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{144 c^3}-\frac{\left (5 b^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 x)}\right )}{48 c^3}+\frac{\left (5 b^2 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{144 c^3}\\ &=-\frac{5 b^2 x \sqrt{a+b \sin ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right )^{5/2}+\frac{15 b^{5/2} \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}-\frac{5 b^{5/2} \sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{144 c^3}-\frac{15 b^{5/2} \sqrt{\frac{\pi }{2}} C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{16 c^3}+\frac{5 b^{5/2} \sqrt{\frac{\pi }{6}} C\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{3 a}{b}\right )}{144 c^3}\\ \end{align*}

Mathematica [C]  time = 0.273454, size = 228, normalized size = 0.64 \[ \frac{b^3 e^{-\frac{3 i a}{b}} \left (-81 e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{7}{2},-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-81 e^{\frac{4 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{7}{2},\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\sqrt{3} \left (\sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{7}{2},-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac{6 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{7}{2},\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )\right )}{648 c^3 \sqrt{a+b \sin ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(a + b*ArcSin[c*x])^(5/2),x]

[Out]

(b^3*(-81*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[7/2, ((-I)*(a + b*ArcSin[c*x]))/b] - 81*E^(
((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[7/2, (I*(a + b*ArcSin[c*x]))/b] + Sqrt[3]*(Sqrt[((-I)*(a +
b*ArcSin[c*x]))/b]*Gamma[7/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b
]*Gamma[7/2, ((3*I)*(a + b*ArcSin[c*x]))/b])))/(648*c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])

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Maple [B]  time = 0.118, size = 792, normalized size = 2.2 \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(c*x))^(5/2),x)

[Out]

1/864/c^3/(a+b*arcsin(c*x))^(1/2)*(-5*3^(1/2)*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(3*a/b)*
FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b^3+5*3^(1/2)*(1/b)^(1/2)*Pi^(1/2)*2^
(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(3*a/b)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)
/b)*b^3+405*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2
)*(a+b*arcsin(c*x))^(1/2)/b)*b^3-405*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(a/b)*FresnelC(2^
(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b^3+216*arcsin(c*x)^3*sin((a+b*arcsin(c*x))/b-a/b)*b^3-7
2*arcsin(c*x)^3*sin(3*(a+b*arcsin(c*x))/b-3*a/b)*b^3+648*arcsin(c*x)^2*sin((a+b*arcsin(c*x))/b-a/b)*a*b^2+540*
arcsin(c*x)^2*cos((a+b*arcsin(c*x))/b-a/b)*b^3-216*arcsin(c*x)^2*sin(3*(a+b*arcsin(c*x))/b-3*a/b)*a*b^2-60*arc
sin(c*x)^2*cos(3*(a+b*arcsin(c*x))/b-3*a/b)*b^3+648*arcsin(c*x)*sin((a+b*arcsin(c*x))/b-a/b)*a^2*b-810*arcsin(
c*x)*sin((a+b*arcsin(c*x))/b-a/b)*b^3+1080*arcsin(c*x)*cos((a+b*arcsin(c*x))/b-a/b)*a*b^2-216*arcsin(c*x)*sin(
3*(a+b*arcsin(c*x))/b-3*a/b)*a^2*b+30*arcsin(c*x)*sin(3*(a+b*arcsin(c*x))/b-3*a/b)*b^3-120*arcsin(c*x)*cos(3*(
a+b*arcsin(c*x))/b-3*a/b)*a*b^2+216*sin((a+b*arcsin(c*x))/b-a/b)*a^3-810*sin((a+b*arcsin(c*x))/b-a/b)*a*b^2+54
0*cos((a+b*arcsin(c*x))/b-a/b)*a^2*b-72*sin(3*(a+b*arcsin(c*x))/b-3*a/b)*a^3+30*sin(3*(a+b*arcsin(c*x))/b-3*a/
b)*a*b^2-60*cos(3*(a+b*arcsin(c*x))/b-3*a/b)*a^2*b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac{5}{2}} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arcsin(c*x))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*arcsin(c*x) + a)^(5/2)*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(c*x))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*asin(c*x))**(5/2),x)

[Out]

Timed out

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Giac [C]  time = 5.10383, size = 3357, normalized size = 9.38 \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(c*x))^(5/2),x, algorithm="giac")

[Out]

