3.12.4 \(\int \frac {x^2}{(c+a^2 c x^2)^{5/2} \text {ArcTan}(a x)^{5/2}} \, dx\) [1104]

Optimal. Leaf size=224 \[ -\frac {2 x^2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{3/2}}-\frac {8 x}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\text {ArcTan}(a x)}}+\frac {4 x^3}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\text {ArcTan}(a x)}}-\frac {\sqrt {2 \pi } \sqrt {1+a^2 x^2} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {6 \pi } \sqrt {1+a^2 x^2} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}} \]

[Out]

-2/3*x^2/a/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(3/2)-1/3*FresnelC(2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi
^(1/2)*(a^2*x^2+1)^(1/2)/a^3/c^2/(a^2*c*x^2+c)^(1/2)+FresnelC(6^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*6^(1/2)*Pi^(
1/2)*(a^2*x^2+1)^(1/2)/a^3/c^2/(a^2*c*x^2+c)^(1/2)-8/3*x/a^2/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2)+4/3*x^3/c
/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2)

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Rubi [A]
time = 0.81, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5088, 5062, 5091, 5090, 4491, 3385, 3433, 5025, 5024, 3393} \begin {gather*} -\frac {2 x^2}{3 a c \text {ArcTan}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}-\frac {8 x}{3 a^2 c \sqrt {\text {ArcTan}(a x)} \left (a^2 c x^2+c\right )^{3/2}}+\frac {4 x^3}{3 c \sqrt {\text {ArcTan}(a x)} \left (a^2 c x^2+c\right )^{3/2}}-\frac {\sqrt {2 \pi } \sqrt {a^2 x^2+1} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{3 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {6 \pi } \sqrt {a^2 x^2+1} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{a^3 c^2 \sqrt {a^2 c x^2+c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x^2)/(3*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)) - (8*x)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*
x]]) + (4*x^3)/(3*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/P
i]*Sqrt[ArcTan[a*x]]])/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[6*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqr
t[ArcTan[a*x]]])/(a^3*c^2*Sqrt[c + a^2*c*x^2])

Rule 3385

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 3393

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 3433

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

Rule 4491

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 5024

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 5025

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 5062

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[
(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1))), x] - Dist[f*(m/(b*c*(p + 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 + 2, 0] && LtQ[p, -1]

Rule 5088

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

Rule 5090

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 5091

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^(q + 1
/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]), Int[x^m*(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] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] &&  !(IntegerQ[q] || GtQ[d, 0])

Rubi steps

\begin {align*} \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 x^2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac {4 \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (2 a) \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2 x^2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}+\frac {4 x^3}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}-4 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx-\frac {16}{3} \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}\\ &=-\frac {2 x^2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}+\frac {4 x^3}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \int \frac {x^2}{\left (1+a^2 x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \int \frac {x^2}{\left (1+a^2 x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (8 \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}{3 a^2 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x^2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}+\frac {4 x^3}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos ^3(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x^2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}+\frac {4 x^3}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {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 )}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\cos (x)}{4 \sqrt {x}}-\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (16 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\cos (x)}{4 \sqrt {x}}-\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x^2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}+\frac {4 x^3}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}+\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x^2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}+\frac {4 x^3}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}+\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 x^2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac {8 x}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}+\frac {4 x^3}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}-\frac {\sqrt {2 \pi } \sqrt {1+a^2 x^2} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {6 \pi } \sqrt {1+a^2 x^2} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.59, size = 311, normalized size = 1.39 \begin {gather*} \frac {-\left (1+a^2 x^2\right )^{3/2} (-i \text {ArcTan}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},-i \text {ArcTan}(a x)\right )+\frac {-4 a^2 x^2 \sqrt {i \text {ArcTan}(a x)}+16 i a x (i \text {ArcTan}(a x))^{3/2}-8 i a^3 x^3 (i \text {ArcTan}(a x))^{3/2}+\left (1+a^2 x^2\right )^{3/2} \text {ArcTan}(a x)^2 \text {Gamma}\left (\frac {1}{2},i \text {ArcTan}(a x)\right )-3 i \sqrt {3} \left (1+a^2 x^2\right )^{3/2} \text {ArcTan}(a x) \sqrt {\text {ArcTan}(a x)^2} \text {Gamma}\left (\frac {1}{2},-3 i \text {ArcTan}(a x)\right )-3 \sqrt {3+3 a^2 x^2} \text {ArcTan}(a x)^2 \text {Gamma}\left (\frac {1}{2},3 i \text {ArcTan}(a x)\right )-3 a^2 x^2 \sqrt {3+3 a^2 x^2} \text {ArcTan}(a x)^2 \text {Gamma}\left (\frac {1}{2},3 i \text {ArcTan}(a x)\right )}{\sqrt {i \text {ArcTan}(a x)}}}{6 a^3 c^2 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((1 + a^2*x^2)^(3/2)*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-I)*ArcTan[a*x]]) + (-4*a^2*x^2*Sqrt[I*ArcTan[a*x]
] + (16*I)*a*x*(I*ArcTan[a*x])^(3/2) - (8*I)*a^3*x^3*(I*ArcTan[a*x])^(3/2) + (1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2
*Gamma[1/2, I*ArcTan[a*x]] - (3*I)*Sqrt[3]*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]*Sqrt[ArcTan[a*x]^2]*Gamma[1/2, (-3*
I)*ArcTan[a*x]] - 3*Sqrt[3 + 3*a^2*x^2]*ArcTan[a*x]^2*Gamma[1/2, (3*I)*ArcTan[a*x]] - 3*a^2*x^2*Sqrt[3 + 3*a^2
*x^2]*ArcTan[a*x]^2*Gamma[1/2, (3*I)*ArcTan[a*x]])/Sqrt[I*ArcTan[a*x]])/(6*a^3*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*
c*x^2]*ArcTan[a*x]^(3/2))

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Maple [F]
time = 7.81, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3063 deep

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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