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

Optimal. Leaf size=511 \[ \frac{5 \sqrt{\frac{\pi }{2}} d^2 \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{\sqrt{\frac{3 \pi }{2}} d^2 \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{3 \sqrt{\frac{5 \pi }{2}} d^2 \sin \left (\frac{5 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{\sqrt{\frac{7 \pi }{2}} d^2 \sin \left (\frac{7 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{14}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{5 \sqrt{\frac{\pi }{2}} d^2 \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{\sqrt{\frac{3 \pi }{2}} d^2 \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{3 \sqrt{\frac{5 \pi }{2}} d^2 \cos \left (\frac{5 a}{b}\right ) S\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{\sqrt{\frac{7 \pi }{2}} d^2 \cos \left (\frac{7 a}{b}\right ) S\left (\frac{\sqrt{\frac{14}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]

[Out]

(-2*d^2*x^2*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSin[c*x]]) - (5*d^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/P
i]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) + (d^2*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]
*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) + (3*d^2*Sqrt[(5*Pi)/2]*Cos[(5*a)/b]*FresnelS[(Sqrt[10/Pi
]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) + (d^2*Sqrt[(7*Pi)/2]*Cos[(7*a)/b]*FresnelS[(Sqrt[14/Pi]
*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) + (5*d^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSi
n[c*x]])/Sqrt[b]]*Sin[a/b])/(16*b^(3/2)*c^3) - (d^2*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]
])/Sqrt[b]]*Sin[(3*a)/b])/(16*b^(3/2)*c^3) - (3*d^2*Sqrt[(5*Pi)/2]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x
]])/Sqrt[b]]*Sin[(5*a)/b])/(16*b^(3/2)*c^3) - (d^2*Sqrt[(7*Pi)/2]*FresnelC[(Sqrt[14/Pi]*Sqrt[a + b*ArcSin[c*x]
])/Sqrt[b]]*Sin[(7*a)/b])/(16*b^(3/2)*c^3)

________________________________________________________________________________________

Rubi [A]  time = 2.12356, antiderivative size = 511, normalized size of antiderivative = 1., number of steps used = 42, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {4721, 4723, 4406, 3306, 3305, 3351, 3304, 3352} \[ \frac{5 \sqrt{\frac{\pi }{2}} d^2 \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{\sqrt{\frac{3 \pi }{2}} d^2 \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{3 \sqrt{\frac{5 \pi }{2}} d^2 \sin \left (\frac{5 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{\sqrt{\frac{7 \pi }{2}} d^2 \sin \left (\frac{7 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{14}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{5 \sqrt{\frac{\pi }{2}} d^2 \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{\sqrt{\frac{3 \pi }{2}} d^2 \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{3 \sqrt{\frac{5 \pi }{2}} d^2 \cos \left (\frac{5 a}{b}\right ) S\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{\sqrt{\frac{7 \pi }{2}} d^2 \cos \left (\frac{7 a}{b}\right ) S\left (\frac{\sqrt{\frac{14}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^2*x^2*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSin[c*x]]) - (5*d^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/P
i]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) + (d^2*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]
*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) + (3*d^2*Sqrt[(5*Pi)/2]*Cos[(5*a)/b]*FresnelS[(Sqrt[10/Pi
]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) + (d^2*Sqrt[(7*Pi)/2]*Cos[(7*a)/b]*FresnelS[(Sqrt[14/Pi]
*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) + (5*d^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSi
n[c*x]])/Sqrt[b]]*Sin[a/b])/(16*b^(3/2)*c^3) - (d^2*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]
])/Sqrt[b]]*Sin[(3*a)/b])/(16*b^(3/2)*c^3) - (3*d^2*Sqrt[(5*Pi)/2]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x
]])/Sqrt[b]]*Sin[(5*a)/b])/(16*b^(3/2)*c^3) - (d^2*Sqrt[(7*Pi)/2]*FresnelC[(Sqrt[14/Pi]*Sqrt[a + b*ArcSin[c*x]
])/Sqrt[b]]*Sin[(7*a)/b])/(16*b^(3/2)*c^3)

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 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]

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]

