3.873 \(\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{(c+a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=256 \[ \frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^4 c^3}-\frac{15 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{256 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{15 x^4 \sqrt{\tan ^{-1}(a x)}}{256 c^3 \left (a^2 x^2+1\right )^2}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (a^2 x^2+1\right )}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^4 c^3 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}-\frac{135 \sqrt{\tan ^{-1}(a x)}}{2048 a^4 c^3} \]

[Out]

(-135*Sqrt[ArcTan[a*x]])/(2048*a^4*c^3) - (15*x^4*Sqrt[ArcTan[a*x]])/(256*c^3*(1 + a^2*x^2)^2) + (45*Sqrt[ArcT
an[a*x]])/(256*a^4*c^3*(1 + a^2*x^2)) + (5*x^3*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)^2) + (15*x*ArcTan[a*
x]^(3/2))/(64*a^3*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x]^(5/2))/(32*a^4*c^3) + (x^4*ArcTan[a*x]^(5/2))/(4*c^3*(1
+ a^2*x^2)^2) + (15*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a^4*c^3) - (15*Sqrt[Pi]*Fresnel
C[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(256*a^4*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.491625, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4944, 4940, 4936, 4930, 4904, 3312, 3304, 3352, 4970} \[ \frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^4 c^3}-\frac{15 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{256 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{15 x^4 \sqrt{\tan ^{-1}(a x)}}{256 c^3 \left (a^2 x^2+1\right )^2}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (a^2 x^2+1\right )}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^4 c^3 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}-\frac{135 \sqrt{\tan ^{-1}(a x)}}{2048 a^4 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-135*Sqrt[ArcTan[a*x]])/(2048*a^4*c^3) - (15*x^4*Sqrt[ArcTan[a*x]])/(256*c^3*(1 + a^2*x^2)^2) + (45*Sqrt[ArcT
an[a*x]])/(256*a^4*c^3*(1 + a^2*x^2)) + (5*x^3*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)^2) + (15*x*ArcTan[a*
x]^(3/2))/(64*a^3*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x]^(5/2))/(32*a^4*c^3) + (x^4*ArcTan[a*x]^(5/2))/(4*c^3*(1
+ a^2*x^2)^2) + (15*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a^4*c^3) - (15*Sqrt[Pi]*Fresnel
C[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(256*a^4*c^3)

Rule 4944

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] - Dist[(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 4940

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*ArcTan[c*x])^(p - 1))/(c*d*m^2), x] + (Dist[(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] - Dist[(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] - Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p
)/(c^2*d*m), x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && G
tQ[p, 1]

Rule 4936

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] + (Dist[(b*p)/(2*c), Int[(x*(a + b*ArcTan[c*x])^(p - 1))/(d + e*x^2)^
2, x], x] - Simp[(x*(a + b*ArcTan[c*x])^p)/(2*c^2*d*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c
^2*d] && GtQ[p, 0]

Rule 4930

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] - Dist[(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 4904

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^q/c, Subst[Int[(a
+ b*x)^p/Cos[x]^(2*(q + 1)), x], x, ArcTan[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 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[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 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4970

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[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])

