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

Optimal. Leaf size=373 \[ \frac{\sqrt{\frac{\pi }{2}} d^2 \cos \left (\frac{4 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c^2}+\frac{\sqrt{3 \pi } d^2 \cos \left (\frac{6 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^2}+\frac{5 \sqrt{\pi } d^2 \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{8 b^{3/2} c^2}+\frac{5 \sqrt{\pi } d^2 \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{8 b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d^2 \sin \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c^2}+\frac{\sqrt{3 \pi } d^2 \sin \left (\frac{6 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^2}-\frac{2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]

[Out]

(-2*d^2*x*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSin[c*x]]) + (d^2*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2
/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^2) + (d^2*Sqrt[3*Pi]*Cos[(6*a)/b]*FresnelC[(2*Sqrt[3/Pi]*Sq
rt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*b^(3/2)*c^2) + (5*d^2*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[
c*x]])/(Sqrt[b]*Sqrt[Pi])])/(8*b^(3/2)*c^2) + (5*d^2*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sq
rt[Pi])]*Sin[(2*a)/b])/(8*b^(3/2)*c^2) + (d^2*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[
b]]*Sin[(4*a)/b])/(b^(3/2)*c^2) + (d^2*Sqrt[3*Pi]*FresnelS[(2*Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin
[(6*a)/b])/(8*b^(3/2)*c^2)

________________________________________________________________________________________

Rubi [A]  time = 1.40087, antiderivative size = 373, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37, Rules used = {4721, 4661, 3312, 3306, 3305, 3351, 3304, 3352, 4723, 4406} \[ \frac{\sqrt{\frac{\pi }{2}} d^2 \cos \left (\frac{4 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c^2}+\frac{\sqrt{3 \pi } d^2 \cos \left (\frac{6 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^2}+\frac{5 \sqrt{\pi } d^2 \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{8 b^{3/2} c^2}+\frac{5 \sqrt{\pi } d^2 \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{8 b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d^2 \sin \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c^2}+\frac{\sqrt{3 \pi } d^2 \sin \left (\frac{6 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^2}-\frac{2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^2*x*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSin[c*x]]) + (d^2*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2
/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^2) + (d^2*Sqrt[3*Pi]*Cos[(6*a)/b]*FresnelC[(2*Sqrt[3/Pi]*Sq
rt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*b^(3/2)*c^2) + (5*d^2*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[
c*x]])/(Sqrt[b]*Sqrt[Pi])])/(8*b^(3/2)*c^2) + (5*d^2*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sq
rt[Pi])]*Sin[(2*a)/b])/(8*b^(3/2)*c^2) + (d^2*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[
b]]*Sin[(4*a)/b])/(b^(3/2)*c^2) + (d^2*Sqrt[3*Pi]*FresnelS[(2*Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin
[(6*a)/b])/(8*b^(3/2)*c^2)

Rule 4721

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
((f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*(a + b*ArcSin[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(f*m*d^IntPar
t[p]*(d + e*x^2)^FracPart[p])/(b*c*(n + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/
2)*(a + b*ArcSin[c*x])^(n + 1), x], x] + Dist[(c*(m + 2*p + 1)*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*f*(n +
1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x])
/; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[2*p, 0]

Rule 4661

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c, Subst[Int[(
a + b*x)^n*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] && I
GtQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 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 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 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 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rubi steps

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

Mathematica [C]  time = 3.07019, size = 509, normalized size = 1.36 \[ \frac{d^2 \left (\frac{i e^{-\frac{6 i a}{b}} \left (11 \sqrt{2} e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-11 \sqrt{2} e^{\frac{8 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-8 e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+8 e^{\frac{10 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt{6} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\sqrt{6} e^{\frac{12 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+10 i e^{\frac{6 i a}{b}} \sin \left (2 \sin ^{-1}(c x)\right )+8 i e^{\frac{6 i a}{b}} \sin \left (4 \sin ^{-1}(c x)\right )+2 i e^{\frac{6 i a}{b}} \sin \left (6 \sin ^{-1}(c x)\right )\right )}{b \sqrt{a+b \sin ^{-1}(c x)}}+64 \sqrt{\pi } \left (\frac{1}{b}\right )^{3/2} \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi }}\right )+64 \sqrt{\pi } \left (\frac{1}{b}\right )^{3/2} \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi }}\right )\right )}{32 c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^2*(64*(b^(-1))^(3/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[Pi]] + 64
*(b^(-1))^(3/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[Pi]]*Sin[(2*a)/b] + (I*(11*Sqr
t[2]*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c*x]))/b] - 11*Sqrt[2
]*E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c*x]))/b] - 8*E^(((2*I)*a)/b
)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-4*I)*(a + b*ArcSin[c*x]))/b] + 8*E^(((10*I)*a)/b)*Sqrt[(I*(
a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((4*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[6]*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*
Gamma[1/2, ((-6*I)*(a + b*ArcSin[c*x]))/b] + Sqrt[6]*E^(((12*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/
2, ((6*I)*(a + b*ArcSin[c*x]))/b] + (10*I)*E^(((6*I)*a)/b)*Sin[2*ArcSin[c*x]] + (8*I)*E^(((6*I)*a)/b)*Sin[4*Ar
cSin[c*x]] + (2*I)*E^(((6*I)*a)/b)*Sin[6*ArcSin[c*x]]))/(b*E^(((6*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])))/(32*c^2)

