3.5.32 \(\int \frac {x^2 (d-c^2 d x^2)}{(a+b \text {ArcSin}(c x))^{3/2}} \, dx\) [432]

Optimal. Leaf size=591 \[ -\frac {2 d x^2 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {ArcSin}(c x)}}-\frac {5 d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^3}+\frac {d \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}-\frac {5 d \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {d \sqrt {\frac {2 \pi }{3}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}+\frac {d \sqrt {\frac {5 \pi }{2}} \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {5 d \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 b^{3/2} c^3}-\frac {d \sqrt {2 \pi } \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} c^3}+\frac {5 d \sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^{3/2} c^3}-\frac {d \sqrt {\frac {2 \pi }{3}} \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{b^{3/2} c^3}-\frac {d \sqrt {\frac {5 \pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{4 b^{3/2} c^3} \]

[Out]

1/8*d*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/b^(3/2)/c^3-1/8*d
*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(3*a/b)*6^(1/2)*Pi^(1/2)/b^(3/2)/c^3-1/4*d*cos(
a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/c^3+1/4*d*FresnelC(2^
(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2)/b^(3/2)/c^3+1/8*d*cos(5*a/b)*Fresnel
S(10^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*10^(1/2)*Pi^(1/2)/b^(3/2)/c^3-1/8*d*FresnelC(10^(1/2)/Pi^
(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(5*a/b)*10^(1/2)*Pi^(1/2)/b^(3/2)/c^3-2*d*x^2*(-c^2*x^2+1)^(3/2)/b/c
/(a+b*arcsin(c*x))^(1/2)

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Rubi [A]
time = 1.06, antiderivative size = 591, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {4799, 4809, 4491, 3387, 3386, 3432, 3385, 3433} \begin {gather*} -\frac {\sqrt {2 \pi } d \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}+\frac {5 \sqrt {\frac {\pi }{2}} d \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^3}-\frac {\sqrt {\frac {2 \pi }{3}} d \sin \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}+\frac {5 \sqrt {\frac {\pi }{6}} d \sin \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}-\frac {\sqrt {\frac {5 \pi }{2}} d \sin \left (\frac {5 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {\sqrt {2 \pi } d \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}-\frac {5 \sqrt {\frac {\pi }{2}} d \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^3}+\frac {\sqrt {\frac {2 \pi }{3}} d \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^3}-\frac {5 \sqrt {\frac {\pi }{6}} d \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {\sqrt {\frac {5 \pi }{2}} d \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}-\frac {2 d x^2 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {ArcSin}(c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d*x^2*(1 - c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSin[c*x]]) - (5*d*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*S
qrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*b^(3/2)*c^3) + (d*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*Arc
Sin[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) - (5*d*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]]
)/Sqrt[b]])/(4*b^(3/2)*c^3) + (d*Sqrt[(2*Pi)/3]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqr
t[b]])/(b^(3/2)*c^3) + (d*Sqrt[(5*Pi)/2]*Cos[(5*a)/b]*FresnelS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])
/(4*b^(3/2)*c^3) + (5*d*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*b^(3/2)
*c^3) - (d*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c^3) + (5*d*Sq
rt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(4*b^(3/2)*c^3) - (d*Sqrt[(2*Pi)
/3]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(3/2)*c^3) - (d*Sqrt[(5*Pi)/2]*Fre
snelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(5*a)/b])/(4*b^(3/2)*c^3)

Rule 3385

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 3386

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 3387

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 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 4491

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 4799

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/(b*c*(n
+ 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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)/(b*f*(n + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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[2*p, 0] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]

Rule 4809

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*c
^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x],
x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m,
0]

