3.10.10 \(\int \frac {x^2 \arctan (a x)^{5/2}}{(c+a^2 c x^2)^{5/2}} \, dx\) [910]

3.10.10.1 Optimal result

Integrand size = 26, antiderivative size = 295 \[ \int \frac {x^2 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {5 x^3 \sqrt {\arctan (a x)}}{36 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 x \sqrt {\arctan (a x)}}{6 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {5 x^2 \arctan (a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^{3/2}}{9 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^3 \arctan (a x)^{5/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{16 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {5 \sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{144 a^3 c^2 \sqrt {c+a^2 c x^2}} \]

output

5/18*x^2*arctan(a*x)^(3/2)/a/c/(a^2*c*x^2+c)^(3/2)+1/3*x^3*arctan(a*x)^(5/ 
2)/c/(a^2*c*x^2+c)^(3/2)+5/9*arctan(a*x)^(3/2)/a^3/c^2/(a^2*c*x^2+c)^(1/2) 
-5/864*FresnelS(6^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*6^(1/2)*Pi^(1/2)*(a^2* 
x^2+1)^(1/2)/a^3/c^2/(a^2*c*x^2+c)^(1/2)+15/32*FresnelS(2^(1/2)/Pi^(1/2)*a 
rctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)/a^3/c^2/(a^2*c*x^2+c) 
^(1/2)-5/36*x^3*arctan(a*x)^(1/2)/c/(a^2*c*x^2+c)^(3/2)-5/6*x*arctan(a*x)^ 
(1/2)/a^2/c^2/(a^2*c*x^2+c)^(1/2)
 

3.10.10.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.96 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.97 \[ \int \frac {x^2 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {-24 \arctan (a x) \left (5 a x \left (6+7 a^2 x^2\right )-10 \left (2+3 a^2 x^2\right ) \arctan (a x)-12 a^3 x^3 \arctan (a x)^2\right )+35 \sqrt {6 \pi } \left (1+a^2 x^2\right )^{3/2} \sqrt {\arctan (a x)} \left (3 \sqrt {3} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )\right )-15 \left (1+a^2 x^2\right )^{3/2} \left (3 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )+3 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )+\sqrt {3} \left (\sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-3 i \arctan (a x)\right )+\sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},3 i \arctan (a x)\right )\right )\right )}{864 a^3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \]

input

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

output

(-24*ArcTan[a*x]*(5*a*x*(6 + 7*a^2*x^2) - 10*(2 + 3*a^2*x^2)*ArcTan[a*x] - 
 12*a^3*x^3*ArcTan[a*x]^2) + 35*Sqrt[6*Pi]*(1 + a^2*x^2)^(3/2)*Sqrt[ArcTan 
[a*x]]*(3*Sqrt[3]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - FresnelS[Sqrt[6 
/Pi]*Sqrt[ArcTan[a*x]]]) - 15*(1 + a^2*x^2)^(3/2)*(3*Sqrt[(-I)*ArcTan[a*x] 
]*Gamma[1/2, (-I)*ArcTan[a*x]] + 3*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan 
[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + 
Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(864*a^3*c*(c + a^2*c 
*x^2)^(3/2)*Sqrt[ArcTan[a*x]])
 

3.10.10.3 Rubi [A] (verified)

Time = 2.71 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.17, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {5479, 5475, 5465, 5440, 5439, 3042, 3777, 25, 3042, 3786, 3832, 5506, 5505, 3042, 3793, 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.

\(\displaystyle \int \frac {x^2 \arctan (a x)^{5/2}}{\left (a^2 c x^2+c\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5479

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

\(\Big \downarrow \) 5475

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (\frac {2 \int \frac {x \arctan (a x)^{3/2}}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {1}{12} \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5465

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (\frac {2 \left (\frac {3 \int \frac {\sqrt {\arctan (a x)}}{\left (a^2 c x^2+c\right )^{3/2}}dx}{2 a}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {1}{12} \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5440

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (\frac {2 \left (\frac {3 \sqrt {a^2 x^2+1} \int \frac {\sqrt {\arctan (a x)}}{\left (a^2 x^2+1\right )^{3/2}}dx}{2 a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {1}{12} \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5439

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (\frac {2 \left (\frac {3 \sqrt {a^2 x^2+1} \int \frac {\sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}d\arctan (a x)}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {1}{12} \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (\frac {2 \left (\frac {3 \sqrt {a^2 x^2+1} \int \sqrt {\arctan (a x)} \sin \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {1}{12} \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (\frac {2 \left (\frac {3 \sqrt {a^2 x^2+1} \left (\frac {1}{2} \int -\frac {a x}{\sqrt {a^2 x^2+1} \sqrt {\arctan (a x)}}d\arctan (a x)+\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {1}{12} \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (\frac {2 \left (\frac {3 \sqrt {a^2 x^2+1} \left (\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}-\frac {1}{2} \int \frac {a x}{\sqrt {a^2 x^2+1} \sqrt {\arctan (a x)}}d\arctan (a x)\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {1}{12} \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (\frac {2 \left (\frac {3 \sqrt {a^2 x^2+1} \left (\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}-\frac {1}{2} \int \frac {\sin (\arctan (a x))}{\sqrt {\arctan (a x)}}d\arctan (a x)\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {1}{12} \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (\frac {2 \left (\frac {3 \sqrt {a^2 x^2+1} \left (\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}-\int \frac {a x}{\sqrt {a^2 x^2+1}}d\sqrt {\arctan (a x)}\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {1}{12} \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (-\frac {1}{12} \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx+\frac {2 \left (\frac {3 \sqrt {a^2 x^2+1} \left (\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}-\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5506

