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

Optimal. Leaf size=754 \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^2 \sin \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^5}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e^2 \sin \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^5}+\frac {\sqrt {\frac {\pi }{10}} \sqrt {b} e^2 \sin \left (\frac {5 a}{b}\right ) C\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{80 c^5}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^5}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e^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 )}{16 c^5}-\frac {\sqrt {\frac {\pi }{10}} \sqrt {b} e^2 \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{80 c^5}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d e \sin \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c^3}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} d e \sin \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{6 c^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d e \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c^3}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} d e \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{6 c^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d^2 \sin \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{c}+d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)} \]

[Out]

-1/800*e^2*cos(5*a/b)*FresnelS(10^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*b^(1/2)*10^(1/2)*Pi^(1/2)/c^
5+1/800*e^2*FresnelC(10^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(5*a/b)*b^(1/2)*10^(1/2)*Pi^(1/2)/c
^5+1/36*d*e*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*b^(1/2)*6^(1/2)*Pi^(1/2)/c^3
+1/96*e^2*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*b^(1/2)*6^(1/2)*Pi^(1/2)/c^5-1
/36*d*e*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(3*a/b)*b^(1/2)*6^(1/2)*Pi^(1/2)/c^3-1/9
6*e^2*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(3*a/b)*b^(1/2)*6^(1/2)*Pi^(1/2)/c^5-1/2*d
^2*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/c-1/4*d*e*cos(
a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/c^3-1/16*e^2*cos(a/b)
*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/c^5+1/2*d^2*FresnelC(2^(1
/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)/c+1/4*d*e*FresnelC(2^(1/2)/Pi^
(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)/c^3+1/16*e^2*FresnelC(2^(1/2)/Pi^(1/2
)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)/c^5+d^2*x*(a+b*arcsin(c*x))^(1/2)+2/3*d*e
*x^3*(a+b*arcsin(c*x))^(1/2)+1/5*e^2*x^5*(a+b*arcsin(c*x))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 2.26, antiderivative size = 754, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4667, 4619, 4723, 3306, 3305, 3351, 3304, 3352, 4629, 3312} \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d e \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c^3}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} d e \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 )}{6 c^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d e \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c^3}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} d e \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{6 c^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^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 )}{8 c^5}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e^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 )}{16 c^5}+\frac {\sqrt {\frac {\pi }{10}} \sqrt {b} e^2 \sin \left (\frac {5 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{80 c^5}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c^5}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e^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 )}{16 c^5}-\frac {\sqrt {\frac {\pi }{10}} \sqrt {b} e^2 \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{80 c^5}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d^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 )}{c}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{c}+d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

d^2*x*Sqrt[a + b*ArcSin[c*x]] + (2*d*e*x^3*Sqrt[a + b*ArcSin[c*x]])/3 + (e^2*x^5*Sqrt[a + b*ArcSin[c*x]])/5 -
(Sqrt[b]*d^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/c - (Sqrt[b]*d*e*Sqrt
[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*c^3) - (Sqrt[b]*e^2*Sqrt[Pi/2]*Cos[
a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*c^5) + (Sqrt[b]*d*e*Sqrt[Pi/6]*Cos[(3*a)/b]*Fr
esnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(6*c^3) + (Sqrt[b]*e^2*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[
(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*c^5) - (Sqrt[b]*e^2*Sqrt[Pi/10]*Cos[(5*a)/b]*FresnelS[(Sqrt
[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(80*c^5) + (Sqrt[b]*d^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*
ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/c + (Sqrt[b]*d*e*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqr
t[b]]*Sin[a/b])/(2*c^3) + (Sqrt[b]*e^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a
/b])/(8*c^5) - (Sqrt[b]*d*e*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(6
*c^3) - (Sqrt[b]*e^2*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(16*c^5)
+ (Sqrt[b]*e^2*Sqrt[Pi/10]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(5*a)/b])/(80*c^5)

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 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 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 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])
)

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 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 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 4667

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[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])

