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

3.1.82.1 Optimal result

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

output

1/3*x^3*arccos(a*x)^(3/2)+1/144*FresnelS(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2 
))*6^(1/2)*Pi^(1/2)/a^3+3/16*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))* 
2^(1/2)*Pi^(1/2)/a^3-1/3*(-a^2*x^2+1)^(1/2)*arccos(a*x)^(1/2)/a^3-1/6*x^2* 
(-a^2*x^2+1)^(1/2)*arccos(a*x)^(1/2)/a
 

3.1.82.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.85 \[ \int x^2 \arccos (a x)^{3/2} \, dx=-\frac {27 \sqrt {-i \arccos (a x)} \Gamma \left (\frac {5}{2},-i \arccos (a x)\right )+27 \sqrt {i \arccos (a x)} \Gamma \left (\frac {5}{2},i \arccos (a x)\right )+\sqrt {3} \left (\sqrt {-i \arccos (a x)} \Gamma \left (\frac {5}{2},-3 i \arccos (a x)\right )+\sqrt {i \arccos (a x)} \Gamma \left (\frac {5}{2},3 i \arccos (a x)\right )\right )}{216 a^3 \sqrt {\arccos (a x)}} \]

input

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

output

-1/216*(27*Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-I)*ArcCos[a*x]] + 27*Sqrt[I 
*ArcCos[a*x]]*Gamma[5/2, I*ArcCos[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcCos[a*x]]* 
Gamma[5/2, (-3*I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*Gamma[5/2, (3*I)*ArcC 
os[a*x]]))/(a^3*Sqrt[ArcCos[a*x]])
 

3.1.82.3 Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.31, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5141, 5211, 5147, 4906, 2009, 5183, 5135, 3042, 3786, 3832}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 \arccos (a x)^{3/2} \, dx\)

\(\Big \downarrow \) 5141

\(\displaystyle \frac {1}{2} a \int \frac {x^3 \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx+\frac {1}{3} x^3 \arccos (a x)^{3/2}\)

\(\Big \downarrow \) 5211

\(\displaystyle \frac {1}{2} a \left (\frac {2 \int \frac {x \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {\int \frac {x^2}{\sqrt {\arccos (a x)}}dx}{6 a}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )+\frac {1}{3} x^3 \arccos (a x)^{3/2}\)

\(\Big \downarrow \) 5147

\(\displaystyle \frac {1}{2} a \left (\frac {2 \int \frac {x \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int \frac {a^2 x^2 \sqrt {1-a^2 x^2}}{\sqrt {\arccos (a x)}}d\arccos (a x)}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )+\frac {1}{3} x^3 \arccos (a x)^{3/2}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {1}{2} a \left (\frac {2 \int \frac {x \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int \left (\frac {\sin (3 \arccos (a x))}{4 \sqrt {\arccos (a x)}}+\frac {\sqrt {1-a^2 x^2}}{4 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )+\frac {1}{3} x^3 \arccos (a x)^{3/2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} a \left (\frac {2 \int \frac {x \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )+\frac {1}{3} x^3 \arccos (a x)^{3/2}\)

\(\Big \downarrow \) 5183

\(\displaystyle \frac {1}{2} a \left (\frac {2 \left (-\frac {\int \frac {1}{\sqrt {\arccos (a x)}}dx}{2 a}-\frac {\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{a^2}\right )}{3 a^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )+\frac {1}{3} x^3 \arccos (a x)^{3/2}\)

\(\Big \downarrow \) 5135

\(\displaystyle \frac {1}{2} a \left (\frac {2 \left (\frac {\int \frac {\sqrt {1-a^2 x^2}}{\sqrt {\arccos (a x)}}d\arccos (a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{a^2}\right )}{3 a^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )+\frac {1}{3} x^3 \arccos (a x)^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} a \left (\frac {2 \left (\frac {\int \frac {\sin (\arccos (a x))}{\sqrt {\arccos (a x)}}d\arccos (a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{a^2}\right )}{3 a^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )+\frac {1}{3} x^3 \arccos (a x)^{3/2}\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {1}{2} a \left (\frac {2 \left (\frac {\int \sqrt {1-a^2 x^2}d\sqrt {\arccos (a x)}}{a^2}-\frac {\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{a^2}\right )}{3 a^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )+\frac {1}{3} x^3 \arccos (a x)^{3/2}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {1}{2} a \left (\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^4}+\frac {2 \left (\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^2}-\frac {\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{3 a^2}\right )+\frac {1}{3} x^3 \arccos (a x)^{3/2}\)

