3.452 \(\int \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2} \, dx\)

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.249361, antiderivative size = 247, normalized size of antiderivative = 1., 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 = {4647, 4641, 4629, 4707, 4635, 4406, 12, 3305, 3351} \[ \frac{15 \sqrt{\pi } \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}-\frac{5 a x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt{1-a^2 x^2}}+\frac{5 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt{1-a^2 x^2}}-\frac{15}{32} x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \]

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 4647

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[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcSin[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(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 4641

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

Rule 4629

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 4707

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

Rule 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 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 12

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

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]

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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [C]  time = 0.123455, size = 158, normalized size = 0.64 \[ \frac{\sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \left (35 i \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-2 i \sin ^{-1}(a x)\right )-35 i \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},2 i \sin ^{-1}(a x)\right )+64 \left (7 a x \sqrt{1-a^2 x^2}+2 \sin ^{-1}(a x)\right ) \left (\sin ^{-1}(a x)^2\right )^{3/2}\right )}{896 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[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])

________________________________________________________________________________________

Maple [F]  time = 0.234, size = 0, normalized size = 0. \begin{align*} \int \sqrt{-{a}^{2}c{x}^{2}+c} \left ( \arcsin \left ( ax \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

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)

________________________________________________________________________________________

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

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

________________________________________________________________________________________

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{\frac{5}{2}}\,{d x} \end{align*}

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]

integrate(sqrt(-a^2*c*x^2 + c)*arcsin(a*x)^(5/2), x)