________________________________________________________________________________________

Maple [A]  time = 0.086, size = 426, normalized size = 1.1 \begin{align*}{\frac{{d}^{2}}{16\,b{c}^{2}} \left ( 2\,\sqrt{3}\sqrt{a+b\arcsin \left ( cx \right ) }\cos \left ( 6\,{\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{6}\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{\pi }\sqrt{{b}^{-1}}+2\,\sqrt{3}\sqrt{a+b\arcsin \left ( cx \right ) }\sin \left ( 6\,{\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{6}\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{\pi }\sqrt{{b}^{-1}}+8\,\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }\cos \left ( 4\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{\pi }\sqrt{{b}^{-1}}+8\,\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }\sin \left ( 4\,{\frac{a}{b}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{\pi }\sqrt{{b}^{-1}}+10\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arcsin \left ( cx \right ) }\cos \left ( 2\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) +10\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arcsin \left ( cx \right ) }\sin \left ( 2\,{\frac{a}{b}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) -5\,\sin \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ) -4\,\sin \left ( 4\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-4\,{\frac{a}{b}} \right ) -\sin \left ( 6\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-6\,{\frac{a}{b}} \right ) \right ){\frac{1}{\sqrt{a+b\arcsin \left ( cx \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16/c^2*d^2/b*(2*3^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(6*a/b)*FresnelC(2^(1/2)/Pi^(1/2)*6^(1/2)/(1/b)^(1/2)*(a+
b*arcsin(c*x))^(1/2)/b)*Pi^(1/2)*(1/b)^(1/2)+2*3^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(6*a/b)*FresnelS(2^(1/2)/Pi^
(1/2)*6^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*Pi^(1/2)*(1/b)^(1/2)+8*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*co
s(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*Pi^(1/2)*(1/b)^(1/2)+8*2^(1/2)*(a+
b*arcsin(c*x))^(1/2)*sin(4*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*Pi^(1/2)*(1
/b)^(1/2)+10*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(2*a/b)*FresnelC(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcs
in(c*x))^(1/2)/b)+10*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(2*a/b)*FresnelS(2/Pi^(1/2)/(1/b)^(1/2)*(
a+b*arcsin(c*x))^(1/2)/b)-5*sin(2*(a+b*arcsin(c*x))/b-2*a/b)-4*sin(4*(a+b*arcsin(c*x))/b-4*a/b)-sin(6*(a+b*arc
sin(c*x))/b-6*a/b))/(a+b*arcsin(c*x))^(1/2)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d^{2} \left (\int \frac{x}{a \sqrt{a + b \operatorname{asin}{\left (c x \right )}} + b \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \operatorname{asin}{\left (c x \right )}}\, dx + \int - \frac{2 c^{2} x^{3}}{a \sqrt{a + b \operatorname{asin}{\left (c x \right )}} + b \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \operatorname{asin}{\left (c x \right )}}\, dx + \int \frac{c^{4} x^{5}}{a \sqrt{a + b \operatorname{asin}{\left (c x \right )}} + b \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \operatorname{asin}{\left (c x \right )}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**2*(Integral(x/(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + b*asin(c*x))*asin(c*x)), x) + Integral(-2*c**2*x**3/(a*
sqrt(a + b*asin(c*x)) + b*sqrt(a + b*asin(c*x))*asin(c*x)), x) + Integral(c**4*x**5/(a*sqrt(a + b*asin(c*x)) +
b*sqrt(a + b*asin(c*x))*asin(c*x)), x))

________________________________________________________________________________________

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

[Out]

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