3.10.9 \(\int \frac {x^3 \arctan (a x)^{5/2}}{(c+a^2 c x^2)^{5/2}} \, dx\) [909]

3.10.9.1 Optimal result

Integrand size = 26, antiderivative size = 350 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {45 \sqrt {\arctan (a x)}}{16 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 x \arctan (a x)^{3/2}}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \arctan (a x)^{5/2}}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {5 \sqrt {1+a^2 x^2} \sqrt {\arctan (a x)} \cos (3 \arctan (a x))}{144 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {45 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{16 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {5 \sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{144 a^4 c^2 \sqrt {c+a^2 c x^2}} \]

output

5/18*x^3*arctan(a*x)^(3/2)/a/c/(a^2*c*x^2+c)^(3/2)-1/3*x^2*arctan(a*x)^(5/ 
2)/a^2/c/(a^2*c*x^2+c)^(3/2)+5/3*x*arctan(a*x)^(3/2)/a^3/c^2/(a^2*c*x^2+c) 
^(1/2)-2/3*arctan(a*x)^(5/2)/a^4/c^2/(a^2*c*x^2+c)^(1/2)+5/864*FresnelC(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^4/c 
^2/(a^2*c*x^2+c)^(1/2)-45/32*FresnelC(2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))* 
2^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)/a^4/c^2/(a^2*c*x^2+c)^(1/2)+45/16*arcta 
n(a*x)^(1/2)/a^4/c^2/(a^2*c*x^2+c)^(1/2)-5/144*cos(3*arctan(a*x))*(a^2*x^2 
+1)^(1/2)*arctan(a*x)^(1/2)/a^4/c^2/(a^2*c*x^2+c)^(1/2)
 

3.10.9.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {4800 \arctan (a x)+5040 a^2 x^2 \arctan (a x)+2880 a x \arctan (a x)^2+3360 a^3 x^3 \arctan (a x)^2-1152 \arctan (a x)^3-1728 a^2 x^2 \arctan (a x)^3+1215 i \left (1+a^2 x^2\right )^{3/2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )-1215 i \left (1+a^2 x^2\right )^{3/2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )-5 i \sqrt {3+3 a^2 x^2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-3 i \arctan (a x)\right )-5 i a^2 x^2 \sqrt {3+3 a^2 x^2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-3 i \arctan (a x)\right )+5 i \sqrt {3+3 a^2 x^2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},3 i \arctan (a x)\right )+5 i a^2 x^2 \sqrt {3+3 a^2 x^2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},3 i \arctan (a x)\right )}{1728 a^4 c^2 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \]

input

Integrate[(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2),x]
 

output

(4800*ArcTan[a*x] + 5040*a^2*x^2*ArcTan[a*x] + 2880*a*x*ArcTan[a*x]^2 + 33 
60*a^3*x^3*ArcTan[a*x]^2 - 1152*ArcTan[a*x]^3 - 1728*a^2*x^2*ArcTan[a*x]^3 
 + (1215*I)*(1 + a^2*x^2)^(3/2)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*Arc 
Tan[a*x]] - (1215*I)*(1 + a^2*x^2)^(3/2)*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I* 
ArcTan[a*x]] - (5*I)*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, 
 (-3*I)*ArcTan[a*x]] - (5*I)*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[ 
a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + (5*I)*Sqrt[3 + 3*a^2*x^2]*Sqrt[I*Ar 
cTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]] + (5*I)*a^2*x^2*Sqrt[3 + 3*a^2*x^ 
2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])/(1728*a^4*c^2*(1 + a 
^2*x^2)*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])
 

3.10.9.3 Rubi [A] (verified)

Time = 2.52 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5475, 5465, 5433, 5440, 5439, 3042, 3785, 3833, 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.

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

\(\Big \downarrow \) 5475

\(\displaystyle \frac {2 \int \frac {x \arctan (a x)^{5/2}}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {5}{12} \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (a^2 c x^2+c\right )^{5/2}}dx-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5465

\(\displaystyle \frac {2 \left (\frac {5 \int \frac {\arctan (a x)^{3/2}}{\left (a^2 c x^2+c\right )^{3/2}}dx}{2 a}-\frac {\arctan (a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {5}{12} \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (a^2 c x^2+c\right )^{5/2}}dx-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5433

\(\displaystyle \frac {2 \left (\frac {5 \left (-\frac {3}{4} \int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \sqrt {\arctan (a x)}}dx+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\arctan (a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {5}{12} \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (a^2 c x^2+c\right )^{5/2}}dx-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5440

\(\displaystyle \frac {2 \left (\frac {5 \left (-\frac {3 \sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{3/2} \sqrt {\arctan (a x)}}dx}{4 c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\arctan (a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {5}{12} \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (a^2 c x^2+c\right )^{5/2}}dx-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5439