Rubi steps

\begin {align*} \int \left (d+e x^2\right )^2 \sqrt {a+b \sin ^{-1}(c x)} \, dx &=\int \left (d^2 \sqrt {a+b \sin ^{-1}(c x)}+2 d e x^2 \sqrt {a+b \sin ^{-1}(c x)}+e^2 x^4 \sqrt {a+b \sin ^{-1}(c x)}\right ) \, dx\\ &=d^2 \int \sqrt {a+b \sin ^{-1}(c x)} \, dx+(2 d e) \int x^2 \sqrt {a+b \sin ^{-1}(c x)} \, dx+e^2 \int x^4 \sqrt {a+b \sin ^{-1}(c x)} \, dx\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}-\frac {1}{2} \left (b c d^2\right ) \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx-\frac {1}{3} (b c d e) \int \frac {x^3}{\sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx-\frac {1}{10} \left (b c e^2\right ) \int \frac {x^5}{\sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}-\frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 c}-\frac {(b d e) \operatorname {Subst}\left (\int \frac {\sin ^3(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 c^3}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {\sin ^5(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{10 c^5}\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}-\frac {(b d e) \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 )}{3 c^3}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \left (\frac {5 \sin (x)}{8 \sqrt {a+b x}}-\frac {5 \sin (3 x)}{16 \sqrt {a+b x}}+\frac {\sin (5 x)}{16 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{10 c^5}-\frac {\left (b d^2 \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 )}{2 c}+\frac {\left (b d^2 \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 )}{2 c}\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}+\frac {(b d e) \operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{12 c^3}-\frac {(b d e) \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {\sin (5 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{160 c^5}+\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^5}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^5}-\frac {\left (d^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 )}{c}+\frac {\left (d^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 )}{c}\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}-\frac {\sqrt {b} d^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 )}{c}+\frac {\sqrt {b} d^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 )}{c}-\frac {\left (b d e \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 )}{4 c^3}-\frac {\left (b e^2 \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 )}{16 c^5}+\frac {\left (b d e \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 )}{12 c^3}+\frac {\left (b e^2 \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 )}{32 c^5}-\frac {\left (b e^2 \cos \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}+5 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{160 c^5}+\frac {\left (b d e \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 )}{4 c^3}+\frac {\left (b e^2 \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 )}{16 c^5}-\frac {\left (b d e \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 )}{12 c^3}-\frac {\left (b e^2 \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 )}{32 c^5}+\frac {\left (b e^2 \sin \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {5 a}{b}+5 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{160 c^5}\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}-\frac {\sqrt {b} d^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 )}{c}+\frac {\sqrt {b} d^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 )}{c}-\frac {\left (d e \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 )}{2 c^3}-\frac {\left (e^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 )}{8 c^5}+\frac {\left (d e \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 )}{6 c^3}+\frac {\left (e^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 )}{16 c^5}-\frac {\left (e^2 \cos \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {5 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{80 c^5}+\frac {\left (d e \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 )}{2 c^3}+\frac {\left (e^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 )}{8 c^5}-\frac {\left (d e \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 )}{6 c^3}-\frac {\left (e^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 )}{16 c^5}+\frac {\left (e^2 \sin \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {5 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{80 c^5}\\ &=d^2 x \sqrt {a+b \sin ^{-1}(c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \sin ^{-1}(c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \sin ^{-1}(c x)}-\frac {\sqrt {b} d^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 )}{c}-\frac {\sqrt {b} d e \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 )}{2 c^3}-\frac {\sqrt {b} e^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 )}{8 c^5}+\frac {\sqrt {b} d e \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 )}{6 c^3}+\frac {\sqrt {b} e^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 )}{16 c^5}-\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{10}} \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{80 c^5}+\frac {\sqrt {b} d^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 )}{c}+\frac {\sqrt {b} d e \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 )}{2 c^3}+\frac {\sqrt {b} e^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 )}{8 c^5}-\frac {\sqrt {b} d e \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 )}{6 c^3}-\frac {\sqrt {b} e^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 )}{16 c^5}+\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{10}} C\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{80 c^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 1.68, size = 400, normalized size = 0.53 \[ \frac {b e^{-\frac {5 i a}{b}} \left (-e \left (25 \sqrt {3} e^{\frac {2 i a}{b}} \left (8 c^2 d+3 e\right ) \sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+25 \sqrt {3} e^{\frac {8 i a}{b}} \left (8 c^2 d+3 e\right ) \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-9 \sqrt {5} e \left (\sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac {10 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )\right )+450 e^{\frac {4 i a}{b}} \left (8 c^4 d^2+4 c^2 d e+e^2\right ) \sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+450 e^{\frac {6 i a}{b}} \left (8 c^4 d^2+4 c^2 d e+e^2\right ) \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{7200 c^5 \sqrt {a+b \sin ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*(450*(8*c^4*d^2 + 4*c^2*d*e + e^2)*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a +
b*ArcSin[c*x]))/b] + 450*(8*c^4*d^2 + 4*c^2*d*e + e^2)*E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[
3/2, (I*(a + b*ArcSin[c*x]))/b] - e*(25*Sqrt[3]*(8*c^2*d + 3*e)*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x])
)/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + 25*Sqrt[3]*(8*c^2*d + 3*e)*E^(((8*I)*a)/b)*Sqrt[(I*(a + b*Ar
cSin[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcSin[c*x]))/b] - 9*Sqrt[5]*e*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gam
ma[3/2, ((-5*I)*(a + b*ArcSin[c*x]))/b] + E^(((10*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((5*I)*(
a + b*ArcSin[c*x]))/b]))))/(7200*c^5*E^(((5*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

