\(\int x^2 (a+b \arccos (c x))^{5/2} \, dx\) [183]

Optimal result

Integrand size = 16, antiderivative size = 358 \[ \int x^2 (a+b \arccos (c x))^{5/2} \, dx=-\frac {5 b^2 x \sqrt {a+b \arccos (c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \arccos (c x)}-\frac {5 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{16 c^3}+\frac {5 b^{5/2} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{144 c^3}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{16 c^3}+\frac {5 b^{5/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{144 c^3} \] Output:

-5/6*b^2*x*(a+b*arccos(c*x))^(1/2)/c^2-5/36*b^2*x^3*(a+b*arccos(c*x))^(1/2 
)-5/9*b*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^(3/2)/c^3-5/18*b*x^2*(-c^2*x^ 
2+1)^(1/2)*(a+b*arccos(c*x))^(3/2)/c+1/3*x^3*(a+b*arccos(c*x))^(5/2)+15/32 
*b^(5/2)*2^(1/2)*Pi^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arccos(c 
*x))^(1/2)/b^(1/2))/c^3+5/864*b^(5/2)*6^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelC 
(6^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))/c^3+15/32*b^(5/2)*2^(1/ 
2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin 
(a/b)/c^3+5/864*b^(5/2)*6^(1/2)*Pi^(1/2)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*ar 
ccos(c*x))^(1/2)/b^(1/2))*sin(3*a/b)/c^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.09 (sec) , antiderivative size = 956, normalized size of antiderivative = 2.67 \[ \int x^2 (a+b \arccos (c x))^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

((-1/72*I)*a^2*b*(-9*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Ga 
mma[3/2, ((-I)*(a + b*ArcCos[c*x]))/b] + 9*E^(((4*I)*a)/b)*Sqrt[(I*(a + b* 
ArcCos[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b] + Sqrt[3]*(-(Sqrt[( 
(-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcCos[c*x]))/b]) + 
 E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b* 
ArcCos[c*x]))/b])))/(c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcCos[c*x]]) - (a*Sqr 
t[b]*(18*Sqrt[b]*Sqrt[a + b*ArcCos[c*x]]*(3*Sqrt[1 - c^2*x^2] - 2*c*x*ArcC 
os[c*x]) - 9*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]) - 9*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqr 
t[a + b*ArcCos[c*x]])/Sqrt[b]]*(2*a*Cos[a/b] - 3*b*Sin[a/b]) - Sqrt[6*Pi]* 
FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*(b*Cos[(3*a)/b] + 2 
*a*Sin[(3*a)/b]) - Sqrt[6*Pi]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]] 
)/Sqrt[b]]*(2*a*Cos[(3*a)/b] - b*Sin[(3*a)/b]) + 6*Sqrt[b]*Sqrt[a + b*ArcC 
os[c*x]]*(-2*ArcCos[c*x]*Cos[3*ArcCos[c*x]] + Sin[3*ArcCos[c*x]])))/(72*c^ 
3) - (Sqrt[b]*(27*(2*Sqrt[b]*Sqrt[a + b*ArcCos[c*x]]*(-2*Sqrt[1 - c^2*x^2] 
*(a - 5*b*ArcCos[c*x]) - b*c*x*(-15 + 4*ArcCos[c*x]^2)) + Sqrt[2*Pi]*Fresn 
elC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*((4*a^2 - 15*b^2)*Cos[a/ 
b] - 12*a*b*Sin[a/b]) + Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[ 
c*x]])/Sqrt[b]]*(12*a*b*Cos[a/b] + (4*a^2 - 15*b^2)*Sin[a/b])) + Sqrt[6*Pi 
]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*((12*a^2 - 5*b...
 

Rubi [A] (verified)

Time = 2.95 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.32, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5141, 5211, 5141, 5183, 5131, 5225, 3042, 3787, 25, 3042, 3785, 3786, 3793, 2009, 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.

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

\(\Big \downarrow \) 5141

\(\displaystyle \frac {5}{6} b c \int \frac {x^3 (a+b \arccos (c x))^{3/2}}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 5211

\(\displaystyle \frac {5}{6} b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))^{3/2}}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {b \int x^2 \sqrt {a+b \arccos (c x)}dx}{2 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 5141

\(\displaystyle \frac {5}{6} b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))^{3/2}}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {b \left (\frac {1}{6} b c \int \frac {x^3}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx+\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}\right )}{2 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 5183

\(\displaystyle \frac {5}{6} b c \left (\frac {2 \left (-\frac {3 b \int \sqrt {a+b \arccos (c x)}dx}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {b \left (\frac {1}{6} b c \int \frac {x^3}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx+\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}\right )}{2 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 5131

\(\displaystyle \frac {5}{6} b c \left (\frac {2 \left (-\frac {3 b \left (\frac {1}{2} b c \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx+x \sqrt {a+b \arccos (c x)}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {b \left (\frac {1}{6} b c \int \frac {x^3}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}dx+\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}\right )}{2 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 5225

\(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\cos ^3\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3787

\(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3785

\(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )^3}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {2 \sin \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\int \left (\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arccos (c x))}{b}\right )}{4 \sqrt {a+b \arccos (c x)}}+\frac {3 \cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{4 \sqrt {a+b \arccos (c x)}}\right )d(a+b \arccos (c x))}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {2 \sin \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{6} b c \left (\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {2 \sin \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^3}\right )}{2 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {5}{6} b c \left (\frac {2 \left (-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}\right )}{3 c^2}-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^3}\right )}{2 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^3}\right )}{2 c}+\frac {2 \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \arccos (c x)}-\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}\right )}{2 c}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{5/2}\)