Rubi steps

\begin {align*} \int \frac {x^2 \left (d-c^2 d x^2\right )}{\left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d x^2 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {(4 d) \int \frac {x \sqrt {1-c^2 x^2}}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac {(10 c d) \int \frac {x^3 \sqrt {1-c^2 x^2}}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d x^2 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {(4 d) \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}-\frac {(10 d) \text {Subst}\left (\int \frac {\cos ^2(x) \sin ^3(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {2 d x^2 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {(4 d) \text {Subst}\left (\int \left (\frac {\sin (x)}{4 \sqrt {a+b x}}+\frac {\sin (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}-\frac {(10 d) \text {Subst}\left (\int \left (\frac {\sin (x)}{8 \sqrt {a+b x}}+\frac {\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 {2 d x^2 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}-\frac {(5 d) \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}+\frac {(5 d) \text {Subst}\left (\int \frac {\sin (5 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}+\frac {d \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}+\frac {d \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}-\frac {(5 d) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac {2 d x^2 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {\left (d \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}-\frac {\left (5 d \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^3}-\frac {\left (5 d \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}+\frac {\left (d \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}+\frac {\left (5 d \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}+5 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}-\frac {\left (d \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}+\frac {\left (5 d \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^3}+\frac {\left (5 d \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}-\frac {\left (d \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}-\frac {\left (5 d \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {5 a}{b}+5 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}\\ &=-\frac {2 d x^2 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {\left (2 d \cos \left (\frac {a}{b}\right )\right ) \text {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 (5 d \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 b^2 c^3}-\frac {\left (5 d \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^3}+\frac {\left (2 d \cos \left (\frac {3 a}{b}\right )\right ) \text {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 (5 d \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {5 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^3}-\frac {\left (2 d \sin \left (\frac {a}{b}\right )\right ) \text {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 (5 d \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 b^2 c^3}+\frac {\left (5 d \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^3}-\frac {\left (2 d \sin \left (\frac {3 a}{b}\right )\right ) \text {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 {\left (5 d \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {5 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^3}\\ &=-\frac {2 d x^2 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}-\frac {5 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 )}{2 b^{3/2} c^3}+\frac {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^3}-\frac {5 d \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 )}{4 b^{3/2} c^3}+\frac {d \sqrt {\frac {2 \pi }{3}} \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 {d \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 )}{4 b^{3/2} c^3}+\frac {5 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 )}{2 b^{3/2} c^3}-\frac {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^3}+\frac {5 d \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 )}{4 b^{3/2} c^3}-\frac {d \sqrt {\frac {2 \pi }{3}} 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}-\frac {d \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 )}{4 b^{3/2} c^3}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.02, size = 514, normalized size = 0.87 \begin {gather*} \frac {d e^{-\frac {5 i (a+b \text {ArcSin}(c x))}{b}} \left (e^{\frac {5 i a}{b}}+e^{\frac {5 i a}{b}+2 i \text {ArcSin}(c x)}-2 e^{\frac {5 i a}{b}+4 i \text {ArcSin}(c x)}-2 e^{\frac {5 i a}{b}+6 i \text {ArcSin}(c x)}+e^{\frac {5 i a}{b}+8 i \text {ArcSin}(c x)}+e^{\frac {5 i (a+2 b \text {ArcSin}(c x))}{b}}+2 e^{\frac {4 i a}{b}+5 i \text {ArcSin}(c x)} \sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )+2 e^{\frac {6 i a}{b}+5 i \text {ArcSin}(c x)} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {i (a+b \text {ArcSin}(c x))}{b}\right )-\sqrt {3} e^{\frac {2 i a}{b}+5 i \text {ArcSin}(c x)} \sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )-\sqrt {3} e^{\frac {8 i a}{b}+5 i \text {ArcSin}(c x)} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )-\sqrt {5} e^{5 i \text {ArcSin}(c x)} \sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )-\sqrt {5} e^{\frac {5 i (2 a+b \text {ArcSin}(c x))}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )\right )}{16 b c^3 \sqrt {a+b \text {ArcSin}(c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(E^(((5*I)*a)/b) + E^(((5*I)*a)/b + (2*I)*ArcSin[c*x]) - 2*E^(((5*I)*a)/b + (4*I)*ArcSin[c*x]) - 2*E^(((5*I
)*a)/b + (6*I)*ArcSin[c*x]) + E^(((5*I)*a)/b + (8*I)*ArcSin[c*x]) + E^(((5*I)*(a + 2*b*ArcSin[c*x]))/b) + 2*E^
(((4*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b]
+ 2*E^(((6*I)*a)/b + (5*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^(((2*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*Arc
Sin[c*x]))/b] - Sqrt[3]*E^(((8*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((3*I)*
(a + b*ArcSin[c*x]))/b] - Sqrt[5]*E^((5*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-5*I)*
(a + b*ArcSin[c*x]))/b] - Sqrt[5]*E^(((5*I)*(2*a + b*ArcSin[c*x]))/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/
2, ((5*I)*(a + b*ArcSin[c*x]))/b]))/(16*b*c^3*E^(((5*I)*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*ArcSin[c*x]])

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Maple [A]
time = 0.36, size = 447, normalized size = 0.76

method	result	size
default	\(-\frac {d \left (\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {5 a}{b}\right ) \mathrm {S}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right )+\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {5 a}{b}\right ) \FresnelC \left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right )-2 \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}-2 \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}+\sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}+\sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}+2 \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right )-\cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right )-\cos \left (-\frac {5 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {5 a}{b}\right )\right )}{8 c^{3} b \sqrt {a +b \arcsin \left (c x \right )}}\)	\(447\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/c^3*d/b*(2^(1/2)*Pi^(1/2)*(-5/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(5*a/b)*FresnelS(5*2^(1/2)/Pi^(1/2)/(-5
/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)+2^(1/2)*Pi^(1/2)*(-5/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(5*a/b)*FresnelC
(5*2^(1/2)/Pi^(1/2)/(-5/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)-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)*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)-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)*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)+(a+b*arc
sin(c*x))^(1/2)*cos(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(-3/b)^(1/2)*Pi
^(1/2)*2^(1/2)+(a+b*arcsin(c*x))^(1/2)*sin(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(
1/2)/b)*(-3/b)^(1/2)*Pi^(1/2)*2^(1/2)+2*cos(-(a+b*arcsin(c*x))/b+a/b)-cos(-3*(a+b*arcsin(c*x))/b+3*a/b)-cos(-5
*(a+b*arcsin(c*x))/b+5*a/b))/(a+b*arcsin(c*x))^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - d \left (\int \left (- \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 )}}\right )\, dx + \int \frac {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\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,\left (d-c^2\,d\,x^2\right )}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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