________________________________________________________________________________________

giac [C]  time = 4.59, size = 3371, normalized size = 4.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*sqrt(pi)*a*b^2*d^2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*ar
csin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/4*I*sqrt(2)*sqrt(pi)*
b^3*d^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^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/2*sqrt(2)*sqrt(pi)*a*b^2*d^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^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) - 1/4*I*sqrt(2)*sqrt(pi)*b^3*d^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^3/sqrt(abs(b)) + b^2
*sqrt(abs(b)))*c) - sqrt(pi)*a*b*d^2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqr
t(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*c) - sqr
t(pi)*a*b*d^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)*sqr
t(abs(b))/b)*e^(-I*a/b)/((-I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*c) + 1/4*sqrt(2)*sqrt(pi)*a*b^
2*d*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 + 1)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 1/8*I*sqrt(2)*sqrt(pi)*b^3*d*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 + 1)/(
(I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 1/4*sqrt(2)*sqrt(pi)*a*b^2*d*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 + 1)/((-I*b^3/sqrt(abs(b)
) + b^2*sqrt(abs(b)))*c^3) - 1/8*I*sqrt(2)*sqrt(pi)*b^3*d*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 + 1)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b
)))*c^3) - 1/2*I*sqrt(b*arcsin(c*x) + a)*d^2*e^(I*arcsin(c*x))/c + 1/2*I*sqrt(b*arcsin(c*x) + a)*d^2*e^(-I*arc
sin(c*x))/c - 1/2*sqrt(pi)*a*b^(3/2)*d*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 + 1)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c^3) - 1/12*I*sqrt(pi)
*b^(5/2)*d*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)/ab
s(b))*e^(3*I*a/b + 1)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c^3) - 1/2*sqrt(pi)*a*b^(3/2)*d*erf(-1/2*sqrt(6)*s
qrt(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 + 1)/((sqrt
(6)*b^2 - I*sqrt(6)*b^3/abs(b))*c^3) + 1/12*I*sqrt(pi)*b^(5/2)*d*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 + 1)/((sqrt(6)*b^2 - I*sqrt(6)*b^3/abs
(b))*c^3) + 1/2*sqrt(pi)*a*b*d*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 + 1)/((sqrt(6)*b^(3/2) + I*sqrt(6)*b^(5/2)/abs(b))*c^3) - 1/2*sqrt(pi)*a*
b*d*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 + 1)/((I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*c^3) - 1/2*sqrt(pi)*a*b*d*erf(1/2*I*s
qrt(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 +
1)/((-I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*c^3) + 1/2*sqrt(pi)*a*b*d*erf(-1/2*sqrt(6)*sqrt(b*a
rcsin(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 + 1)/((sqrt(6)*b^(
3/2) - I*sqrt(6)*b^(5/2)/abs(b))*c^3) + 1/16*sqrt(2)*sqrt(pi)*a*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 + 2)/((I*b^3/sqrt(abs(b)) + b^2*s
qrt(abs(b)))*c^5) + 1/32*I*sqrt(2)*sqrt(pi)*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 + 2)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^5) +
1/16*sqrt(2)*sqrt(pi)*a*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 + 2)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^5) - 1/32*I*sqrt(2)*sqrt
