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

Optimal. Leaf size=295 \[ -\frac {5 x \sqrt {\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {5 x^2 \tan ^{-1}(a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \tan ^{-1}(a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5 x^3 \sqrt {\tan ^{-1}(a x)}}{36 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{16 a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {5 \sqrt {\frac {\pi }{6}} \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{144 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {5 \tan ^{-1}(a x)^{3/2}}{9 a^3 c^2 \sqrt {a^2 c x^2+c}} \]

[Out]

5/18*x^2*arctan(a*x)^(3/2)/a/c/(a^2*c*x^2+c)^(3/2)+1/3*x^3*arctan(a*x)^(5/2)/c/(a^2*c*x^2+c)^(3/2)+5/9*arctan(
a*x)^(3/2)/a^3/c^2/(a^2*c*x^2+c)^(1/2)-5/864*FresnelS(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)+15/32*FresnelS(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)-5/36*x^3*arctan(a*x)^(1/2)/c/(a^2*c*x^2+c)^(3/2)-5/6*x*arctan(a
*x)^(1/2)/a^2/c^2/(a^2*c*x^2+c)^(1/2)

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Rubi [A]  time = 0.78, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {4944, 4940, 4930, 4905, 4904, 3296, 3305, 3351, 4971, 4970, 3312} \[ \frac {15 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{16 a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {5 \sqrt {\frac {\pi }{6}} \sqrt {a^2 x^2+1} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{144 a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {5 x \sqrt {\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {5 \tan ^{-1}(a x)^{3/2}}{9 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x^3 \tan ^{-1}(a x)^{5/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {5 x^3 \sqrt {\tan ^{-1}(a x)}}{36 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^2 \tan ^{-1}(a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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 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 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 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 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 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 4971

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

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Mathematica [C]  time = 1.13, size = 287, normalized size = 0.97 \[ \frac {35 \sqrt {6 \pi } \left (a^2 x^2+1\right )^{3/2} \sqrt {\tan ^{-1}(a x)} \left (3 \sqrt {3} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )-S\left (\sqrt {\frac {6}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )\right )-15 \left (a^2 x^2+1\right )^{3/2} \left (3 \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-i \tan ^{-1}(a x)\right )+3 \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},i \tan ^{-1}(a x)\right )+\sqrt {3} \left (\sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 i \tan ^{-1}(a x)\right )+\sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},3 i \tan ^{-1}(a x)\right )\right )\right )-24 \tan ^{-1}(a x) \left (-12 a^3 x^3 \tan ^{-1}(a x)^2+5 a x \left (7 a^2 x^2+6\right )-10 \left (3 a^2 x^2+2\right ) \tan ^{-1}(a x)\right )}{864 a^3 c \left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-24*ArcTan[a*x]*(5*a*x*(6 + 7*a^2*x^2) - 10*(2 + 3*a^2*x^2)*ArcTan[a*x] - 12*a^3*x^3*ArcTan[a*x]^2) + 35*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]]]) - 15*(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]])))/(864*a^3*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [F]  time = 8.79, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \arctan \left (a x \right )^{\frac {5}{2}}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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