Input:

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

Output:

(x^3*(a + b*ArcCos[c*x])^(5/2))/3 + (5*b*c*(-1/3*(x^2*Sqrt[1 - c^2*x^2]*(a 
 + b*ArcCos[c*x])^(3/2))/c^2 + (2*(-((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]*Cos[ 
a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]] + Sqrt[b]*Sqrt 
[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(2 
*c)))/(2*c)))/(3*c^2) - (b*((x^3*Sqrt[a + b*ArcCos[c*x]])/3 - ((3*Sqrt[b]* 
Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]] 
)/2 + (Sqrt[b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*Arc 
Cos[c*x]])/Sqrt[b]])/2 + (3*Sqrt[b]*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a 
 + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/2 + (Sqrt[b]*Sqrt[Pi/6]*FresnelS[(Sq 
rt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/2)/(6*c^3)))/(2*c 
)))/6
 

Defintions of rubi rules used

rule 25

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

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 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 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

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 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 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]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(818\) vs. \(2(278)=556\).

Time = 0.28 (sec) , antiderivative size = 819, normalized size of antiderivative = 2.29

method	result	size
default	\(\text {Expression too large to display}\)	\(819\)

Input:

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

Output:

-1/864/c^3*b*(36*arccos(c*x)^2*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*cos(-3*(a+b*a 
rccos(c*x))/b+3*a/b)*(a+b*arccos(c*x))^(1/2)*b^2+108*arccos(c*x)^2*2^(1/2) 
*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(-(a+b*arccos(c*x))/b+a/ 
b)*b^2+5*Pi*(-1/b)^(1/2)*(-3/b)^(1/2)*cos(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/ 
2)/(-3/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*b^3-5*Pi*(-1/b)^(1/2)*(-3/b)^(1 
/2)*sin(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arccos(c*x))^ 
(1/2)/b)*b^3+72*arccos(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*cos(-3*(a+b*arcc 
os(c*x))/b+3*a/b)*(a+b*arccos(c*x))^(1/2)*a*b+30*arccos(c*x)*2^(1/2)*Pi^(1 
/2)*(-1/b)^(1/2)*sin(-3*(a+b*arccos(c*x))/b+3*a/b)*(a+b*arccos(c*x))^(1/2) 
*b^2+216*arccos(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2) 
*cos(-(a+b*arccos(c*x))/b+a/b)*a*b+270*arccos(c*x)*2^(1/2)*Pi^(1/2)*(-1/b) 
^(1/2)*(a+b*arccos(c*x))^(1/2)*sin(-(a+b*arccos(c*x))/b+a/b)*b^2+36*2^(1/2 
)*Pi^(1/2)*(-1/b)^(1/2)*cos(-3*(a+b*arccos(c*x))/b+3*a/b)*(a+b*arccos(c*x) 
)^(1/2)*a^2-15*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*cos(-3*(a+b*arccos(c*x))/b+3* 
a/b)*(a+b*arccos(c*x))^(1/2)*b^2+30*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*sin(-3*( 
a+b*arccos(c*x))/b+3*a/b)*(a+b*arccos(c*x))^(1/2)*a*b+108*2^(1/2)*Pi^(1/2) 
*(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(-(a+b*arccos(c*x))/b+a/b)*a^2-40 
5*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(-(a+b*arccos(c 
*x))/b+a/b)*b^2+270*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)* 
sin(-(a+b*arccos(c*x))/b+a/b)*a*b+405*Pi*b^2*FresnelS(2^(1/2)/Pi^(1/2)/...
 

Fricas [F(-2)]

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

integrate(x^2*(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)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.17 (sec) , antiderivative size = 3096, normalized size of antiderivative = 8.65 \[ \int x^2 (a+b \arccos (c x))^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/576*(72*sqrt(pi)*a^2*b^(7/2)*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sq 
rt(b) - 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/ 
(sqrt(6)*b^3 + I*sqrt(6)*b^4/abs(b)) + 36*I*sqrt(pi)*a*b^(9/2)*erf(-1/2*sq 
rt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + 
 a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*b^3 + I*sqrt(6)*b^4/abs(b)) - 10* 
sqrt(pi)*b^(11/2)*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) - 1/2*I 
*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*b^3 
+ I*sqrt(6)*b^4/abs(b)) - 72*I*sqrt(2)*sqrt(pi)*a^3*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^3/sqrt(abs(b)) + b^2*sqrt(abs(b))) + 72*I 
*sqrt(2)*sqrt(pi)*a^3*b^2*erf(1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(a 
bs(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))) + 72*sqrt(pi)*a^2*b^(7/2)*erf(-1/2* 
sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) 
 + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*b^3 - I*sqrt(6)*b^4/abs(b)) - 
36*I*sqrt(pi)*a*b^(9/2)*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) + 
 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt( 
6)*b^3 - I*sqrt(6)*b^4/abs(b)) - 10*sqrt(pi)*b^(11/2)*erf(-1/2*sqrt(6)*sqr 
t(b*arccos(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt( 
b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*b^3 - I*sqrt(6)*b^4/abs(b)) - 216*I*sq...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

2*int(sqrt(acos(c*x)*b + a)*acos(c*x)*x**2,x)*a*b + int(sqrt(acos(c*x)*b + 
 a)*acos(c*x)**2*x**2,x)*b**2 + int(sqrt(acos(c*x)*b + a)*x**2,x)*a**2