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

Optimal. Leaf size=293 \[ -\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{16 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{144 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{36 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

(5*Sqrt[ArcTan[a*x]])/(36*a^2*c*(c + a^2*c*x^2)^(3/2)) + (5*Sqrt[ArcTan[a*x]])/(6*a^2*c^2*Sqrt[c + a^2*c*x^2])
+ (5*x*ArcTan[a*x]^(3/2))/(18*a*c*(c + a^2*c*x^2)^(3/2)) + (5*x*ArcTan[a*x]^(3/2))/(9*a*c^2*Sqrt[c + a^2*c*x^
2]) - ArcTan[a*x]^(5/2)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (15*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]
*Sqrt[ArcTan[a*x]]])/(16*a^2*c^2*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sq
rt[ArcTan[a*x]]])/(144*a^2*c^2*Sqrt[c + a^2*c*x^2])

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Rubi [A]  time = 0.450979, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4930, 4900, 4898, 4905, 4904, 3304, 3352, 3312} \[ -\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{16 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{144 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{36 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[ArcTan[a*x]])/(36*a^2*c*(c + a^2*c*x^2)^(3/2)) + (5*Sqrt[ArcTan[a*x]])/(6*a^2*c^2*Sqrt[c + a^2*c*x^2])
+ (5*x*ArcTan[a*x]^(3/2))/(18*a*c*(c + a^2*c*x^2)^(3/2)) + (5*x*ArcTan[a*x]^(3/2))/(9*a*c^2*Sqrt[c + a^2*c*x^
2]) - ArcTan[a*x]^(5/2)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (15*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]
*Sqrt[ArcTan[a*x]]])/(16*a^2*c^2*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sq
rt[ArcTan[a*x]]])/(144*a^2*c^2*Sqrt[c + a^2*c*x^2])

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 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 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 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{x \tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\frac{\tan ^{-1}(a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \int \frac{\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{6 a}\\ &=\frac{5 \sqrt{\tan ^{-1}(a x)}}{36 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac{\tan ^{-1}(a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{72 a}+\frac{5 \int \frac{\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 a c}\\ &=\frac{5 \sqrt{\tan ^{-1}(a x)}}{36 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{9 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{12 a c}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \int \frac{1}{\left (1+a^2 x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{72 a c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{5 \sqrt{\tan ^{-1}(a x)}}{36 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{9 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos ^3(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{72 a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \int \frac{1}{\left (1+a^2 x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{12 a c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{5 \sqrt{\tan ^{-1}(a x)}}{36 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{9 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3 \cos (x)}{4 \sqrt{x}}+\frac{\cos (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{72 a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{12 a^2 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{5 \sqrt{\tan ^{-1}(a x)}}{36 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{9 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{288 a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{96 a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{6 a^2 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{5 \sqrt{\tan ^{-1}(a x)}}{36 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{9 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 \sqrt{\frac{\pi }{2}} \sqrt{1+a^2 x^2} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{6 a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{144 a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (5 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{48 a^2 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{5 \sqrt{\tan ^{-1}(a x)}}{36 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{9 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{1+a^2 x^2} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{16 a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{1+a^2 x^2} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{144 a^2 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [C]  time = 0.5335, size = 356, normalized size = 1.22 \[ \frac{5 i a^2 x^2 \sqrt{3 a^2 x^2+3} \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \tan ^{-1}(a x)\right )-5 i a^2 x^2 \sqrt{3 a^2 x^2+3} \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \tan ^{-1}(a x)\right )+405 i \left (a^2 x^2+1\right )^{3/2} \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \tan ^{-1}(a x)\right )-405 i \left (a^2 x^2+1\right )^{3/2} \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \tan ^{-1}(a x)\right )+5 i \sqrt{3 a^2 x^2+3} \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \tan ^{-1}(a x)\right )-5 i \sqrt{3 a^2 x^2+3} \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \tan ^{-1}(a x)\right )+960 a^3 x^3 \tan ^{-1}(a x)^2+1440 a^2 x^2 \tan ^{-1}(a x)+1440 a x \tan ^{-1}(a x)^2-576 \tan ^{-1}(a x)^3+1680 \tan ^{-1}(a x)}{1728 a^2 c^2 \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(1680*ArcTan[a*x] + 1440*a^2*x^2*ArcTan[a*x] + 1440*a*x*ArcTan[a*x]^2 + 960*a^3*x^3*ArcTan[a*x]^2 - 576*ArcTan
[a*x]^3 + (405*I)*(1 + a^2*x^2)^(3/2)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] - (405*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]]*Gam
ma[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)*ArcTa
n[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]])/(1728*a^2*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]
*Sqrt[ArcTan[a*x]])

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Maple [F]  time = 0.799, size = 0, normalized size = 0. \begin{align*} \int{x \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(x*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(5/2),x)

[Out]

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

[Out]

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