\(\int \frac {x^2}{(a+b \arccos (c x))^{5/2}} \, dx\) [198]

Optimal result

Integrand size = 16, antiderivative size = 292 \[ \int \frac {x^2}{(a+b \arccos (c x))^{5/2}} \, dx=\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {4 x^3}{b^2 \sqrt {a+b \arccos (c x)}}+\frac {\sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}+\frac {\sqrt {6 \pi } \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b^{5/2} c^3}-\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 b^{5/2} c^3}-\frac {\sqrt {6 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{b^{5/2} c^3} \] Output:

2/3*x^2*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arccos(c*x))^(3/2)-8/3*x/b^2/c^2/(a+b* 
arccos(c*x))^(1/2)+4*x^3/b^2/(a+b*arccos(c*x))^(1/2)+1/3*2^(1/2)*Pi^(1/2)* 
cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))/b^(5/2 
)/c^3+6^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arccos(c* 
x))^(1/2)/b^(1/2))/b^(5/2)/c^3-1/3*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)/b^(5/2)/c^3-6^(1/2)*Pi^(1/2) 
*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(3*a/b)/b^( 
5/2)/c^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.69 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.10 \[ \int \frac {x^2}{(a+b \arccos (c x))^{5/2}} \, dx=-\frac {-b \sqrt {1-c^2 x^2}-(a+b \arccos (c x)) \left (e^{-i \arccos (c x)}+e^{i \arccos (c x)}-e^{-\frac {i a}{b}} \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arccos (c x))}{b}\right )-e^{\frac {i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arccos (c x))}{b}\right )\right )-3 (a+b \arccos (c x)) \left (e^{-3 i \arccos (c x)}+e^{3 i \arccos (c x)}-\sqrt {3} e^{-\frac {3 i a}{b}} \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arccos (c x))}{b}\right )-\sqrt {3} e^{\frac {3 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arccos (c x))}{b}\right )\right )-b \sin (3 \arccos (c x))}{6 b^2 c^3 (a+b \arccos (c x))^{3/2}} \] Input:

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

Output:

-1/6*(-(b*Sqrt[1 - c^2*x^2]) - (a + b*ArcCos[c*x])*(E^((-I)*ArcCos[c*x]) + 
 E^(I*ArcCos[c*x]) - (Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-I)* 
(a + b*ArcCos[c*x]))/b])/E^((I*a)/b) - E^((I*a)/b)*Sqrt[(I*(a + b*ArcCos[c 
*x]))/b]*Gamma[1/2, (I*(a + b*ArcCos[c*x]))/b]) - 3*(a + b*ArcCos[c*x])*(E 
^((-3*I)*ArcCos[c*x]) + E^((3*I)*ArcCos[c*x]) - (Sqrt[3]*Sqrt[((-I)*(a + b 
*ArcCos[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcCos[c*x]))/b])/E^(((3*I)*a) 
/b) - Sqrt[3]*E^(((3*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ( 
(3*I)*(a + b*ArcCos[c*x]))/b]) - b*Sin[3*ArcCos[c*x]])/(b^2*c^3*(a + b*Arc 
Cos[c*x])^(3/2))
 

Rubi [A] (verified)

Time = 2.04 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.46, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5145, 5223, 5135, 25, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833, 5147, 25, 4906, 2009}

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.

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

\(\Big \downarrow \) 5145

\(\displaystyle -\frac {4 \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}dx}{3 b c}+\frac {2 c \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}dx}{b}+\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 5223

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

\(\Big \downarrow \) 5135

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3787

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3832

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 5147

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*x^2*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcCos[c*x])^(3/2)) - (4*((2*x)/(b 
*c*Sqrt[a + b*ArcCos[c*x]]) + (2*(Sqrt[b]*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sq 
rt[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]))/(b^2*c^2)))/(3*b*c 
) + (2*c*((2*x^3)/(b*c*Sqrt[a + b*ArcCos[c*x]]) + (6*((Sqrt[b]*Sqrt[Pi/2]* 
Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/2 + (Sqrt 
[b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/ 
Sqrt[b]])/2 - (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c 
*x]])/Sqrt[b]]*Sin[a/b])/2 - (Sqrt[b]*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt 
[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/2))/(b^2*c^4)))/b
 

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 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 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 5145

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

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 5223

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(672\) vs. \(2(236)=472\).

Time = 0.24 (sec) , antiderivative size = 673, normalized size of antiderivative = 2.30

method	result	size
default	\(\frac {-6 \arccos \left (c x \right ) \sqrt {-\frac {3}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {a +b \arccos \left (c x \right )}\, b -6 \arccos \left (c x \right ) \sqrt {-\frac {3}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {a +b \arccos \left (c x \right )}\, b -2 \arccos \left (c x \right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \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 \arccos \left (c x \right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \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 -6 \sqrt {-\frac {3}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {a +b \arccos \left (c x \right )}\, a -6 \sqrt {-\frac {3}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {a +b \arccos \left (c x \right )}\, a -2 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \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 ) a -2 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \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 ) a +2 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b +6 \arccos \left (c x \right ) \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b -\sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b +2 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a -\sin \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b +6 \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a}{6 c^{3} b^{2} \left (a +b \arccos \left (c x \right )\right )^{\frac {3}{2}}}\)	\(673\)

Input:

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

Output:

1/6/c^3/b^2*(-6*arccos(c*x)*(-3/b)^(1/2)*2^(1/2)*Pi^(1/2)*cos(3*a/b)*Fresn 
elS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*(a+b*arccos 
(c*x))^(1/2)*b-6*arccos(c*x)*(-3/b)^(1/2)*2^(1/2)*Pi^(1/2)*sin(3*a/b)*Fres 
nelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*(a+b*arcco 
s(c*x))^(1/2)*b-2*arccos(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arccos(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*arccos(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(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-6*(-3/b)^(1/2)*2^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelS(3*2^(1/2)/Pi 
^(1/2)/(-3/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*(a+b*arccos(c*x))^(1/2)*a-6 
*(-3/b)^(1/2)*2^(1/2)*Pi^(1/2)*sin(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/ 
b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*(a+b*arccos(c*x))^(1/2)*a-2*2^(1/2)*Pi 
^(1/2)*(-1/b)^(1/2)*(a+b*arccos(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)*a-2*2^(1/2)*Pi^(1/2)*(-1/b)^( 
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)*a+2*arccos(c*x)*cos(-(a+b*arccos(c*x))/b+a/b 
)*b+6*arccos(c*x)*cos(-3*(a+b*arccos(c*x))/b+3*a/b)*b-sin(-(a+b*arccos(c*x 
))/b+a/b)*b+2*cos(-(a+b*arccos(c*x))/b+a/b)*a-sin(-3*(a+b*arccos(c*x))/b+3 
*a/b)*b+6*cos(-3*(a+b*arccos(c*x))/b+3*a/b)*a)/(a+b*arccos(c*x))^(3/2)
 

Fricas [F(-2)]

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

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

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

\[ \int \frac {x^2}{(a+b \arccos (c x))^{5/2}} \, dx=\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}\, x^{2}}{\mathit {acos} \left (c x \right )^{3} b^{3}+3 \mathit {acos} \left (c x \right )^{2} a \,b^{2}+3 \mathit {acos} \left (c x \right ) a^{2} b +a^{3}}d x \] Input:

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

Output:

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