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

Optimal. Leaf size=369 \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c^3}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} e \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{12 c^3}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c^3}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} e \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{12 c^3}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} d \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{c}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} d \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{c}+d x \sqrt{a+b \sin ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sin ^{-1}(c x)} \]

[Out]

d*x*Sqrt[a + b*ArcSin[c*x]] + (e*x^3*Sqrt[a + b*ArcSin[c*x]])/3 - (Sqrt[b]*d*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqr
t[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/c - (Sqrt[b]*e*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*
ArcSin[c*x]])/Sqrt[b]])/(4*c^3) + (Sqrt[b]*e*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*
x]])/Sqrt[b]])/(12*c^3) + (Sqrt[b]*d*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b
])/c + (Sqrt[b]*e*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(4*c^3) - (Sqrt[
b]*e*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(12*c^3)

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Rubi [A]  time = 1.02657, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4667, 4619, 4723, 3306, 3305, 3351, 3304, 3352, 4629, 3312} \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c^3}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} e \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{12 c^3}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c^3}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} e \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{12 c^3}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} d \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{c}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} d \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{c}+d x \sqrt{a+b \sin ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sin ^{-1}(c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

d*x*Sqrt[a + b*ArcSin[c*x]] + (e*x^3*Sqrt[a + b*ArcSin[c*x]])/3 - (Sqrt[b]*d*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqr
t[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/c - (Sqrt[b]*e*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*
ArcSin[c*x]])/Sqrt[b]])/(4*c^3) + (Sqrt[b]*e*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*
x]])/Sqrt[b]])/(12*c^3) + (Sqrt[b]*d*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b
])/c + (Sqrt[b]*e*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(4*c^3) - (Sqrt[
b]*e*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(12*c^3)

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 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 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 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 \left (d+e x^2\right ) \sqrt{a+b \sin ^{-1}(c x)} \, dx &=\int \left (d \sqrt{a+b \sin ^{-1}(c x)}+e x^2 \sqrt{a+b \sin ^{-1}(c x)}\right ) \, dx\\ &=d \int \sqrt{a+b \sin ^{-1}(c x)} \, dx+e \int x^2 \sqrt{a+b \sin ^{-1}(c x)} \, dx\\ &=d x \sqrt{a+b \sin ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sin ^{-1}(c x)}-\frac{1}{2} (b c d) \int \frac{x}{\sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx-\frac{1}{6} (b c e) \int \frac{x^3}{\sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx\\ &=d x \sqrt{a+b \sin ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sin ^{-1}(c x)}-\frac{(b d) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 c}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\sin ^3(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{6 c^3}\\ &=d x \sqrt{a+b \sin ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sin ^{-1}(c x)}-\frac{(b 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 )}{6 c^3}-\frac{\left (b d \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 \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 x \sqrt{a+b \sin ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sin ^{-1}(c x)}+\frac{(b e) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{24 c^3}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 c^3}-\frac{\left (d \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 \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 x \sqrt{a+b \sin ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sin ^{-1}(c x)}-\frac{\sqrt{b} d \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 \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 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 )}{8 c^3}+\frac{\left (b 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 )}{24 c^3}+\frac{\left (b 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 )}{8 c^3}-\frac{\left (b 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 )}{24 c^3}\\ &=d x \sqrt{a+b \sin ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sin ^{-1}(c x)}-\frac{\sqrt{b} d \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 \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 (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 )}{4 c^3}+\frac{\left (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 )}{12 c^3}+\frac{\left (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 )}{4 c^3}-\frac{\left (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 )}{12 c^3}\\ &=d x \sqrt{a+b \sin ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sin ^{-1}(c x)}-\frac{\sqrt{b} d \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} 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 )}{4 c^3}+\frac{\sqrt{b} 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 )}{12 c^3}+\frac{\sqrt{b} d \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} 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 )}{4 c^3}-\frac{\sqrt{b} 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 )}{12 c^3}\\ \end{align*}

Mathematica [C]  time = 0.613829, size = 244, normalized size = 0.66 \[ \frac{b e^{-\frac{3 i a}{b}} \left (9 e^{\frac{2 i a}{b}} \left (4 c^2 d+e\right ) \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+9 e^{\frac{4 i a}{b}} \left (4 c^2 d+e\right ) \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt{3} e \left (\sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac{6 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )\right )}{72 c^3 \sqrt{a+b \sin ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*(9*(4*c^2*d + e)*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b
] + 9*(4*c^2*d + e)*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b] - Sq
rt[3]*e*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[
(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcSin[c*x]))/b])))/(72*c^3*E^(((3*I)*a)/b)*Sqrt[a + b*Arc
Sin[c*x]])

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Maple [A]  time = 0.112, size = 542, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72/c^3*(-36*(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*c^2*d+36*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(a/b)*Fresne
lC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b*c^2*d+3^(1/2)*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*a
rcsin(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*e-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*e-9*(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*e+9*(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*e+72*arcsin(c*x)*sin((a+b*arcsi
n(c*x))/b-a/b)*b*c^2*d+72*sin((a+b*arcsin(c*x))/b-a/b)*a*c^2*d+18*arcsin(c*x)*sin((a+b*arcsin(c*x))/b-a/b)*b*e
-6*arcsin(c*x)*sin(3*(a+b*arcsin(c*x))/b-3*a/b)*b*e+18*sin((a+b*arcsin(c*x))/b-a/b)*a*e-6*sin(3*(a+b*arcsin(c*
x))/b-3*a/b)*a*e)/(a+b*arcsin(c*x))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x^2 + d)*sqrt(b*arcsin(c*x) + a), 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((e*x^2+d)*(a+b*arcsin(c*x))^(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 \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \left (d + e x^{2}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 2.38412, size = 865, normalized size = 2.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*I*sqrt(2)*sqrt(pi)*b^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcs
in(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c) - 1/4*I*sqrt(2)*sqrt(pi)*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)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c) - 1/2*I*sqrt(b*arcsin(c*x) + a)*d*e^(I*arcsin(c*x))/c +
1/2*I*sqrt(b*arcsin(c*x) + a)*d*e^(-I*arcsin(c*x))/c + 1/16*I*sqrt(2)*sqrt(pi)*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 + 1)/((I*b^2/sqrt
(abs(b)) + b*sqrt(abs(b)))*c^3) - 1/16*I*sqrt(2)*sqrt(pi)*b^2*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 + 1)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs
(b)))*c^3) - 1/24*I*sqrt(pi)*b^(3/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*a
rcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b + 1)/((sqrt(6)*b + I*sqrt(6)*b^2/abs(b))*c^3) + 1/24*I*sqrt(pi)*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 + 1)/((sqrt(6)*b - I*sqrt(6)*b^2/abs(b))*c^3) + 1/24*I*sqrt(b*arcsin(c*x) + a)*e^(3*I*arcsin(c*x)
+ 1)/c^3 - 1/8*I*sqrt(b*arcsin(c*x) + a)*e^(I*arcsin(c*x) + 1)/c^3 + 1/8*I*sqrt(b*arcsin(c*x) + a)*e^(-I*arcsi
n(c*x) + 1)/c^3 - 1/24*I*sqrt(b*arcsin(c*x) + a)*e^(-3*I*arcsin(c*x) + 1)/c^3