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (-\frac {\sqrt {a^2 x^2+1} \int \frac {x^3}{\left (a^2 x^2+1\right )^{5/2} \sqrt {\arctan (a x)}}dx}{12 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (\frac {3 \sqrt {a^2 x^2+1} \left (\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}-\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5505

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (-\frac {\sqrt {a^2 x^2+1} \int \frac {a^3 x^3}{\left (a^2 x^2+1\right )^{3/2} \sqrt {\arctan (a x)}}d\arctan (a x)}{12 a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (\frac {3 \sqrt {a^2 x^2+1} \left (\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}-\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (-\frac {\sqrt {a^2 x^2+1} \int \frac {\sin (\arctan (a x))^3}{\sqrt {\arctan (a x)}}d\arctan (a x)}{12 a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (\frac {3 \sqrt {a^2 x^2+1} \left (\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}-\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (-\frac {\sqrt {a^2 x^2+1} \int \left (\frac {3 a x}{4 \sqrt {a^2 x^2+1} \sqrt {\arctan (a x)}}-\frac {\sin (3 \arctan (a x))}{4 \sqrt {\arctan (a x)}}\right )d\arctan (a x)}{12 a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (\frac {3 \sqrt {a^2 x^2+1} \left (\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}-\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^3 \arctan (a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5}{6} a \left (\frac {2 \left (\frac {3 \sqrt {a^2 x^2+1} \left (\frac {a x \sqrt {\arctan (a x)}}{\sqrt {a^2 x^2+1}}-\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^{3/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{12 a^4 c^2 \sqrt {a^2 c x^2+c}}\right )\)

input

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

output

(x^3*ArcTan[a*x]^(5/2))/(3*c*(c + a^2*c*x^2)^(3/2)) - (5*a*((x^3*Sqrt[ArcT 
an[a*x]])/(6*a*c*(c + a^2*c*x^2)^(3/2)) - (x^2*ArcTan[a*x]^(3/2))/(3*a^2*c 
*(c + a^2*c*x^2)^(3/2)) + (2*(-(ArcTan[a*x]^(3/2)/(a^2*c*Sqrt[c + a^2*c*x^ 
2])) + (3*Sqrt[1 + a^2*x^2]*((a*x*Sqrt[ArcTan[a*x]])/Sqrt[1 + a^2*x^2] - S 
qrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]]))/(2*a^2*c*Sqrt[c + a^2*c 
*x^2])))/(3*a^2*c) - (Sqrt[1 + a^2*x^2]*((3*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi] 
*Sqrt[ArcTan[a*x]]])/2 - (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]] 
])/2))/(12*a^4*c^2*Sqrt[c + a^2*c*x^2])))/6
 

3.10.10.3.1 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 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 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 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 5439

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ 
Symbol] :> Simp[d^q/c   Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, Ar 
cTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*( 
q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
 

rule 5440

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ 
Symbol] :> Simp[d^(q + 1/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2])   Int[(1 + 
c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && 
 EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] &&  !(IntegerQ[q] || GtQ[d, 0])
 

rule 5465

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 
1))), x] - Simp[b*(p/(2*c*(q + 1)))   Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) 
^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 
 0] && NeQ[q, -1]
 

rule 5475

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_)*((d_) + (e_.) 
*(x_)^2)^(q_), x_Symbol] :> Simp[b*p*(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*Ar 
cTan[c*x])^(p - 1)/(c*d*m^2)), x] + (-Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(q + 
 1)*((a + b*ArcTan[c*x])^p/(c^2*d*m)), x] + Simp[f^2*((m - 1)/(c^2*d*m)) 
Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Simp[ 
b^2*p*((p - 1)/m^2)   Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 2) 
, x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2* 
q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
 

rule 5479

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + 
 b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1)))   Int[(f*x) 
^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, 
c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] 
&& NeQ[m, -1]
 

rule 5505

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

rule 5506

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[d^(q + 1/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]) 
  Int[x^m*(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, 
d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] &&  !(I 
ntegerQ[q] || GtQ[d, 0])
 

3.10.10.4 Maple [F]

\[\int \frac {x^{2} \arctan \left (a x \right )^{\frac {5}{2}}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]

input

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

output

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

3.10.10.5 Fricas [F(-2)]

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

input

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

output

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

3.10.10.6 Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.10.10.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

input

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

output

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

3.10.10.8 Giac [F]

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

input

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

output

sage0*x
 

3.10.10.9 Mupad [F(-1)]

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

input

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

output

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