Rubi steps

\begin{align*} \int \frac{x^2 \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^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{\left (4 d^2\right ) \int \frac{x \left (1-c^2 x^2\right )^{3/2}}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac{\left (14 c d^2\right ) \int \frac{x^3 \left (1-c^2 x^2\right )^{3/2}}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{\left (4 d^2\right ) \operatorname{Subst}\left (\int \frac{\cos ^4(x) \sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}-\frac{\left (14 d^2\right ) \operatorname{Subst}\left (\int \frac{\cos ^4(x) \sin ^3(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{\left (4 d^2\right ) \operatorname{Subst}\left (\int \left (\frac{\sin (x)}{8 \sqrt{a+b x}}+\frac{3 \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 )}{b c^3}-\frac{\left (14 d^2\right ) \operatorname{Subst}\left (\int \left (\frac{3 \sin (x)}{64 \sqrt{a+b x}}+\frac{3 \sin (3 x)}{64 \sqrt{a+b x}}-\frac{\sin (5 x)}{64 \sqrt{a+b x}}-\frac{\sin (7 x)}{64 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{\left (7 d^2\right ) \operatorname{Subst}\left (\int \frac{\sin (5 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{32 b c^3}+\frac{\left (7 d^2\right ) \operatorname{Subst}\left (\int \frac{\sin (7 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{32 b c^3}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\sin (5 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^3}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 b c^3}-\frac{\left (21 d^2\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{32 b c^3}-\frac{\left (21 d^2\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{32 b c^3}+\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac{2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{\left (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 b c^3}-\frac{\left (21 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 )}{32 b c^3}-\frac{\left (21 d^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 b c^3}+\frac{\left (3 d^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 )}{4 b c^3}+\frac{\left (7 d^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 )}{32 b c^3}+\frac{\left (d^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 )}{4 b c^3}+\frac{\left (7 d^2 \cos \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{7 a}{b}+7 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{32 b c^3}-\frac{\left (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 b c^3}+\frac{\left (21 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 )}{32 b c^3}+\frac{\left (21 d^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 b c^3}-\frac{\left (3 d^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 )}{4 b c^3}-\frac{\left (7 d^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 )}{32 b c^3}-\frac{\left (d^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 )}{4 b c^3}-\frac{\left (7 d^2 \sin \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{7 a}{b}+7 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{32 b c^3}\\ &=-\frac{2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\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 )}{b^2 c^3}-\frac{\left (21 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 )}{16 b^2 c^3}-\frac{\left (21 d^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 b^2 c^3}+\frac{\left (3 d^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 )}{2 b^2 c^3}+\frac{\left (7 d^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 )}{16 b^2 c^3}+\frac{\left (d^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 )}{2 b^2 c^3}+\frac{\left (7 d^2 \cos \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{7 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{16 b^2 c^3}-\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 )}{b^2 c^3}+\frac{\left (21 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 )}{16 b^2 c^3}+\frac{\left (21 d^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 b^2 c^3}-\frac{\left (3 d^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 )}{2 b^2 c^3}-\frac{\left (7 d^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 )}{16 b^2 c^3}-\frac{\left (d^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 )}{2 b^2 c^3}-\frac{\left (7 d^2 \sin \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{7 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{16 b^2 c^3}\\ &=-\frac{2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{5 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 )}{16 b^{3/2} c^3}+\frac{d^2 \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 )}{16 b^{3/2} c^3}+\frac{3 d^2 \sqrt{\frac{5 \pi }{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 )}{16 b^{3/2} c^3}+\frac{d^2 \sqrt{\frac{7 \pi }{2}} \cos \left (\frac{7 a}{b}\right ) S\left (\frac{\sqrt{\frac{14}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{5 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 )}{16 b^{3/2} c^3}-\frac{d^2 \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 )}{16 b^{3/2} c^3}-\frac{3 d^2 \sqrt{\frac{5 \pi }{2}} C\left (\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{5 a}{b}\right )}{16 b^{3/2} c^3}-\frac{d^2 \sqrt{\frac{7 \pi }{2}} C\left (\frac{\sqrt{\frac{14}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{7 a}{b}\right )}{16 b^{3/2} c^3}\\ \end{align*}

