3.2.85 \(\int (a+b \arccos (c x))^{5/2} \, dx\) [185]

3.2.85.1 Optimal result

Integrand size = 12, antiderivative size = 179 \[ \int (a+b \arccos (c x))^{5/2} \, dx=-\frac {15}{4} b^2 x \sqrt {a+b \arccos (c x)}-\frac {5 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c}+x (a+b \arccos (c x))^{5/2}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{4 c}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 c} \]

output

x*(a+b*arccos(c*x))^(5/2)+15/8*b^(5/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)* 
(a+b*arccos(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/c+15/8*b^(5/2)*FresnelS( 
2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2 
)/c-5/2*b*(a+b*arccos(c*x))^(3/2)*(-c^2*x^2+1)^(1/2)/c-15/4*b^2*x*(a+b*arc 
cos(c*x))^(1/2)
 

3.2.85.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.40 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.08 \[ \int (a+b \arccos (c x))^{5/2} \, dx=\frac {\sqrt {b} e^{-\frac {i a}{b}} \left (\left (4 a^2+15 b^2\right ) \left (1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arccos (c x)} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-i \left (4 a^2+15 b^2\right ) \left (-1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arccos (c x)} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-4 \sqrt {b} \left (e^{\frac {i a}{b}} (a+b \arccos (c x)) \left (5 \left (3 b c x+2 a \sqrt {1-c^2 x^2}\right )+\left (-8 a c x+10 b \sqrt {1-c^2 x^2}\right ) \arccos (c x)-4 b c x \arccos (c x)^2\right )-2 i a^2 \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arccos (c x))}{b}\right )+2 i a^2 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arccos (c x))}{b}\right )\right )\right )}{16 c \sqrt {a+b \arccos (c x)}} \]

input

Integrate[(a + b*ArcCos[c*x])^(5/2),x]
 

output

(Sqrt[b]*((4*a^2 + 15*b^2)*(1 + E^(((2*I)*a)/b))*Sqrt[2*Pi]*Sqrt[a + b*Arc 
Cos[c*x]]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]] - I*(4*a^ 
2 + 15*b^2)*(-1 + E^(((2*I)*a)/b))*Sqrt[2*Pi]*Sqrt[a + b*ArcCos[c*x]]*Fres 
nelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]] - 4*Sqrt[b]*(E^((I*a)/b 
)*(a + b*ArcCos[c*x])*(5*(3*b*c*x + 2*a*Sqrt[1 - c^2*x^2]) + (-8*a*c*x + 1 
0*b*Sqrt[1 - c^2*x^2])*ArcCos[c*x] - 4*b*c*x*ArcCos[c*x]^2) - (2*I)*a^2*Sq 
rt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcCos[c*x]))/b] 
+ (2*I)*a^2*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, (I* 
(a + b*ArcCos[c*x]))/b])))/(16*c*E^((I*a)/b)*Sqrt[a + b*ArcCos[c*x]])
 

3.2.85.3 Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5131, 5183, 5131, 5225, 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.

\(\displaystyle \int (a+b \arccos (c x))^{5/2} \, dx\)

\(\Big \downarrow \) 5131

\(\displaystyle \frac {5}{2} b c \int \frac {x (a+b \arccos (c x))^{3/2}}{\sqrt {1-c^2 x^2}}dx+x (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 5183

\(\displaystyle \frac {5}{2} b c \left (-\frac {3 b \int \sqrt {a+b \arccos (c x)}dx}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )+x (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 5131

\(\displaystyle \frac {5}{2} b c \left (-\frac {3 b \left (\frac {1}{2} b c \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx+x \sqrt {a+b \arccos (c x)}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )+x (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 5225

\(\displaystyle \frac {5}{2} b c \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )+x (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5}{2} b c \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )+x (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3787

\(\displaystyle \frac {5}{2} b c \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )+x (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {5}{2} b c \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )+x (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5}{2} b c \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )+x (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3785

\(\displaystyle \frac {5}{2} b c \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )+x (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {5}{2} b c \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {2 \sin \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )+x (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {5}{2} b c \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )+x (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {5}{2} b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}\right )}{2 c}\right )+x (a+b \arccos (c x))^{5/2}\)

input

Int[(a + b*ArcCos[c*x])^(5/2),x]
 

output

x*(a + b*ArcCos[c*x])^(5/2) + (5*b*c*(-((Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c 
*x])^(3/2))/c^2) - (3*b*(x*Sqrt[a + b*ArcCos[c*x]] - (Sqrt[b]*Sqrt[2*Pi]*C 
os[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]] + Sqrt[b]*S 
qrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b]) 
/(2*c)))/(2*c)))/2
 

3.2.85.3.1 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 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 5225

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(p_.), x_Symbol] :> Simp[(-(b*c^(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c 
^2*x^2)^p]   Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], 
 x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e 
, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

3.2.85.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(139)=278\).

