3.5.42 \(\int (c-a^2 c x^2)^{3/2} \sqrt {\text {ArcSin}(a x)} \, dx\) [442]

Optimal. Leaf size=227 \[ \frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\text {ArcSin}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\text {ArcSin}(a x)}+\frac {c \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}(a x)}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{8 a \sqrt {1-a^2 x^2}} \]

[Out]

1/4*c*arcsin(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-1/128*c*FresnelS(2*2^(1/2)/Pi^(1/2)*arcsin(a
*x)^(1/2))*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-1/8*c*FresnelS(2*arcsin(a*x)^(1/2)/Pi^(1
/2))*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)+1/4*x*(-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(1/2)+3/8*c*x*(
-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4743, 4741, 4737, 4731, 4491, 12, 3386, 3432, 4809} \begin {gather*} -\frac {\sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}(a x)}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {\sqrt {\pi } c \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{4} x \sqrt {\text {ArcSin}(a x)} \left (c-a^2 c x^2\right )^{3/2}+\frac {c \text {ArcSin}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}+\frac {3}{8} c x \sqrt {\text {ArcSin}(a x)} \sqrt {c-a^2 c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*c*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/8 + (x*(c - a^2*c*x^2)^(3/2)*Sqrt[ArcSin[a*x]])/4 + (c*Sqrt[c -
a^2*c*x^2]*ArcSin[a*x]^(3/2))/(4*a*Sqrt[1 - a^2*x^2]) - (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*FresnelS[2*Sqrt[2/Pi
]*Sqrt[ArcSin[a*x]]])/(64*a*Sqrt[1 - a^2*x^2]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2*Sqrt[ArcSin[a*x]]
)/Sqrt[Pi]])/(8*a*Sqrt[1 - a^2*x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 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 4731

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

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4741

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*((
a + b*ArcSin[c*x])^n/2), x] + (Dist[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[x*(a + b*ArcSin[c*x])^(
n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4743

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

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 \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)} \, dx &=\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} (3 c) \int \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)} \, dx-\frac {\left (a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\sin ^{-1}(a x)}} \, dx}{8 \sqrt {1-a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{8 \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a \sqrt {1-a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a \sqrt {1-a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{16 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a \sqrt {1-a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{16 a \sqrt {1-a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.15, size = 166, normalized size = 0.73 \begin {gather*} \frac {c \sqrt {c-a^2 c x^2} \left (32 \text {ArcSin}(a x)^2+8 \sqrt {2} \sqrt {-i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {3}{2},-2 i \text {ArcSin}(a x)\right )+8 \sqrt {2} \sqrt {i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {3}{2},2 i \text {ArcSin}(a x)\right )+\sqrt {-i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {3}{2},-4 i \text {ArcSin}(a x)\right )+\sqrt {i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {3}{2},4 i \text {ArcSin}(a x)\right )\right )}{128 a \sqrt {1-a^2 x^2} \sqrt {\text {ArcSin}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*Sqrt[c - a^2*c*x^2]*(32*ArcSin[a*x]^2 + 8*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (-2*I)*ArcSin[a*x]] + 8
*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (2*I)*ArcSin[a*x]] + Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (-4*I)*ArcSin[a
*x]] + Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (4*I)*ArcSin[a*x]]))/(128*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])

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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\arcsin \left (a x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(1/2),x)

[Out]

int((-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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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((-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(1/2),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*} \int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {\operatorname {asin}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**(3/2)*asin(a*x)**(1/2),x)

[Out]

Integral((-c*(a*x - 1)*(a*x + 1))**(3/2)*sqrt(asin(a*x)), x)

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Giac [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((-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)^(1/2)*(c - a^2*c*x^2)^(3/2),x)

[Out]

int(asin(a*x)^(1/2)*(c - a^2*c*x^2)^(3/2), x)

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