Integrand size = 12, antiderivative size = 179 \[ \int (a+b \arccos (c x))^{5/2} \, dx=-\frac {15}{4} b^2 x \sqrt {a+b \arccos (c x)}-\frac {5 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c}+x (a+b \arccos (c x))^{5/2}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{4 c}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 c} \]
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)
Result contains complex when optimal does not.
Time = 1.40 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.08 \[ \int (a+b \arccos (c x))^{5/2} \, dx=\frac {\sqrt {b} e^{-\frac {i a}{b}} \left (\left (4 a^2+15 b^2\right ) \left (1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arccos (c x)} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-i \left (4 a^2+15 b^2\right ) \left (-1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arccos (c x)} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-4 \sqrt {b} \left (e^{\frac {i a}{b}} (a+b \arccos (c x)) \left (5 \left (3 b c x+2 a \sqrt {1-c^2 x^2}\right )+\left (-8 a c x+10 b \sqrt {1-c^2 x^2}\right ) \arccos (c x)-4 b c x \arccos (c x)^2\right )-2 i a^2 \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arccos (c x))}{b}\right )+2 i a^2 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arccos (c x))}{b}\right )\right )\right )}{16 c \sqrt {a+b \arccos (c x)}} \]
(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]])
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 |
\(\displaystyle \frac {5}{2} b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}\right )}{2 c}\right )+x (a+b \arccos (c x))^{5/2}\) |
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
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]
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]
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]
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]
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]
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]
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]
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]
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 | \(\frac {15 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, b^{3}-15 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, b^{3}+8 \arccos \left (c x \right )^{3} \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+24 \arccos \left (c x \right )^{2} \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+20 \arccos \left (c x \right )^{2} \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+24 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} b -30 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+40 \arccos \left (c x \right ) \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+8 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{3}-30 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+20 \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} b}{8 c \sqrt {a +b \arccos \left (c x \right )}}\) | \(401\) |
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)
Exception generated. \[ \int (a+b \arccos (c x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int (a+b \arccos (c x))^{5/2} \, dx=\int \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \]
\[ \int (a+b \arccos (c x))^{5/2} \, dx=\int { {\left (b \arccos \left (c x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
Result contains complex when optimal does not.
Time = 1.58 (sec) , antiderivative size = 1177, normalized size of antiderivative = 6.58 \[ \int (a+b \arccos (c x))^{5/2} \, dx=\text {Too large to display} \]
-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*...
Timed out. \[ \int (a+b \arccos (c x))^{5/2} \, dx=\int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{5/2} \,d x \]