input

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

output

(x^3*ArcCos[a*x]^(3/2))/3 + (a*(-1/3*(x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a* 
x]])/a^2 + (2*(-((Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/a^2) + (Sqrt[Pi/2]* 
FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a^2))/(3*a^2) + ((Sqrt[Pi/2]*Fresn 
elS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/2 + (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqr 
t[ArcCos[a*x]]])/2)/(6*a^4)))/2
 

3.1.82.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[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 3832

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 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG 
tQ[p, 0]
 

rule 5135

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

rule 5141

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

rule 5147

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

rule 5183

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 
1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   I 
nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[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 5211

Int[((a_.) + ArcCos[(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*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p 
 + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S 
imp[b*f*(n/(c*(m + 2*p + 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*ArcCos[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]
 

3.1.82.4 Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {36 \arccos \left (a x \right )^{2} a x +\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }+12 \arccos \left (a x \right )^{2} \cos \left (3 \arccos \left (a x \right )\right )+27 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }-54 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}-6 \arccos \left (a x \right ) \sin \left (3 \arccos \left (a x \right )\right )}{144 a^{3} \sqrt {\arccos \left (a x \right )}}\)	\(130\)

input

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

output

1/144/a^3*(36*arccos(a*x)^2*a*x+FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a 
*x)^(1/2))*3^(1/2)*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)+12*arccos(a*x)^2*cos 
(3*arccos(a*x))+27*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*ar 
ccos(a*x)^(1/2)*Pi^(1/2)-54*arccos(a*x)*(-a^2*x^2+1)^(1/2)-6*arccos(a*x)*s 
in(3*arccos(a*x)))/arccos(a*x)^(1/2)
 

3.1.82.5 Fricas [F(-2)]

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

input

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

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

3.1.82.6 Sympy [F]

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

input

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

output

Integral(x**2*acos(a*x)**(3/2), x)
 

3.1.82.7 Maxima [F(-2)]

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

input

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

output

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

3.1.82.8 Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.61 \[ \int x^2 \arccos (a x)^{3/2} \, dx=\frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{576 \, a^{3}} - \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{576 \, a^{3}} + \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{3}} - \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{3}} + \frac {i \, \sqrt {\arccos \left (a x\right )} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{48 \, a^{3}} + \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{16 \, a^{3}} - \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{16 \, a^{3}} - \frac {i \, \sqrt {\arccos \left (a x\right )} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{48 \, a^{3}} \]

input

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

output

1/24*arccos(a*x)^(3/2)*e^(3*I*arccos(a*x))/a^3 + 1/8*arccos(a*x)^(3/2)*e^( 
I*arccos(a*x))/a^3 + 1/8*arccos(a*x)^(3/2)*e^(-I*arccos(a*x))/a^3 + 1/24*a 
rccos(a*x)^(3/2)*e^(-3*I*arccos(a*x))/a^3 + (1/576*I - 1/576)*sqrt(6)*sqrt 
(pi)*erf((1/2*I - 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^3 - (1/576*I + 1/576)* 
sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^3 + (3/64 
*I - 3/64)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a 
^3 - (3/64*I + 3/64)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcc 
os(a*x)))/a^3 + 1/48*I*sqrt(arccos(a*x))*e^(3*I*arccos(a*x))/a^3 + 3/16*I* 
sqrt(arccos(a*x))*e^(I*arccos(a*x))/a^3 - 3/16*I*sqrt(arccos(a*x))*e^(-I*a 
rccos(a*x))/a^3 - 1/48*I*sqrt(arccos(a*x))*e^(-3*I*arccos(a*x))/a^3
 

3.1.82.9 Mupad [F(-1)]

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

input

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

output

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