-1/8*I*sqrt(2)*sqrt(pi)*a^2*b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*a
rcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 3/16*sqrt(2)*sqrt(pi
)*a*b^4*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs
(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 1/8*I*sqrt(2)*sqrt(pi)*a^2*b^3*erf(1/2*I*sqr
t(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-
I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 3/16*sqrt(2)*sqrt(pi)*a*b^4*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x)
+ a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^
2*sqrt(abs(b)))*c^3) + 1/12*I*sqrt(pi)*a^2*b^(5/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sq
rt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c^3) - 1/24*sq
rt(pi)*a*b^(7/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt
(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c^3) + 1/8*I*sqrt(2)*sqrt(pi)*a^2*b^2*erf(-1/2*I
*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/
((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c^3) - 3/16*sqrt(2)*sqrt(pi)*a*b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x
) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b)) + b*s
qrt(abs(b)))*c^3) - 15/64*I*sqrt(2)*sqrt(pi)*b^4*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2
*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c^3) - 1/8*I
*sqrt(2)*sqrt(pi)*a^2*b^2*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c
*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c^3) - 3/16*sqrt(2)*sqrt(pi)*a*b^3
*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*
e^(-I*a/b)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c^3) + 15/64*I*sqrt(2)*sqrt(pi)*b^4*erf(1/2*I*sqrt(2)*sqrt(
b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^2/sqrt
(abs(b)) + b*sqrt(abs(b)))*c^3) - 1/12*I*sqrt(pi)*a^2*b^(5/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b)
+ 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/((sqrt(6)*b^2 - I*sqrt(6)*b^3/abs(b))*c^
3) - 1/24*sqrt(pi)*a*b^(7/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(c*
x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/((sqrt(6)*b^2 - I*sqrt(6)*b^3/abs(b))*c^3) + 1/24*I*sqrt(b*arcsin(c*x) +
a)*b^2*arcsin(c*x)^2*e^(3*I*arcsin(c*x))/c^3 - 1/8*I*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c*x)^2*e^(I*arcsin(c*x
))/c^3 + 1/8*I*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c*x)^2*e^(-I*arcsin(c*x))/c^3 - 1/24*I*sqrt(b*arcsin(c*x) +
a)*b^2*arcsin(c*x)^2*e^(-3*I*arcsin(c*x))/c^3 - 1/24*I*sqrt(pi)*a^2*b^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) +
a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*b^(3/2) + I*sqrt(6)*b
^(5/2)/abs(b))*c^3) + 1/24*I*sqrt(pi)*a^2*b^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)
*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/((sqrt(6)*b^(3/2) - I*sqrt(6)*b^(5/2)/abs(b))*c^3) - 1/2
4*I*sqrt(pi)*a^2*b^(3/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) +
a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*b + I*sqrt(6)*b^2/abs(b))*c^3) + 1/24*sqrt(pi)*a*b^(5/2)*erf(-1/2*sq
rt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sq
rt(6)*b + I*sqrt(6)*b^2/abs(b))*c^3) + 5/288*I*sqrt(pi)*b^(7/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(
b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*b + I*sqrt(6)*b^2/abs(b))*c^3
) + 1/24*I*sqrt(pi)*a^2*b^(3/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin
(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/((sqrt(6)*b - I*sqrt(6)*b^2/abs(b))*c^3) + 1/24*sqrt(pi)*a*b^(5/2)*erf
(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*
a/b)/((sqrt(6)*b - I*sqrt(6)*b^2/abs(b))*c^3) - 5/288*I*sqrt(pi)*b^(7/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) +
a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/((sqrt(6)*b - I*sqrt(6)*b^2/a
bs(b))*c^3) + 1/12*I*sqrt(b*arcsin(c*x) + a)*a*b*arcsin(c*x)*e^(3*I*arcsin(c*x))/c^3 - 5/144*sqrt(b*arcsin(c*x
) + a)*b^2*arcsin(c*x)*e^(3*I*arcsin(c*x))/c^3 - 1/4*I*sqrt(b*arcsin(c*x) + a)*a*b*arcsin(c*x)*e^(I*arcsin(c*x
))/c^3 + 5/16*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c*x)*e^(I*arcsin(c*x))/c^3 + 1/4*I*sqrt(b*arcsin(c*x) + a)*a*
b*arcsin(c*x)*e^(-I*arcsin(c*x))/c^3 + 5/16*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c*x)*e^(-I*arcsin(c*x))/c^3 - 1
/12*I*sqrt(b*arcsin(c*x) + a)*a*b*arcsin(c*x)*e^(-3*I*arcsin(c*x))/c^3 - 5/144*sqrt(b*arcsin(c*x) + a)*b^2*arc
sin(c*x)*e^(-3*I*arcsin(c*x))/c^3 + 1/24*I*sqrt(b*arcsin(c*x) + a)*a^2*e^(3*I*arcsin(c*x))/c^3 - 5/144*sqrt(b*
arcsin(c*x) + a)*a*b*e^(3*I*arcsin(c*x))/c^3 - 5/288*I*sqrt(b*arcsin(c*x) + a)*b^2*e^(3*I*arcsin(c*x))/c^3 - 1
/8*I*sqrt(b*arcsin(c*x) + a)*a^2*e^(I*arcsin(c*x))/c^3 + 5/16*sqrt(b*arcsin(c*x) + a)*a*b*e^(I*arcsin(c*x))/c^
3 + 15/32*I*sqrt(b*arcsin(c*x) + a)*b^2*e^(I*arcsin(c*x))/c^3 + 1/8*I*sqrt(b*arcsin(c*x) + a)*a^2*e^(-I*arcsin
(c*x))/c^3 + 5/16*sqrt(b*arcsin(c*x) + a)*a*b*e^(-I*arcsin(c*x))/c^3 - 15/32*I*sqrt(b*arcsin(c*x) + a)*b^2*e^(
-I*arcsin(c*x))/c^3 - 1/24*I*sqrt(b*arcsin(c*x) + a)*a^2*e^(-3*I*arcsin(c*x))/c^3 - 5/144*sqrt(b*arcsin(c*x) +
a)*a*b*e^(-3*I*arcsin(c*x))/c^3 + 5/288*I*sqrt(b*arcsin(c*x) + a)*b^2*e^(-3*I*arcsin(c*x))/c^3