Rubi steps

\begin{align*} \int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{1}{8} (5 a) \int \frac{x^4 \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx\\ &=-\frac{15 x^4 \sqrt{\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{1}{512} (15 a) \int \frac{x^4}{\left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx-\frac{15 \int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a c}\\ &=-\frac{15 x^4 \sqrt{\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \operatorname{Subst}\left (\int \frac{\sin ^4(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a^4 c^3}-\frac{45 \int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{128 a^2 c}\\ &=-\frac{15 x^4 \sqrt{\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}-\frac{\cos (2 x)}{2 \sqrt{x}}+\frac{\cos (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{512 a^4 c^3}-\frac{45 \int \frac{1}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{512 a^3 c}\\ &=\frac{45 \sqrt{\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac{15 x^4 \sqrt{\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{4096 a^4 c^3}-\frac{15 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{1024 a^4 c^3}-\frac{45 \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a^4 c^3}\\ &=\frac{45 \sqrt{\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac{15 x^4 \sqrt{\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{2048 a^4 c^3}-\frac{15 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{512 a^4 c^3}-\frac{45 \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{512 a^4 c^3}\\ &=-\frac{135 \sqrt{\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac{15 x^4 \sqrt{\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^4 c^3}-\frac{15 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{1024 a^4 c^3}-\frac{45 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{1024 a^4 c^3}\\ &=-\frac{135 \sqrt{\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac{15 x^4 \sqrt{\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^4 c^3}-\frac{15 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{1024 a^4 c^3}-\frac{45 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{512 a^4 c^3}\\ &=-\frac{135 \sqrt{\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac{15 x^4 \sqrt{\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^4 c^3}-\frac{15 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{256 a^4 c^3}\\ \end{align*}

Mathematica [C]  time = 0.701617, size = 359, normalized size = 1.4 \[ \frac{510 \sqrt{2 \pi } \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )+\frac{900 i \sqrt{2} \left (a^2 x^2+1\right )^2 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \tan ^{-1}(a x)\right )-900 i \sqrt{2} \left (a^2 x^2+1\right )^2 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \tan ^{-1}(a x)\right )+135 i \left (a^2 x^2+1\right )^2 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )-135 i \left (a^2 x^2+1\right )^2 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )-4080 \sqrt{\pi } \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)} \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )+20480 a^4 x^4 \tan ^{-1}(a x)^3-16320 a^4 x^4 \tan ^{-1}(a x)+51200 a^3 x^3 \tan ^{-1}(a x)^2-24576 a^2 x^2 \tan ^{-1}(a x)^3+5760 a^2 x^2 \tan ^{-1}(a x)+30720 a x \tan ^{-1}(a x)^2-12288 \tan ^{-1}(a x)^3+14400 \tan ^{-1}(a x)}{\left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}}}{131072 a^4 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(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] - 163
20*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]*(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)*Arc
Tan[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)

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Maple [A]  time = 0.12, size = 180, normalized size = 0.7 \begin{align*} -{\frac{\cos \left ( 2\,\arctan \left ( ax \right ) \right ) }{8\,{c}^{3}{a}^{4}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{\cos \left ( 4\,\arctan \left ( ax \right ) \right ) }{32\,{c}^{3}{a}^{4}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{5\,\sin \left ( 2\,\arctan \left ( ax \right ) \right ) }{32\,{c}^{3}{a}^{4}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{5\,\sin \left ( 4\,\arctan \left ( ax \right ) \right ) }{256\,{c}^{3}{a}^{4}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{15\,\cos \left ( 2\,\arctan \left ( ax \right ) \right ) }{128\,{c}^{3}{a}^{4}}\sqrt{\arctan \left ( ax \right ) }}-{\frac{15\,\cos \left ( 4\,\arctan \left ( ax \right ) \right ) }{2048\,{c}^{3}{a}^{4}}\sqrt{\arctan \left ( ax \right ) }}+{\frac{15\,\sqrt{2}\sqrt{\pi }}{8192\,{c}^{3}{a}^{4}}{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) }-{\frac{15\,\sqrt{\pi }}{256\,{c}^{3}{a}^{4}}{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/a^4/c^3*arctan(a*x)^(5/2)*cos(2*arctan(a*x))+1/32/a^4/c^3*arctan(a*x)^(5/2)*cos(4*arctan(a*x))+5/32/a^4/c
^3*arctan(a*x)^(3/2)*sin(2*arctan(a*x))-5/256/a^4/c^3*arctan(a*x)^(3/2)*sin(4*arctan(a*x))+15/128/a^4/c^3*arct
an(a*x)^(1/2)*cos(2*arctan(a*x))-15/2048/a^4/c^3*arctan(a*x)^(1/2)*cos(4*arctan(a*x))+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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \arctan \left (a x\right )^{\frac{5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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