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

Optimal. Leaf size=337 \[ \frac{45 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{16 a c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{144 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^{5/2}}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{3 a c^2 \sqrt{a^2 c x^2+c}}-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{16 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \sin \left (3 \tan ^{-1}(a x)\right )}{144 a c^2 \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

(-45*x*Sqrt[ArcTan[a*x]])/(16*c^2*Sqrt[c + a^2*c*x^2]) + (5*ArcTan[a*x]^(3/2))/(18*a*c*(c + a^2*c*x^2)^(3/2))
+ (5*ArcTan[a*x]^(3/2))/(3*a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^(5/2))/(3*c*(c + a^2*c*x^2)^(3/2)) + (2
*x*ArcTan[a*x]^(5/2))/(3*c^2*Sqrt[c + a^2*c*x^2]) + (45*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[
ArcTan[a*x]]])/(16*a*c^2*Sqrt[c + a^2*c*x^2]) + (5*Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTa
n[a*x]]])/(144*a*c^2*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Sin[3*ArcTan[a*x]])/(144*a*
c^2*Sqrt[c + a^2*c*x^2])

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Rubi [A]  time = 0.414009, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {4900, 4898, 4905, 4904, 3296, 3305, 3351, 3312} \[ \frac{45 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{16 a c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{144 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^{5/2}}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{3 a c^2 \sqrt{a^2 c x^2+c}}-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{16 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \sin \left (3 \tan ^{-1}(a x)\right )}{144 a c^2 \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*x*Sqrt[ArcTan[a*x]])/(16*c^2*Sqrt[c + a^2*c*x^2]) + (5*ArcTan[a*x]^(3/2))/(18*a*c*(c + a^2*c*x^2)^(3/2))
+ (5*ArcTan[a*x]^(3/2))/(3*a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^(5/2))/(3*c*(c + a^2*c*x^2)^(3/2)) + (2
*x*ArcTan[a*x]^(5/2))/(3*c^2*Sqrt[c + a^2*c*x^2]) + (45*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[
ArcTan[a*x]]])/(16*a*c^2*Sqrt[c + a^2*c*x^2]) + (5*Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTa
n[a*x]]])/(144*a*c^2*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Sin[3*ArcTan[a*x]])/(144*a*
c^2*Sqrt[c + a^2*c*x^2])

Rule 4900

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b*p*(d + e*x^2)^(q
+ 1)*(a + b*ArcTan[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q +
1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(
p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e
}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

Rule 4898

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] + (-Dist[b^2*p*(p - 1), Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(
3/2), x], x] + Simp[(x*(a + b*ArcTan[c*x])^p)/(d*Sqrt[d + e*x^2]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e,
c^2*d] && GtQ[p, 1]

Rule 4905

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

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])
)

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac{5 \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5}{12} \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac{2 \int \frac{\tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}\\ &=\frac{5 \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{3 a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^{5/2}}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{5 \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{2 c}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{5/2}} \, dx}{12 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{5 \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{3 a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^{5/2}}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sqrt{x} \cos ^3(x) \, dx,x,\tan ^{-1}(a x)\right )}{12 a c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{5 \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{3 a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^{5/2}}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{4} \sqrt{x} \cos (x)+\frac{1}{4} \sqrt{x} \cos (3 x)\right ) \, dx,x,\tan ^{-1}(a x)\right )}{12 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sqrt{x} \cos (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{5 x \sqrt{\tan ^{-1}(a x)}}{2 c^2 \sqrt{c+a^2 c x^2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{3 a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^{5/2}}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sqrt{x} \cos (3 x) \, dx,x,\tan ^{-1}(a x)\right )}{48 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sqrt{x} \cos (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{16 c^2 \sqrt{c+a^2 c x^2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{3 a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^{5/2}}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{5 \sqrt{1+a^2 x^2} \sqrt{\tan ^{-1}(a x)} \sin \left (3 \tan ^{-1}(a x)\right )}{144 a c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{288 a c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{32 a c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{2 a c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{16 c^2 \sqrt{c+a^2 c x^2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{3 a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^{5/2}}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{5 \sqrt{\frac{\pi }{2}} \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{2 a c^2 \sqrt{c+a^2 c x^2}}-\frac{5 \sqrt{1+a^2 x^2} \sqrt{\tan ^{-1}(a x)} \sin \left (3 \tan ^{-1}(a x)\right )}{144 a c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{144 a c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{16 a c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{16 c^2 \sqrt{c+a^2 c x^2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{3 a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{5/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^{5/2}}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{45 \sqrt{\frac{\pi }{2}} \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{16 a c^2 \sqrt{c+a^2 c x^2}}+\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{144 a c^2 \sqrt{c+a^2 c x^2}}-\frac{5 \sqrt{1+a^2 x^2} \sqrt{\tan ^{-1}(a x)} \sin \left (3 \tan ^{-1}(a x)\right )}{144 a c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [C]  time = 1.0915, size = 293, normalized size = 0.87 \[ \frac{-105 \left (a^2 x^2+1\right )^{3/2} \left (3 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \tan ^{-1}(a x)\right )+3 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \tan ^{-1}(a x)\right )+\sqrt{3} \left (\sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \tan ^{-1}(a x)\right )+\sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \tan ^{-1}(a x)\right )\right )\right )+200 \sqrt{6 \pi } \left (a^2 x^2+1\right )^{3/2} \sqrt{\tan ^{-1}(a x)} \left (3 \sqrt{3} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )-S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )\right )-48 \tan ^{-1}(a x) \left (5 a x \left (20 a^2 x^2+21\right )-12 a x \left (2 a^2 x^2+3\right ) \tan ^{-1}(a x)^2-10 \left (6 a^2 x^2+7\right ) \tan ^{-1}(a x)\right )}{1728 a c \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-48*ArcTan[a*x]*(5*a*x*(21 + 20*a^2*x^2) - 10*(7 + 6*a^2*x^2)*ArcTan[a*x] - 12*a*x*(3 + 2*a^2*x^2)*ArcTan[a*x
]^2) + 200*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]]]) - 105*(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]])))/(1728*a*c*(c + a^2*c*x^2)^(3/2)*Sqrt[
ArcTan[a*x]])

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Maple [F]  time = 0.741, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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