\(\displaystyle \frac {2 \left (\frac {5 \left (-\frac {3 \sqrt {a^2 x^2+1} \int \frac {1}{\sqrt {a^2 x^2+1} \sqrt {\arctan (a x)}}d\arctan (a x)}{4 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\arctan (a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {5}{12} \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (a^2 c x^2+c\right )^{5/2}}dx-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \left (\frac {5 \left (-\frac {3 \sqrt {a^2 x^2+1} \int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )}{\sqrt {\arctan (a x)}}d\arctan (a x)}{4 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\arctan (a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {5}{12} \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (a^2 c x^2+c\right )^{5/2}}dx-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 3785

\(\displaystyle \frac {2 \left (\frac {5 \left (-\frac {3 \sqrt {a^2 x^2+1} \int \frac {1}{\sqrt {a^2 x^2+1}}d\sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\arctan (a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {5}{12} \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (a^2 c x^2+c\right )^{5/2}}dx-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 3833

\(\displaystyle -\frac {5}{12} \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (a^2 c x^2+c\right )^{5/2}}dx+\frac {2 \left (\frac {5 \left (-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\arctan (a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5506

\(\displaystyle -\frac {5 \sqrt {a^2 x^2+1} \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (a^2 x^2+1\right )^{5/2}}dx}{12 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (\frac {5 \left (-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\arctan (a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5505

\(\displaystyle -\frac {5 \sqrt {a^2 x^2+1} \int \frac {a^3 x^3 \sqrt {\arctan (a x)}}{\left (a^2 x^2+1\right )^{3/2}}d\arctan (a x)}{12 a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (\frac {5 \left (-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\arctan (a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 \sqrt {a^2 x^2+1} \int \sqrt {\arctan (a x)} \sin (\arctan (a x))^3d\arctan (a x)}{12 a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (\frac {5 \left (-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\arctan (a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {5 \sqrt {a^2 x^2+1} \int \left (\frac {3 a x \sqrt {\arctan (a x)}}{4 \sqrt {a^2 x^2+1}}-\frac {1}{4} \sqrt {\arctan (a x)} \sin (3 \arctan (a x))\right )d\arctan (a x)}{12 a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (\frac {5 \left (-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\arctan (a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {5 \left (-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\arctan (a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5 \sqrt {a^2 x^2+1} \left (-\frac {3 \sqrt {\arctan (a x)}}{4 \sqrt {a^2 x^2+1}}+\frac {3}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{12} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{12} \sqrt {\arctan (a x)} \cos (3 \arctan (a x))\right )}{12 a^4 c^2 \sqrt {a^2 c x^2+c}}\)

input

Int[(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2),x]
 

output

(5*x^3*ArcTan[a*x]^(3/2))/(18*a*c*(c + a^2*c*x^2)^(3/2)) - (x^2*ArcTan[a*x 
]^(5/2))/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) + (2*(-(ArcTan[a*x]^(5/2)/(a^2*c* 
Sqrt[c + a^2*c*x^2])) + (5*((3*Sqrt[ArcTan[a*x]])/(2*a*c*Sqrt[c + a^2*c*x^ 
2]) + (x*ArcTan[a*x]^(3/2))/(c*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[Pi/2]*Sqrt[1 
 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a*c*Sqrt[c + a^2*c* 
x^2])))/(2*a)))/(3*a^2*c) - (5*Sqrt[1 + a^2*x^2]*((-3*Sqrt[ArcTan[a*x]])/( 
4*Sqrt[1 + a^2*x^2]) + (Sqrt[ArcTan[a*x]]*Cos[3*ArcTan[a*x]])/12 + (3*Sqrt 
[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/4 - (Sqrt[Pi/6]*FresnelC[Sq 
rt[6/Pi]*Sqrt[ArcTan[a*x]]])/12))/(12*a^4*c^2*Sqrt[c + a^2*c*x^2])
 

3.10.9.3.1 Defintions of rubi rules used

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 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 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 5433

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_ 
Symbol] :> Simp[b*p*((a + b*ArcTan[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2])), x] 
 + (Simp[x*((a + b*ArcTan[c*x])^p/(d*Sqrt[d + e*x^2])), x] - Simp[b^2*p*(p 
- 1)   Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[ 
{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]
 

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 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])
 

3.10.9.4 Maple [F]

\[\int \frac {x^{3} \arctan \left (a x \right )^{\frac {5}{2}}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]

input

int(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(5/2),x)
 

output

int(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(5/2),x)
 

3.10.9.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate(x^3*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)
 

3.10.9.6 Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Timed out} \]

input

integrate(x**3*atan(a*x)**(5/2)/(a**2*c*x**2+c)**(5/2),x)
 

output

Timed out
 

3.10.9.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

input

integrate(x^3*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.
 

3.10.9.8 Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

3.10.9.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

input

int((x^3*atan(a*x)^(5/2))/(c + a^2*c*x^2)^(5/2),x)
 

output

int((x^3*atan(a*x)^(5/2))/(c + a^2*c*x^2)^(5/2), x)