3.872 \(\int \frac{x^4 \tan ^{-1}(a x)^{5/2}}{(c+a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=310 \[ \frac{15 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^5 c^3}-\frac{15 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a^5 c^3}+\frac{5 x^4 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac{x^3 \tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}-\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 a^4 c^3 \left (a^2 x^2+1\right )}+\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 a^4 c^3 \left (a^2 x^2+1\right )}-\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a^5 c^3 \left (a^2 x^2+1\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a^5 c^3}+\frac{45 \tan ^{-1}(a x)^{3/2}}{256 a^5 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a^5 c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a^5 c^3} \]

[Out]

(45*x*Sqrt[ArcTan[a*x]])/(128*a^4*c^3*(1 + a^2*x^2)) + (45*ArcTan[a*x]^(3/2))/(256*a^5*c^3) + (5*x^4*ArcTan[a*
x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)^2) - (15*ArcTan[a*x]^(3/2))/(32*a^5*c^3*(1 + a^2*x^2)) - (x^3*ArcTan[a*x]^(5
/2))/(4*a^2*c^3*(1 + a^2*x^2)^2) - (3*x*ArcTan[a*x]^(5/2))/(8*a^4*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(7/2))/(
28*a^5*c^3) + (15*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a^5*c^3) - (15*Sqrt[Pi]*FresnelS[
(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(128*a^5*c^3) + (15*Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(256*a^5*c^3) - (15
*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(2048*a^5*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.486393, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {4940, 4936, 4930, 4892, 4970, 4406, 12, 3305, 3351, 3312, 3296} \[ \frac{15 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^5 c^3}-\frac{15 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a^5 c^3}+\frac{5 x^4 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac{x^3 \tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}-\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 a^4 c^3 \left (a^2 x^2+1\right )}+\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 a^4 c^3 \left (a^2 x^2+1\right )}-\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a^5 c^3 \left (a^2 x^2+1\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a^5 c^3}+\frac{45 \tan ^{-1}(a x)^{3/2}}{256 a^5 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a^5 c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a^5 c^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3,x]

[Out]

(45*x*Sqrt[ArcTan[a*x]])/(128*a^4*c^3*(1 + a^2*x^2)) + (45*ArcTan[a*x]^(3/2))/(256*a^5*c^3) + (5*x^4*ArcTan[a*
x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)^2) - (15*ArcTan[a*x]^(3/2))/(32*a^5*c^3*(1 + a^2*x^2)) - (x^3*ArcTan[a*x]^(5
/2))/(4*a^2*c^3*(1 + a^2*x^2)^2) - (3*x*ArcTan[a*x]^(5/2))/(8*a^4*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(7/2))/(
28*a^5*c^3) + (15*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a^5*c^3) - (15*Sqrt[Pi]*FresnelS[
(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(128*a^5*c^3) + (15*Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(256*a^5*c^3) - (15
*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(2048*a^5*c^3)

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 4892

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

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])

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3351

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

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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rubi steps

\begin{align*} \int \frac{x^4 \tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{5 x^4 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{x^3 \tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{15}{64} \int \frac{x^4 \sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{3 \int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a^2 c}\\ &=\frac{5 x^4 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{x^3 \tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a^5 c^3}-\frac{15 \operatorname{Subst}\left (\int \sqrt{x} \sin ^4(x) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^5 c^3}+\frac{15 \int \frac{x \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a^3 c}\\ &=\frac{5 x^4 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac{x^3 \tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a^5 c^3}-\frac{15 \operatorname{Subst}\left (\int \left (\frac{3 \sqrt{x}}{8}-\frac{1}{2} \sqrt{x} \cos (2 x)+\frac{1}{8} \sqrt{x} \cos (4 x)\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^5 c^3}+\frac{45 \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{64 a^4 c}\\ &=\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{45 \tan ^{-1}(a x)^{3/2}}{256 a^5 c^3}+\frac{5 x^4 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac{x^3 \tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a^5 c^3}-\frac{15 \operatorname{Subst}\left (\int \sqrt{x} \cos (4 x) \, dx,x,\tan ^{-1}(a x)\right )}{512 a^5 c^3}+\frac{15 \operatorname{Subst}\left (\int \sqrt{x} \cos (2 x) \, dx,x,\tan ^{-1}(a x)\right )}{128 a^5 c^3}-\frac{45 \int \frac{x}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{256 a^3 c}\\ &=\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{45 \tan ^{-1}(a x)^{3/2}}{256 a^5 c^3}+\frac{5 x^4 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac{x^3 \tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a^5 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a^5 c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a^5 c^3}+\frac{15 \operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{4096 a^5 c^3}-\frac{15 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a^5 c^3}-\frac{45 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{256 a^5 c^3}\\ &=\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{45 \tan ^{-1}(a x)^{3/2}}{256 a^5 c^3}+\frac{5 x^4 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac{x^3 \tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a^5 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a^5 c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a^5 c^3}+\frac{15 \operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{2048 a^5 c^3}-\frac{15 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{256 a^5 c^3}-\frac{45 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{256 a^5 c^3}\\ &=\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{45 \tan ^{-1}(a x)^{3/2}}{256 a^5 c^3}+\frac{5 x^4 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac{x^3 \tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a^5 c^3}+\frac{15 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^5 c^3}-\frac{15 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{512 a^5 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a^5 c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a^5 c^3}-\frac{45 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a^5 c^3}\\ &=\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{45 \tan ^{-1}(a x)^{3/2}}{256 a^5 c^3}+\frac{5 x^4 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac{x^3 \tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a^5 c^3}+\frac{15 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^5 c^3}-\frac{15 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{512 a^5 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a^5 c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a^5 c^3}-\frac{45 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{256 a^5 c^3}\\ &=\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{45 \tan ^{-1}(a x)^{3/2}}{256 a^5 c^3}+\frac{5 x^4 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac{x^3 \tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a^5 c^3}+\frac{15 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a^5 c^3}-\frac{15 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a^5 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a^5 c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a^5 c^3}\\ \end{align*}

