\(\int (a+b \arccos (c x))^{3/2} \, dx\) [180]

Optimal result

Integrand size = 12, antiderivative size = 159 \[ \int (a+b \arccos (c x))^{3/2} \, dx=-\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2}+\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c} \] Output:

-3/2*b*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^(1/2)/c+x*(a+b*arccos(c*x))^(3 
/2)+3/4*b^(3/2)*2^(1/2)*Pi^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*a 
rccos(c*x))^(1/2)/b^(1/2))/c-3/4*b^(3/2)*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2) 
/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(a/b)/c
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.50 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.82 \[ \int (a+b \arccos (c x))^{3/2} \, dx=\frac {\sqrt {b} \left (2 \sqrt {b} \sqrt {a+b \arccos (c x)} \left (-3 \sqrt {1-c^2 x^2}+2 c x \arccos (c x)\right )+\frac {2 i a \sqrt {b} e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arccos (c x))}{b}\right )-e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arccos (c x))}{b}\right )\right )}{\sqrt {a+b \arccos (c x)}}+\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )+\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )\right )}{4 c} \] Input:

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

Output:

(Sqrt[b]*(2*Sqrt[b]*Sqrt[a + b*ArcCos[c*x]]*(-3*Sqrt[1 - c^2*x^2] + 2*c*x* 
ArcCos[c*x]) + ((2*I)*a*Sqrt[b]*(Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[ 
3/2, ((-I)*(a + b*ArcCos[c*x]))/b] - E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcCos 
[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b]))/(E^((I*a)/b)*Sqrt[a + b 
*ArcCos[c*x]]) + Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/ 
Sqrt[b]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) + Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*S 
qrt[a + b*ArcCos[c*x]])/Sqrt[b]]*(2*a*Cos[a/b] - 3*b*Sin[a/b])))/(4*c)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5131, 5183, 5135, 25, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833}

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.

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

\(\Big \downarrow \) 5131

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 5135

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3787

\(\displaystyle \frac {3}{2} b c \left (\frac {-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\cos \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )+x (a+b \arccos (c x))^{3/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{2} b c \left (\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )+x (a+b \arccos (c x))^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{2} b c \left (\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )+x (a+b \arccos (c x))^{3/2}\)

\(\Big \downarrow \) 3785

\(\displaystyle \frac {3}{2} b c \left (\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )+x (a+b \arccos (c x))^{3/2}\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {3}{2} b c \left (\frac {2 \cos \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )+x (a+b \arccos (c x))^{3/2}\)

\(\Big \downarrow \) 3832

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

\(\Big \downarrow \) 3833

\(\displaystyle \frac {3}{2} b c \left (\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )+x (a+b \arccos (c x))^{3/2}\)

Input:

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

Output:

x*(a + b*ArcCos[c*x])^(3/2) + (3*b*c*(-((Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcC 
os[c*x]])/c^2) + (Sqrt[b]*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a 
+ b*ArcCos[c*x]])/Sqrt[b]] - Sqrt[b]*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[ 
a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c^2)))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

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

rule 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[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 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 3787

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos 
[(d*e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( 
d*e - c*f)/d]   Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d 
, e, f}, x] && ComplexFreeQ[f] && NeQ[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 3833

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 5131

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(277\) vs. \(2(123)=246\).

Time = 0.00 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.75

method	result	size
default	\(\frac {-3 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b^{2}-3 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b^{2}+4 \arccos \left (c x \right )^{2} \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}+8 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a b +6 \arccos \left (c x \right ) \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}+4 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2}+6 \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a b}{4 c \sqrt {a +b \arccos \left (c x \right )}}\)	\(278\)

Input:

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

Output:

1/4/c/(a+b*arccos(c*x))^(1/2)*(-3*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcco 
s(c*x))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos( 
c*x))^(1/2)/b)*b^2-3*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2) 
*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b 
)*b^2+4*arccos(c*x)^2*cos(-(a+b*arccos(c*x))/b+a/b)*b^2+8*arccos(c*x)*cos( 
-(a+b*arccos(c*x))/b+a/b)*a*b+6*arccos(c*x)*sin(-(a+b*arccos(c*x))/b+a/b)* 
b^2+4*cos(-(a+b*arccos(c*x))/b+a/b)*a^2+6*sin(-(a+b*arccos(c*x))/b+a/b)*a* 
b)
 

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

integrate((a+b*acos(c*x))**(3/2),x)
 

Output:

Integral((a + b*acos(c*x))**(3/2), x)
 

Maxima [F]

\[ \int (a+b \arccos (c x))^{3/2} \, dx=\int { {\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}} \,d x } \] Input:

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

Output:

integrate((b*arccos(c*x) + a)^(3/2), x)
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.09 (sec) , antiderivative size = 1159, normalized size of antiderivative = 7.29 \[ \int (a+b \arccos (c x))^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

-I*sqrt(2)*sqrt(pi)*a^2*b^2*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqr 
t(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/ 
((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/2*sqrt(2)*sqrt(pi)*a*b^3*e 
rf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt( 
b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sq 
rt(abs(b)))*c) + I*sqrt(2)*sqrt(pi)*a^2*b^2*erf(1/2*I*sqrt(2)*sqrt(b*arcco 
s(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b) 
)/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/2*sqrt(2) 
*sqrt(pi)*a*b^3*erf(1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1 
/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqr 
t(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/2*I*sqrt(2)*sqrt(pi)*a^2*b*erf(-1/2*I 
*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos( 
c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) 
*c) - 1/2*sqrt(2)*sqrt(pi)*a*b^2*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a 
)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I* 
a/b)/((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c) - 3/8*I*sqrt(2)*sqrt(pi)*b^ 
3*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sq 
rt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b)) + b*s 
qrt(abs(b)))*c) - 1/2*I*sqrt(2)*sqrt(pi)*a^2*b*erf(1/2*I*sqrt(2)*sqrt(b*ar 
ccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(...
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (a+b \arccos (c x))^{3/2} \, dx=\left (\int \sqrt {\mathit {acos} \left (c x \right ) b +a}d x \right ) a +\left (\int \sqrt {\mathit {acos} \left (c x \right ) b +a}\, \mathit {acos} \left (c x \right )d x \right ) b \] Input:

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

Output:

int(sqrt(acos(c*x)*b + a),x)*a + int(sqrt(acos(c*x)*b + a)*acos(c*x),x)*b