\(\int x^2 \arcsin (a x)^{3/2} \, dx\) [82]

Optimal result

Integrand size = 12, antiderivative size = 147 \[ \int x^2 \arcsin (a x)^{3/2} \, dx=\frac {\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{3 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{6 a}+\frac {1}{3} x^3 \arcsin (a x)^{3/2}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{24 a^3} \]

[Out]

1/3*x^3*arcsin(a*x)^(3/2)+1/144*FresnelC(6^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^3-3/16*Fresnel
C(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^3+1/3*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^(1/2)/a^3+1/6*x^
2*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^(1/2)/a

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4725, 4795, 4767, 4719, 3385, 3433, 4731, 4491} \[ \int x^2 \arcsin (a x)^{3/2} \, dx=-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{24 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{6 a}+\frac {\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{3 a^3}+\frac {1}{3} x^3 \arcsin (a x)^{3/2} \]

[In]

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

[Out]

(Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(3*a^3) + (x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(6*a) + (x^3*ArcSin[
a*x]^(3/2))/3 - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^3) + (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi
]*Sqrt[ArcSin[a*x]]])/(24*a^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 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 4719

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

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 4767

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

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*} \text {integral}& = \frac {1}{3} x^3 \arcsin (a x)^{3/2}-\frac {1}{2} a \int \frac {x^3 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{6 a}+\frac {1}{3} x^3 \arcsin (a x)^{3/2}-\frac {1}{12} \int \frac {x^2}{\sqrt {\arcsin (a x)}} \, dx-\frac {\int \frac {x \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx}{3 a} \\ & = \frac {\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{3 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{6 a}+\frac {1}{3} x^3 \arcsin (a x)^{3/2}-\frac {\text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{12 a^3}-\frac {\int \frac {1}{\sqrt {\arcsin (a x)}} \, dx}{6 a^2} \\ & = \frac {\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{3 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{6 a}+\frac {1}{3} x^3 \arcsin (a x)^{3/2}-\frac {\text {Subst}\left (\int \left (\frac {\cos (x)}{4 \sqrt {x}}-\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arcsin (a x)\right )}{12 a^3}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{6 a^3} \\ & = \frac {\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{3 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{6 a}+\frac {1}{3} x^3 \arcsin (a x)^{3/2}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{48 a^3}+\frac {\text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{48 a^3}-\frac {\text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{3 a^3} \\ & = \frac {\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{3 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{6 a}+\frac {1}{3} x^3 \arcsin (a x)^{3/2}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^3}-\frac {\text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{24 a^3}+\frac {\text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{24 a^3} \\ & = \frac {\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{3 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{6 a}+\frac {1}{3} x^3 \arcsin (a x)^{3/2}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{24 a^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.06 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.93 \[ \int x^2 \arcsin (a x)^{3/2} \, dx=\frac {\sqrt {\arcsin (a x)} \left (27 \sqrt {i \arcsin (a x)} \Gamma \left (\frac {5}{2},-i \arcsin (a x)\right )+27 \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {5}{2},i \arcsin (a x)\right )-\sqrt {3} \left (\sqrt {i \arcsin (a x)} \Gamma \left (\frac {5}{2},-3 i \arcsin (a x)\right )+\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {5}{2},3 i \arcsin (a x)\right )\right )\right )}{216 a^3 \sqrt {\arcsin (a x)^2}} \]

[In]

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

[Out]

(Sqrt[ArcSin[a*x]]*(27*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-I)*ArcSin[a*x]] + 27*Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2,
I*ArcSin[a*x]] - Sqrt[3]*(Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-3*I)*ArcSin[a*x]] + Sqrt[(-I)*ArcSin[a*x]]*Gamma[5
/2, (3*I)*ArcSin[a*x]])))/(216*a^3*Sqrt[ArcSin[a*x]^2])

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {-36 a x \arcsin \left (a x \right )^{2}-\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+12 \arcsin \left (a x \right )^{2} \sin \left (3 \arcsin \left (a x \right )\right )+27 \,\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+6 \arcsin \left (a x \right ) \cos \left (3 \arcsin \left (a x \right )\right )-54 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{144 a^{3} \sqrt {\arcsin \left (a x \right )}}\)	\(131\)

[In]

int(x^2*arcsin(a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/144/a^3/arcsin(a*x)^(1/2)*(-36*a*x*arcsin(a*x)^2-FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))*3^(1/
2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)+12*arcsin(a*x)^2*sin(3*arcsin(a*x))+27*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(
a*x)^(1/2))*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)+6*arcsin(a*x)*cos(3*arcsin(a*x))-54*arcsin(a*x)*(-a^2*x^2+1)^(1
/2))

Fricas [F(-2)]

Exception generated. \[ \int x^2 \arcsin (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^2*arcsin(a*x)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int x^2 \arcsin (a x)^{3/2} \, dx=\int x^{2} \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}\, dx \]

[In]

integrate(x**2*asin(a*x)**(3/2),x)

[Out]

Integral(x**2*asin(a*x)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int x^2 \arcsin (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.61 \[ \int x^2 \arcsin (a x)^{3/2} \, dx=\frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{24 \, a^{3}} - \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arcsin \left (a x\right )\right )}}{8 \, a^{3}} + \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{8 \, a^{3}} - \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{24 \, a^{3}} - \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{576 \, a^{3}} + \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{576 \, a^{3}} + \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{64 \, a^{3}} - \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{64 \, a^{3}} - \frac {\sqrt {\arcsin \left (a x\right )} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{48 \, a^{3}} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{16 \, a^{3}} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{16 \, a^{3}} - \frac {\sqrt {\arcsin \left (a x\right )} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{48 \, a^{3}} \]

[In]

integrate(x^2*arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

1/24*I*arcsin(a*x)^(3/2)*e^(3*I*arcsin(a*x))/a^3 - 1/8*I*arcsin(a*x)^(3/2)*e^(I*arcsin(a*x))/a^3 + 1/8*I*arcsi
n(a*x)^(3/2)*e^(-I*arcsin(a*x))/a^3 - 1/24*I*arcsin(a*x)^(3/2)*e^(-3*I*arcsin(a*x))/a^3 - (1/576*I + 1/576)*sq
rt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^3 + (1/576*I - 1/576)*sqrt(6)*sqrt(pi)*erf(-(1/2
*I + 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^3 + (3/64*I + 3/64)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arc
sin(a*x)))/a^3 - (3/64*I - 3/64)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^3 - 1/48*sqr
t(arcsin(a*x))*e^(3*I*arcsin(a*x))/a^3 + 3/16*sqrt(arcsin(a*x))*e^(I*arcsin(a*x))/a^3 + 3/16*sqrt(arcsin(a*x))
*e^(-I*arcsin(a*x))/a^3 - 1/48*sqrt(arcsin(a*x))*e^(-3*I*arcsin(a*x))/a^3

Mupad [F(-1)]

Timed out. \[ \int x^2 \arcsin (a x)^{3/2} \, dx=\int x^2\,{\mathrm {asin}\left (a\,x\right )}^{3/2} \,d x \]

[In]

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

[Out]

int(x^2*asin(a*x)^(3/2), x)