3.700 \(\int \frac{d+e x^2}{(a+b \sin ^{-1}(c x))^{3/2}} \, dx\)

Optimal. Leaf size=394 \[ \frac{\sqrt{\frac{\pi }{2}} e \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c^3}-\frac{\sqrt{\frac{3 \pi }{2}} e \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c^3}-\frac{\sqrt{\frac{\pi }{2}} e \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c^3}+\frac{\sqrt{\frac{3 \pi }{2}} e \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c^3}+\frac{2 \sqrt{2 \pi } d \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}-\frac{2 \sqrt{2 \pi } d \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}-\frac{2 d \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{2 e x^2 \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]

[Out]

(-2*d*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*x]]) - (2*e*x^2*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*
x]]) - (e*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) - (2*d*Sqr
t[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c) + (e*Sqrt[(3*Pi)/2]*Cos[(
3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) + (e*Sqrt[Pi/2]*FresnelC[(Sqrt[2
/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c^3) + (2*d*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a +
b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c) - (e*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x
]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(3/2)*c^3)

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Rubi [A]  time = 0.797611, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {4667, 4621, 4723, 3306, 3305, 3351, 3304, 3352, 4631} \[ \frac{\sqrt{\frac{\pi }{2}} e \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c^3}-\frac{\sqrt{\frac{3 \pi }{2}} e \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c^3}-\frac{\sqrt{\frac{\pi }{2}} e \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c^3}+\frac{\sqrt{\frac{3 \pi }{2}} e \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c^3}+\frac{2 \sqrt{2 \pi } d \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}-\frac{2 \sqrt{2 \pi } d \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}-\frac{2 d \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{2 e x^2 \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)/(a + b*ArcSin[c*x])^(3/2),x]

[Out]

(-2*d*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*x]]) - (2*e*x^2*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*
x]]) - (e*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) - (2*d*Sqr
t[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c) + (e*Sqrt[(3*Pi)/2]*Cos[(
3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) + (e*Sqrt[Pi/2]*FresnelC[(Sqrt[2
/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c^3) + (2*d*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a +
b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c) - (e*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x
]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(3/2)*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 4621

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

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 4631

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

Rubi steps

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

Mathematica [C]  time = 1.17343, size = 417, normalized size = 1.06 \[ \frac{e^{-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}} \left (\left (4 c^2 d+e\right ) e^{\frac{2 i a}{b}+3 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\left (4 c^2 d+e\right ) e^{\frac{4 i a}{b}+3 i \sin ^{-1}(c x)} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt{3} e e^{3 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt{3} e e^{3 i \left (\frac{2 a}{b}+\sin ^{-1}(c x)\right )} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-4 c^2 d e^{\frac{3 i a}{b}+2 i \sin ^{-1}(c x)}-4 c^2 d e^{\frac{3 i a}{b}+4 i \sin ^{-1}(c x)}-e e^{\frac{3 i a}{b}+2 i \sin ^{-1}(c x)}-e e^{\frac{3 i a}{b}+4 i \sin ^{-1}(c x)}+e e^{\frac{3 i \left (a+2 b \sin ^{-1}(c x)\right )}{b}}+e e^{\frac{3 i a}{b}}\right )}{4 b c^3 \sqrt{a+b \sin ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d + e*x^2)/(a + b*ArcSin[c*x])^(3/2),x]

[Out]

(e*E^(((3*I)*a)/b) - 4*c^2*d*E^(((3*I)*a)/b + (2*I)*ArcSin[c*x]) - e*E^(((3*I)*a)/b + (2*I)*ArcSin[c*x]) - 4*c
^2*d*E^(((3*I)*a)/b + (4*I)*ArcSin[c*x]) - e*E^(((3*I)*a)/b + (4*I)*ArcSin[c*x]) + e*E^(((3*I)*(a + 2*b*ArcSin
[c*x]))/b) + (4*c^2*d + e)*E^(((2*I)*a)/b + (3*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (
(-I)*(a + b*ArcSin[c*x]))/b] + (4*c^2*d + e)*E^(((4*I)*a)/b + (3*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/
b]*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*e*E^((3*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*
Gamma[1/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*e*E^((3*I)*((2*a)/b + ArcSin[c*x]))*Sqrt[(I*(a + b*ArcSin
[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c*x]))/b])/(4*b*c^3*E^(((3*I)*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*Ar
cSin[c*x]])

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Maple [A]  time = 0.113, size = 446, normalized size = 1.1 \begin{align*}{\frac{1}{2\,b{c}^{3}} \left ( -4\,\sqrt{\pi }\sqrt{a+b\arcsin \left ( cx \right ) }\cos \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{2}\sqrt{{b}^{-1}}{c}^{2}d+4\,\sqrt{\pi }\sqrt{a+b\arcsin \left ( cx \right ) }\sin \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{2}\sqrt{{b}^{-1}}{c}^{2}d+\sqrt{\pi }\sqrt{a+b\arcsin \left ( cx \right ) }\cos \left ( 3\,{\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) \sqrt{2}\sqrt{{b}^{-1}}\sqrt{3}e-\sqrt{\pi }\sqrt{a+b\arcsin \left ( cx \right ) }\sin \left ( 3\,{\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) \sqrt{2}\sqrt{{b}^{-1}}\sqrt{3}e-\sqrt{\pi }\sqrt{a+b\arcsin \left ( cx \right ) }\cos \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) \sqrt{2}\sqrt{{b}^{-1}}e+\sqrt{\pi }\sqrt{a+b\arcsin \left ( cx \right ) }\sin \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) \sqrt{2}\sqrt{{b}^{-1}}e-4\,\cos \left ({\frac{a+b\arcsin \left ( cx \right ) }{b}}-{\frac{a}{b}} \right ){c}^{2}d+\cos \left ( 3\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-3\,{\frac{a}{b}} \right ) e-\cos \left ({\frac{a+b\arcsin \left ( cx \right ) }{b}}-{\frac{a}{b}} \right ) e \right ){\frac{1}{\sqrt{a+b\arcsin \left ( cx \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)/(a+b*arcsin(c*x))^(3/2),x)

[Out]

1/2/c^3/b*(-4*Pi^(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)*2^(1/2)*(1/b)^(1/2)*c^2*d+4*Pi^(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)*2^(1/2)*(1/b)^(1/2)*c^2*d+Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(3*a/b)*Fres
nelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*2^(1/2)*(1/b)^(1/2)*3^(1/2)*e-Pi^(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)*2^(1
/2)*(1/b)^(1/2)*3^(1/2)*e-Pi^(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)*2^(1/2)*(1/b)^(1/2)*e+Pi^(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)*2^(1/2)*(1/b)^(1/2)*e-4*cos((a+b*arcsin(c*x))/b-a/b)*c^2*d+cos(3*(a+
b*arcsin(c*x))/b-3*a/b)*e-cos((a+b*arcsin(c*x))/b-a/b)*e)/(a+b*arcsin(c*x))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e x^{2} + d}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac{3}{2}}}\,{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))^(3/2),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)/(b*arcsin(c*x) + a)^(3/2), 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((e*x^2+d)/(a+b*arcsin(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d + e x^{2}}{\left (a + b \operatorname{asin}{\left (c x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)/(a+b*asin(c*x))**(3/2),x)

[Out]

Integral((d + e*x**2)/(a + b*asin(c*x))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e x^{2} + d}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac{3}{2}}}\,{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))^(3/2),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)/(b*arcsin(c*x) + a)^(3/2), x)