3.5.52 \(\int \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^{5/2} \, dx\) [452]

Optimal. Leaf size=247 \[ -\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\text {ArcSin}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}} \]

[Out]

1/2*x*arcsin(a*x)^(5/2)*(-a^2*c*x^2+c)^(1/2)+5/16*arcsin(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-
5/8*a*x^2*arcsin(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)+1/7*arcsin(a*x)^(7/2)*(-a^2*c*x^2+c)^(1/2)
/a/(-a^2*x^2+1)^(1/2)+15/128*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)-15/32*x*(-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4741, 4737, 4725, 4795, 4731, 4491, 12, 3386, 3432} \begin {gather*} \frac {15 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}}+\frac {\text {ArcSin}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \text {ArcSin}(a x)^{5/2} \sqrt {c-a^2 c x^2}-\frac {5 a x^2 \text {ArcSin}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{8 \sqrt {1-a^2 x^2}}+\frac {5 \text {ArcSin}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{16 a \sqrt {1-a^2 x^2}}-\frac {15}{32} x \sqrt {\text {ArcSin}(a x)} \sqrt {c-a^2 c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(5/2),x]

[Out]

(-15*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/32 + (5*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/(16*a*Sqrt[1 - a^
2*x^2]) - (5*a*x^2*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/(8*Sqrt[1 - a^2*x^2]) + (x*Sqrt[c - a^2*c*x^2]*ArcSi
n[a*x]^(5/2))/2 + (Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(7/2))/(7*a*Sqrt[1 - a^2*x^2]) + (15*Sqrt[Pi]*Sqrt[c - a^2*
c*x^2]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(128*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 4725

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcSin[c*x])^n/(m
+ 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 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 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps

\begin {align*} \int \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2} \, dx &=\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \int \frac {\sin ^{-1}(a x)^{5/2}}{\sqrt {1-a^2 x^2}} \, dx}{2 \sqrt {1-a^2 x^2}}-\frac {\left (5 a \sqrt {c-a^2 c x^2}\right ) \int x \sin ^{-1}(a x)^{3/2} \, dx}{4 \sqrt {1-a^2 x^2}}\\ &=-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 a^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{32 \sqrt {1-a^2 x^2}}+\frac {\left (15 a \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx}{64 \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \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 )}{64 a \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \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 )}{64 a \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \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 )}{128 a \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \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 )}{64 a \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.08, size = 158, normalized size = 0.64 \begin {gather*} \frac {\sqrt {c-a^2 c x^2} \sqrt {\text {ArcSin}(a x)} \left (64 \left (\text {ArcSin}(a x)^2\right )^{3/2} \left (7 a x \sqrt {1-a^2 x^2}+2 \text {ArcSin}(a x)\right )+35 i \sqrt {2} \sqrt {i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {5}{2},-2 i \text {ArcSin}(a x)\right )-35 i \sqrt {2} \sqrt {-i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {5}{2},2 i \text {ArcSin}(a x)\right )\right )}{896 a \sqrt {1-a^2 x^2} \sqrt {\text {ArcSin}(a x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(5/2),x]

[Out]

(Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]]*(64*(ArcSin[a*x]^2)^(3/2)*(7*a*x*Sqrt[1 - a^2*x^2] + 2*ArcSin[a*x]) + (
35*I)*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-2*I)*ArcSin[a*x]] - (35*I)*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma
[5/2, (2*I)*ArcSin[a*x]]))/(896*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]^2])

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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \sqrt {-a^{2} c \,x^{2}+c}\, \arcsin \left (a x \right )^{\frac {5}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(5/2),x)

[Out]

int((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(5/2),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)^(1/2)*arcsin(a*x)^(5/2),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)^(1/2)*arcsin(a*x)^(5/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**(1/2)*asin(a*x)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4368 deep

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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)^(1/2)*arcsin(a*x)^(5/2),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 {\mathrm {asin}\left (a\,x\right )}^{5/2}\,\sqrt {c-a^2\,c\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)^(5/2)*(c - a^2*c*x^2)^(1/2),x)

[Out]

int(asin(a*x)^(5/2)*(c - a^2*c*x^2)^(1/2), x)

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