Time = 2.08 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.24

method	result	size
default	\(\frac {15 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, b^{3}-15 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, b^{3}+8 \arccos \left (c x \right )^{3} \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+24 \arccos \left (c x \right )^{2} \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+20 \arccos \left (c x \right )^{2} \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+24 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} b -30 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+40 \arccos \left (c x \right ) \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+8 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{3}-30 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+20 \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} b}{8 c \sqrt {a +b \arccos \left (c x \right )}}\)	\(401\)

input

int((a+b*arccos(c*x))^(5/2),x,method=_RETURNVERBOSE)
 

output

1/8/c/(a+b*arccos(c*x))^(1/2)*(15*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^ 
(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1 
/2)/b)*2^(1/2)*b^3-15*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*sin(a/ 
b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*2^(1/ 
2)*b^3+8*arccos(c*x)^3*cos(-(a+b*arccos(c*x))/b+a/b)*b^3+24*arccos(c*x)^2* 
cos(-(a+b*arccos(c*x))/b+a/b)*a*b^2+20*arccos(c*x)^2*sin(-(a+b*arccos(c*x) 
)/b+a/b)*b^3+24*arccos(c*x)*cos(-(a+b*arccos(c*x))/b+a/b)*a^2*b-30*arccos( 
c*x)*cos(-(a+b*arccos(c*x))/b+a/b)*b^3+40*arccos(c*x)*sin(-(a+b*arccos(c*x 
))/b+a/b)*a*b^2+8*cos(-(a+b*arccos(c*x))/b+a/b)*a^3-30*cos(-(a+b*arccos(c* 
x))/b+a/b)*a*b^2+20*sin(-(a+b*arccos(c*x))/b+a/b)*a^2*b)
 

3.2.85.5 Fricas [F(-2)]

Exception generated. \[ \int (a+b \arccos (c x))^{5/2} \, dx=\text {Exception raised: TypeError} \]

input

integrate((a+b*arccos(c*x))^(5/2),x, algorithm="fricas")
 

output

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

3.2.85.6 Sympy [F]

\[ \int (a+b \arccos (c x))^{5/2} \, dx=\int \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \]

input

integrate((a+b*acos(c*x))**(5/2),x)
 

output

Integral((a + b*acos(c*x))**(5/2), x)
 

3.2.85.7 Maxima [F]

\[ \int (a+b \arccos (c x))^{5/2} \, dx=\int { {\left (b \arccos \left (c x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

input

integrate((a+b*arccos(c*x))^(5/2),x, algorithm="maxima")
 

output

integrate((b*arccos(c*x) + a)^(5/2), x)
 

3.2.85.8 Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.58 (sec) , antiderivative size = 1177, normalized size of antiderivative = 6.58 \[ \int (a+b \arccos (c x))^{5/2} \, dx=\text {Too large to display} \]

input

integrate((a+b*arccos(c*x))^(5/2),x, algorithm="giac")
 

output

-1/2*I*sqrt(2)*sqrt(pi)*a^3*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^4/sqrt(abs(b)) + b^3*sqrt(abs(b)))*c) + 1/2*I*sqrt(2)*sqrt(pi)*a 
^3*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^4/sqrt(abs(b)) 
 + b^3*sqrt(abs(b)))*c) + 3/2*sqrt(2)*sqrt(pi)*a^2*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/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 3 
/2*sqrt(2)*sqrt(pi)*a^2*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/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) - 3/2*sqrt(2)*sqrt(pi)*a^2*b^ 
2*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) - 15/16*sqrt(2)*sqrt(pi)*b^4*erf(-1/2*I*sqrt(2)*sqrt(b*arc 
cos(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) - 3/2*sqrt(2)*s 
qrt(pi)*a^2*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/sqr 
t(abs(b)) + b*sqrt(abs(b)))*c) - 15/16*sqrt(2)*sqrt(pi)*b^4*erf(1/2*I*sqrt 
(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*...
 

3.2.85.9 Mupad [F(-1)]

Timed out. \[ \int (a+b \arccos (c x))^{5/2} \, dx=\int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{5/2} \,d x \]

input

int((a + b*acos(c*x))^(5/2),x)
 

output

int((a + b*acos(c*x))^(5/2), x)