Mathematica [C]  time = 2.75583, size = 686, normalized size = 1.34 \[ \frac{d^2 e^{-\frac{7 i \left (a+b \sin ^{-1}(c x)\right )}{b}} \left (5 e^{\frac{6 i a}{b}+7 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+5 e^{\frac{8 i a}{b}+7 i \sin ^{-1}(c x)} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt{3} e^{\frac{4 i a}{b}+7 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt{3} e^{\frac{10 i a}{b}+7 i \sin ^{-1}(c x)} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-3 \sqrt{5} e^{\frac{2 i a}{b}+7 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-3 \sqrt{5} e^{\frac{12 i a}{b}+7 i \sin ^{-1}(c x)} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt{7} e^{7 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{7 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt{7} e^{\frac{7 i \left (2 a+b \sin ^{-1}(c x)\right )}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{7 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+3 e^{\frac{7 i a}{b}+2 i \sin ^{-1}(c x)}+e^{\frac{7 i a}{b}+4 i \sin ^{-1}(c x)}-5 e^{\frac{7 i a}{b}+6 i \sin ^{-1}(c x)}-5 e^{\frac{7 i a}{b}+8 i \sin ^{-1}(c x)}+e^{\frac{7 i a}{b}+10 i \sin ^{-1}(c x)}+3 e^{\frac{7 i a}{b}+12 i \sin ^{-1}(c x)}+e^{\frac{7 i \left (a+2 b \sin ^{-1}(c x)\right )}{b}}+e^{\frac{7 i a}{b}}\right )}{64 b c^3 \sqrt{a+b \sin ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^2*(E^(((7*I)*a)/b) + 3*E^(((7*I)*a)/b + (2*I)*ArcSin[c*x]) + E^(((7*I)*a)/b + (4*I)*ArcSin[c*x]) - 5*E^(((7
*I)*a)/b + (6*I)*ArcSin[c*x]) - 5*E^(((7*I)*a)/b + (8*I)*ArcSin[c*x]) + E^(((7*I)*a)/b + (10*I)*ArcSin[c*x]) +
3*E^(((7*I)*a)/b + (12*I)*ArcSin[c*x]) + E^(((7*I)*(a + 2*b*ArcSin[c*x]))/b) + 5*E^(((6*I)*a)/b + (7*I)*ArcSi
n[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 5*E^(((8*I)*a)/b + (7*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^(((4*I)*a)/b +
(7*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^
(((10*I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c*x]))/b] -
3*Sqrt[5]*E^(((2*I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-5*I)*(a + b*Ar
cSin[c*x]))/b] - 3*Sqrt[5]*E^(((12*I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((5
*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[7]*E^((7*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-7
*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[7]*E^(((7*I)*(2*a + b*ArcSin[c*x]))/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamm
a[1/2, ((7*I)*(a + b*ArcSin[c*x]))/b]))/(64*b*c^3*E^(((7*I)*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*ArcSin[c*x]])

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Maple [A]  time = 0.112, size = 590, normalized size = 1.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32/c^3*d^2/b*(3*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(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)-3*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*5^
(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*sin(5*a/b)-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)+3^(1/2)*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(3*a/b)*FresnelS(2^(1/2)/Pi^(1/
2)*3^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)+(a+b*arcsin(c*x))^(1/2)*2^(1/2)*cos(7*a/b)*FresnelS(2^(1/2)/
Pi^(1/2)*7^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(1/b)^(1/2)*Pi^(1/2)*7^(1/2)-(a+b*arcsin(c*x))^(1/2)*2
^(1/2)*sin(7*a/b)*FresnelC(2^(1/2)/Pi^(1/2)*7^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(1/b)^(1/2)*Pi^(1/2
)*7^(1/2)-5*(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)+5*(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)+3*cos(5*(a+b*arcsin(c*x))/b-5*a/b)+cos(7*(a+b*arcsin(c*x))/b-7
*a/b)-5*cos((a+b*arcsin(c*x))/b-a/b)+cos(3*(a+b*arcsin(c*x))/b-3*a/b))/(a+b*arcsin(c*x))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} d x^{2} - d\right )}^{2} x^{2}}{{\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^2*(-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^2/(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(x^2*(-c^2*d*x^2+d)^2/(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*} d^{2} \left (\int \frac{x^{2}}{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^{4}}{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^{6}}{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**2*(-c**2*d*x**2+d)**2/(a+b*asin(c*x))**(3/2),x)

[Out]

d**2*(Integral(x**2/(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + b*asin(c*x))*asin(c*x)), x) + Integral(-2*c**2*x**4/
(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + b*asin(c*x))*asin(c*x)), x) + Integral(c**4*x**6/(a*sqrt(a + b*asin(c*x)
) + b*sqrt(a + b*asin(c*x))*asin(c*x)), x))

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