(pi)*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 + 2)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^5) - 1/32*sqrt(6)*sqrt(pi)*a*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
+ 2)/((b^2 + I*b^3/abs(b))*c^5) - 1/32*sqrt(6)*sqrt(pi)*a*b^(3/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqr
t(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b + 2)/((b^2 - I*b^3/abs(b))*c^5) + 1/1
2*I*sqrt(b*arcsin(c*x) + a)*d*e^(3*I*arcsin(c*x) + 1)/c^3 - 1/4*I*sqrt(b*arcsin(c*x) + a)*d*e^(I*arcsin(c*x) +
1)/c^3 + 1/4*I*sqrt(b*arcsin(c*x) + a)*d*e^(-I*arcsin(c*x) + 1)/c^3 - 1/12*I*sqrt(b*arcsin(c*x) + a)*d*e^(-3*
I*arcsin(c*x) + 1)/c^3 + 1/16*sqrt(pi)*a*b^(3/2)*erf(-1/2*sqrt(10)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqr
t(10)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(5*I*a/b + 2)/((sqrt(10)*b^2 + I*sqrt(10)*b^3/abs(b))*c^5) + 1
/160*I*sqrt(pi)*b^(5/2)*erf(-1/2*sqrt(10)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(10)*sqrt(b*arcsin(c*x)
+ a)*sqrt(b)/abs(b))*e^(5*I*a/b + 2)/((sqrt(10)*b^2 + I*sqrt(10)*b^3/abs(b))*c^5) - 1/32*I*sqrt(pi)*b^(5/2)*er
f(-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 + 2)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c^5) + 1/32*I*sqrt(pi)*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 + 2)/((sqrt(6)*b^2 - I*sq
rt(6)*b^3/abs(b))*c^5) + 1/16*sqrt(pi)*a*b^(3/2)*erf(-1/2*sqrt(10)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqr
t(10)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-5*I*a/b + 2)/((sqrt(10)*b^2 - I*sqrt(10)*b^3/abs(b))*c^5) -
1/160*I*sqrt(pi)*b^(5/2)*erf(-1/2*sqrt(10)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(10)*sqrt(b*arcsin(c*x)
+ a)*sqrt(b)/abs(b))*e^(-5*I*a/b + 2)/((sqrt(10)*b^2 - I*sqrt(10)*b^3/abs(b))*c^5) - 1/16*sqrt(pi)*a*b*erf(-1
/2*sqrt(10)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(10)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(5*I*a/
b + 2)/((sqrt(10)*b^(3/2) + I*sqrt(10)*b^(5/2)/abs(b))*c^5) - 1/8*sqrt(pi)*a*b*erf(-1/2*I*sqrt(2)*sqrt(b*arcsi
n(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b + 2)/((I*sqrt(2)*b^2/s
qrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*c^5) - 1/8*sqrt(pi)*a*b*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(a
bs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b + 2)/((-I*sqrt(2)*b^2/sqrt(abs(b)) + sq
rt(2)*b*sqrt(abs(b)))*c^5) - 1/16*sqrt(pi)*a*b*erf(-1/2*sqrt(10)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(
10)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-5*I*a/b + 2)/((sqrt(10)*b^(3/2) - I*sqrt(10)*b^(5/2)/abs(b))*c
^5) + 3/16*sqrt(pi)*a*sqrt(b)*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 + 2)/((sqrt(6)*b + I*sqrt(6)*b^2/abs(b))*c^5) + 3/16*sqrt(pi)*a*sqrt(b)*er
f(-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 + 2)/((sqrt(6)*b - I*sqrt(6)*b^2/abs(b))*c^5) - 1/160*I*sqrt(b*arcsin(c*x) + a)*e^(5*I*arcsin(c*x) + 2)/c
^5 + 1/32*I*sqrt(b*arcsin(c*x) + a)*e^(3*I*arcsin(c*x) + 2)/c^5 - 1/16*I*sqrt(b*arcsin(c*x) + a)*e^(I*arcsin(c
*x) + 2)/c^5 + 1/16*I*sqrt(b*arcsin(c*x) + a)*e^(-I*arcsin(c*x) + 2)/c^5 - 1/32*I*sqrt(b*arcsin(c*x) + a)*e^(-
3*I*arcsin(c*x) + 2)/c^5 + 1/160*I*sqrt(b*arcsin(c*x) + a)*e^(-5*I*arcsin(c*x) + 2)/c^5