Mathematica [C]  time = 0.598959, size = 287, normalized size = 0.93 \[ \frac{3360 \sqrt{2} \left (a^2 x^2+1\right )^2 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \tan ^{-1}(a x)\right )+3360 \sqrt{2} \left (a^2 x^2+1\right )^2 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \tan ^{-1}(a x)\right )-105 \left (a^2 x^2+1\right )^2 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )-105 \left (a^2 x^2+1\right )^2 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )+38080 a^4 x^4 \tan ^{-1}(a x)^2-71680 a^3 x^3 \tan ^{-1}(a x)^3+57120 a^3 x^3 \tan ^{-1}(a x)-13440 a^2 x^2 \tan ^{-1}(a x)^2+12288 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^4-43008 a x \tan ^{-1}(a x)^3+50400 a x \tan ^{-1}(a x)-33600 \tan ^{-1}(a x)^2}{114688 a^5 c^3 \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3,x]

[Out]

(50400*a*x*ArcTan[a*x] + 57120*a^3*x^3*ArcTan[a*x] - 33600*ArcTan[a*x]^2 - 13440*a^2*x^2*ArcTan[a*x]^2 + 38080
*a^4*x^4*ArcTan[a*x]^2 - 43008*a*x*ArcTan[a*x]^3 - 71680*a^3*x^3*ArcTan[a*x]^3 + 12288*(1 + a^2*x^2)^2*ArcTan[
a*x]^4 + 3360*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + 3360*Sqrt[2]*(1
+ a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - 105*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Ga
mma[1/2, (-4*I)*ArcTan[a*x]] - 105*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(114688*
a^5*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])

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Maple [A]  time = 0.122, size = 194, normalized size = 0.6 \begin{align*}{\frac{3}{28\,{c}^{3}{a}^{5}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{7}{2}}}}-{\frac{\sin \left ( 2\,\arctan \left ( ax \right ) \right ) }{4\,{c}^{3}{a}^{5}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{\sin \left ( 4\,\arctan \left ( ax \right ) \right ) }{32\,{c}^{3}{a}^{5}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{5\,\cos \left ( 2\,\arctan \left ( ax \right ) \right ) }{16\,{c}^{3}{a}^{5}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{5\,\cos \left ( 4\,\arctan \left ( ax \right ) \right ) }{256\,{c}^{3}{a}^{5}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{15\,\sin \left ( 2\,\arctan \left ( ax \right ) \right ) }{64\,{c}^{3}{a}^{5}}\sqrt{\arctan \left ( ax \right ) }}-{\frac{15\,\sin \left ( 4\,\arctan \left ( ax \right ) \right ) }{2048\,{c}^{3}{a}^{5}}\sqrt{\arctan \left ( ax \right ) }}+{\frac{15\,\sqrt{2}\sqrt{\pi }}{8192\,{c}^{3}{a}^{5}}{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) }-{\frac{15\,\sqrt{\pi }}{128\,{c}^{3}{a}^{5}}{\it FresnelS} \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^4*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^3,x)

[Out]

3/28*arctan(a*x)^(7/2)/a^5/c^3-1/4/a^5/c^3*arctan(a*x)^(5/2)*sin(2*arctan(a*x))+1/32/a^5/c^3*arctan(a*x)^(5/2)
*sin(4*arctan(a*x))-5/16/a^5/c^3*arctan(a*x)^(3/2)*cos(2*arctan(a*x))+5/256/a^5/c^3*arctan(a*x)^(3/2)*cos(4*ar
ctan(a*x))+15/64*sin(2*arctan(a*x))*arctan(a*x)^(1/2)/a^5/c^3-15/2048*sin(4*arctan(a*x))*arctan(a*x)^(1/2)/a^5
/c^3+15/8192*FresnelS(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^5/c^3-15/128*FresnelS(2*arctan(
a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a^5/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^4*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^4*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**4*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^{4} \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^4*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

integrate(x^4*arctan(a*x)^(5/2)/(a^2*c*x^2 + c)^3, x)