Integrand size = 24, antiderivative size = 256 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {135 \sqrt {\arctan (a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\arctan (a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\arctan (a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \arctan (a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4096 a^4 c^3}-\frac {15 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{256 a^4 c^3} \]
5/32*x^3*arctan(a*x)^(3/2)/a/c^3/(a^2*x^2+1)^2+15/64*x*arctan(a*x)^(3/2)/a ^3/c^3/(a^2*x^2+1)-3/32*arctan(a*x)^(5/2)/a^4/c^3+1/4*x^4*arctan(a*x)^(5/2 )/c^3/(a^2*x^2+1)^2+15/8192*FresnelC(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2)) *2^(1/2)*Pi^(1/2)/a^4/c^3-15/256*FresnelC(2*arctan(a*x)^(1/2)/Pi^(1/2))*Pi ^(1/2)/a^4/c^3-135/2048*arctan(a*x)^(1/2)/a^4/c^3-15/256*x^4*arctan(a*x)^( 1/2)/c^3/(a^2*x^2+1)^2+45/256*arctan(a*x)^(1/2)/a^4/c^3/(a^2*x^2+1)
Result contains complex when optimal does not.
Time = 0.60 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.40 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {510 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {14400 \arctan (a x)+5760 a^2 x^2 \arctan (a x)-16320 a^4 x^4 \arctan (a x)+30720 a x \arctan (a x)^2+51200 a^3 x^3 \arctan (a x)^2-12288 \arctan (a x)^3-24576 a^2 x^2 \arctan (a x)^3+20480 a^4 x^4 \arctan (a x)^3-4080 \sqrt {\pi } \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+900 i \sqrt {2} \left (1+a^2 x^2\right )^2 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )-900 i \sqrt {2} \left (1+a^2 x^2\right )^2 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )+135 i \left (1+a^2 x^2\right )^2 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )-135 i \left (1+a^2 x^2\right )^2 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{\left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}}{131072 a^4 c^3} \]
(510*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + (14400*ArcTan[a *x] + 5760*a^2*x^2*ArcTan[a*x] - 16320*a^4*x^4*ArcTan[a*x] + 30720*a*x*Arc Tan[a*x]^2 + 51200*a^3*x^3*ArcTan[a*x]^2 - 12288*ArcTan[a*x]^3 - 24576*a^2 *x^2*ArcTan[a*x]^3 + 20480*a^4*x^4*ArcTan[a*x]^3 - 4080*Sqrt[Pi]*(1 + a^2* x^2)^2*Sqrt[ArcTan[a*x]]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + (900*I )*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[ a*x]] - (900*I)*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2* I)*ArcTan[a*x]] + (135*I)*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2 , (-4*I)*ArcTan[a*x]] - (135*I)*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[ 1/2, (4*I)*ArcTan[a*x]])/((1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]))/(131072*a^4* c^3)
Time = 1.62 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.17, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {5479, 27, 5475, 5471, 5465, 5439, 3042, 3793, 2009, 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.
| \(\displaystyle \int \frac {x^3 \arctan (a x)^{5/2}}{\left (a^2 c x^2+c\right )^3} \, dx\) |
| \(\Big \downarrow \) 5479 |
| \(\displaystyle \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {5}{8} a \int \frac {x^4 \arctan (a x)^{3/2}}{c^3 \left (a^2 x^2+1\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {5 a \int \frac {x^4 \arctan (a x)^{3/2}}{\left (a^2 x^2+1\right )^3}dx}{8 c^3}\) |
| \(\Big \downarrow \) 5475 |
| \(\displaystyle \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {5 a \left (\frac {3 \int \frac {x^2 \arctan (a x)^{3/2}}{\left (a^2 x^2+1\right )^2}dx}{4 a^2}-\frac {3}{64} \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{8 c^3}\) |
| \(\Big \downarrow \) 5471 |
| \(\displaystyle \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {5 a \left (-\frac {3}{64} \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx+\frac {3 \left (\frac {3 \int \frac {x \sqrt {\arctan (a x)}}{\left (a^2 x^2+1\right )^2}dx}{4 a}+\frac {\arctan (a x)^{5/2}}{5 a^3}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{8 c^3}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {5 a \left (-\frac {3}{64} \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx+\frac {3 \left (\frac {3 \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx}{4 a}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}+\frac {\arctan (a x)^{5/2}}{5 a^3}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{8 c^3}\) |
| \(\Big \downarrow \) 5439 |
| \(\displaystyle \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {5 a \left (-\frac {3}{64} \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx+\frac {3 \left (\frac {3 \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}d\arctan (a x)}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}+\frac {\arctan (a x)^{5/2}}{5 a^3}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{8 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {5 a \left (-\frac {3}{64} \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx+\frac {3 \left (\frac {3 \left (\frac {\int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^2}{\sqrt {\arctan (a x)}}d\arctan (a x)}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}+\frac {\arctan (a x)^{5/2}}{5 a^3}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{8 c^3}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {5 a \left (-\frac {3}{64} \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx+\frac {3 \left (\frac {3 \left (\frac {\int \left (\frac {\cos (2 \arctan (a x))}{2 \sqrt {\arctan (a x)}}+\frac {1}{2 \sqrt {\arctan (a x)}}\right )d\arctan (a x)}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}+\frac {\arctan (a x)^{5/2}}{5 a^3}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{8 c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {5 a \left (-\frac {3}{64} \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 \left (a^2 x^2+1\right )^2}+\frac {3 \left (\frac {\arctan (a x)^{5/2}}{5 a^3}+\frac {3 \left (\frac {\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {\arctan (a x)}}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}\right )}{8 c^3}\) |