________________________________________________________________________________________

maple [A]  time = 0.50, size = 1137, normalized size = 1.51 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7200/c^5*(-200*(a+b*arcsin(c*x))^(1/2)*(1/b)^(1/2)*2^(1/2)*Pi^(1/2)*3^(1/2)*cos(3*a/b)*FresnelS(2^(1/2)/Pi^
(1/2)*3^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b*c^2*d*e+200*(a+b*arcsin(c*x))^(1/2)*(1/b)^(1/2)*2^(1/2)
*Pi^(1/2)*3^(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*c^2*d*
e-75*(a+b*arcsin(c*x))^(1/2)*(1/b)^(1/2)*2^(1/2)*Pi^(1/2)*3^(1/2)*cos(3*a/b)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)
/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b*e^2+75*(a+b*arcsin(c*x))^(1/2)*(1/b)^(1/2)*2^(1/2)*Pi^(1/2)*3^(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*e^2+1200*arcsin(c*x)*sin
(3*(a+b*arcsin(c*x))/b-3*a/b)*b*c^2*d*e+1200*sin(3*(a+b*arcsin(c*x))/b-3*a/b)*a*c^2*d*e+450*arcsin(c*x)*sin(3*
(a+b*arcsin(c*x))/b-3*a/b)*b*e^2+450*sin(3*(a+b*arcsin(c*x))/b-3*a/b)*a*e^2+3600*(a+b*arcsin(c*x))^(1/2)*(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(c*x))^(1/2)/b)*b*c^4*d^2-36
00*(a+b*arcsin(c*x))^(1/2)*(1/b)^(1/2)*2^(1/2)*Pi^(1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*ar
csin(c*x))^(1/2)/b)*b*c^4*d^2+1800*(a+b*arcsin(c*x))^(1/2)*(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(c*x))^(1/2)/b)*b*c^2*d*e-1800*(a+b*arcsin(c*x))^(1/2)*(1/b)^(1/2)*2^(1/2)
*Pi^(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*c^2*d*e+450*(a+b*arcsin(
c*x))^(1/2)*(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(c*x))^(1/2
)/b)*b*e^2-450*(a+b*arcsin(c*x))^(1/2)*(1/b)^(1/2)*2^(1/2)*Pi^(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*e^2-7200*arcsin(c*x)*sin((a+b*arcsin(c*x))/b-a/b)*b*c^4*d^2-7200*sin((a+b*ar
csin(c*x))/b-a/b)*a*c^4*d^2-3600*arcsin(c*x)*sin((a+b*arcsin(c*x))/b-a/b)*b*c^2*d*e-3600*sin((a+b*arcsin(c*x))
/b-a/b)*a*c^2*d*e-900*arcsin(c*x)*sin((a+b*arcsin(c*x))/b-a/b)*b*e^2-900*sin((a+b*arcsin(c*x))/b-a/b)*a*e^2+9*
(a+b*arcsin(c*x))^(1/2)*(1/b)^(1/2)*2^(1/2)*Pi^(1/2)*5^(1/2)*cos(5*a/b)*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)/(1/b
)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b*e^2-9*(a+b*arcsin(c*x))^(1/2)*(1/b)^(1/2)*2^(1/2)*Pi^(1/2)*5^(1/2)*sin(5*
a/b)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b*e^2-90*arcsin(c*x)*sin(5*(a+b*
arcsin(c*x))/b-5*a/b)*b*e^2-90*sin(5*(a+b*arcsin(c*x))/b-5*a/b)*a*e^2)/(a+b*arcsin(c*x))^(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x^2 + d)^2*sqrt(b*arcsin(c*x) + a), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________