| \(\Big \downarrow \) 5505 |
| \(\displaystyle \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {5 a \left (-\frac {3 \int \frac {a^4 x^4}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}d\arctan (a x)}{64 a^5}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 \left (a^2 x^2+1\right )^2}+\frac {3 \left (\frac {\arctan (a x)^{5/2}}{5 a^3}+\frac {3 \left (\frac {\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {\arctan (a x)}}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}\right )}{8 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {5 a \left (-\frac {3 \int \frac {\sin (\arctan (a x))^4}{\sqrt {\arctan (a x)}}d\arctan (a x)}{64 a^5}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 \left (a^2 x^2+1\right )^2}+\frac {3 \left (\frac {\arctan (a x)^{5/2}}{5 a^3}+\frac {3 \left (\frac {\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {\arctan (a x)}}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}\right )}{8 c^3}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {5 a \left (-\frac {3 \int \left (-\frac {\cos (2 \arctan (a x))}{2 \sqrt {\arctan (a x)}}+\frac {\cos (4 \arctan (a x))}{8 \sqrt {\arctan (a x)}}+\frac {3}{8 \sqrt {\arctan (a x)}}\right )d\arctan (a x)}{64 a^5}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 \left (a^2 x^2+1\right )^2}+\frac {3 \left (\frac {\arctan (a x)^{5/2}}{5 a^3}+\frac {3 \left (\frac {\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {\arctan (a x)}}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}\right )}{8 c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^4 \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {5 a \left (-\frac {3 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arctan (a x)}\right )}{64 a^5}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 \left (a^2 x^2+1\right )^2}+\frac {3 \left (\frac {\arctan (a x)^{5/2}}{5 a^3}+\frac {3 \left (\frac {\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {\arctan (a x)}}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}\right )}{8 c^3}\) |
(x^4*ArcTan[a*x]^(5/2))/(4*c^3*(1 + a^2*x^2)^2) - (5*a*((3*x^4*Sqrt[ArcTan [a*x]])/(32*a*(1 + a^2*x^2)^2) - (x^3*ArcTan[a*x]^(3/2))/(4*a^2*(1 + a^2*x ^2)^2) - (3*((3*Sqrt[ArcTan[a*x]])/4 + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*S qrt[ArcTan[a*x]]])/8 - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]]) /2))/(64*a^5) + (3*(-1/2*(x*ArcTan[a*x]^(3/2))/(a^2*(1 + a^2*x^2)) + ArcTa n[a*x]^(5/2)/(5*a^3) + (3*(-1/2*Sqrt[ArcTan[a*x]]/(a^2*(1 + a^2*x^2)) + (S qrt[ArcTan[a*x]] + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/2)/ (4*a^2)))/(4*a)))/(4*a^2)))/(8*c^3)
3.9.73.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]))
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])
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]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^2)/((d_) + (e_.)*(x_)^2) ^2, x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)), x] + (-Simp[x*((a + b*ArcTan[c*x])^p/(2*c^2*d*(d + e*x^2))), x] + Simp[b*(p/( 2*c)) Int[x*((a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ [{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
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]
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]
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])
Time = 2.56 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.61
\[\frac {-1024 \arctan \left (a x \right )^{\frac {5}{2}} \cos \left (2 \arctan \left (a x \right )\right ) \sqrt {\pi }+256 \arctan \left (a x \right )^{\frac {5}{2}} \cos \left (4 \arctan \left (a x \right )\right ) \sqrt {\pi }+1280 \arctan \left (a x \right )^{\frac {3}{2}} \sin \left (2 \arctan \left (a x \right )\right ) \sqrt {\pi }-160 \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sin \left (4 \arctan \left (a x \right )\right )+15 \pi \sqrt {2}\, \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+960 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \cos \left (2 \arctan \left (a x \right )\right )-60 \cos \left (4 \arctan \left (a x \right )\right ) \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }-480 \pi \,\operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )}{8192 c^{3} a^{4} \sqrt {\pi }}\]
1/8192/c^3/a^4*(-1024*arctan(a*x)^(5/2)*cos(2*arctan(a*x))*Pi^(1/2)+256*ar ctan(a*x)^(5/2)*cos(4*arctan(a*x))*Pi^(1/2)+1280*arctan(a*x)^(3/2)*sin(2*a rctan(a*x))*Pi^(1/2)-160*arctan(a*x)^(3/2)*Pi^(1/2)*sin(4*arctan(a*x))+15* Pi*2^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))+960*arctan(a*x)^ (1/2)*Pi^(1/2)*cos(2*arctan(a*x))-60*cos(4*arctan(a*x))*arctan(a*x)^(1/2)* Pi^(1/2)-480*Pi*FresnelC(2*arctan(a*x)^(1/2)/Pi^(1/2)))/Pi^(1/2)
Exception generated. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Exception generated. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{\frac {5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]