\(\int \frac {(c+a^2 c x^2)^{5/2} \arctan (a x)}{x^3} \, dx\) [222]

Optimal result

Integrand size = 22, antiderivative size = 364 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=-\frac {a c^2 \sqrt {c+a^2 c x^2}}{2 x}-\frac {1}{6} a^3 c^2 x \sqrt {c+a^2 c x^2}+2 a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {5 a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {13}{6} a^2 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {5 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {5 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}} \]

[Out]

1/3*a^2*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)-13/6*a^2*c^(5/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))-5*a^2*c^3*
arctan(a*x)*arctanh((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+5/2*I*a^2*c^3*polyl
og(2,-(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-5/2*I*a^2*c^3*polylog(2,(1+I*a*x)
^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-1/2*a*c^2*(a^2*c*x^2+c)^(1/2)/x-1/6*a^3*c^2*x*(a
^2*c*x^2+c)^(1/2)+2*a^2*c^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)-1/2*c^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5070, 5066, 5082, 270, 5078, 5074, 223, 212, 5050, 201} \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=-\frac {5 a^2 c^3 \sqrt {a^2 x^2+1} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+2 a^2 c^2 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {c^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{2 x^2}+\frac {1}{3} a^2 c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}-\frac {13}{6} a^2 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )+\frac {5 i a^2 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {5 i a^2 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {a c^2 \sqrt {a^2 c x^2+c}}{2 x}-\frac {1}{6} a^3 c^2 x \sqrt {a^2 c x^2+c} \]

[In]

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

[Out]

-1/2*(a*c^2*Sqrt[c + a^2*c*x^2])/x - (a^3*c^2*x*Sqrt[c + a^2*c*x^2])/6 + 2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[
a*x] - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*x^2) + (a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/3 - (5*a^2*c^
3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (13*a^2*c^(5/2
)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/6 + (((5*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I
*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (((5*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]
/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 5050

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 5066

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

Rule 5070

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

Rule 5074

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

Rule 5078

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

Rule 5082

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

Rubi steps \begin{align*} \text {integral}& = c \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^3} \, dx+\left (a^2 c\right ) \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx \\ & = c^2 \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x^3} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x} \, dx\right )+\left (a^4 c^2\right ) \int x \sqrt {c+a^2 c x^2} \arctan (a x) \, dx \\ & = -\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {1}{3} \left (a^3 c^2\right ) \int \sqrt {c+a^2 c x^2} \, dx-c^3 \int \frac {\arctan (a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx+\left (a c^3\right ) \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)+\left (a^2 c^3\right ) \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx-\left (a^3 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx\right ) \\ & = -\frac {a c^2 \sqrt {c+a^2 c x^2}}{x}-\frac {1}{6} a^3 c^2 x \sqrt {c+a^2 c x^2}-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {1}{2} \left (a c^3\right ) \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{2} \left (a^2 c^3\right ) \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{6} \left (a^3 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\left (a^3 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\right ) \\ & = -\frac {a c^2 \sqrt {c+a^2 c x^2}}{2 x}-\frac {1}{6} a^3 c^2 x \sqrt {c+a^2 c x^2}-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {2 a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-a^2 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}\right )-\frac {1}{6} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{x \sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {a c^2 \sqrt {c+a^2 c x^2}}{2 x}-\frac {1}{6} a^3 c^2 x \sqrt {c+a^2 c x^2}-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {1}{6} a^2 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {2 a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-a^2 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 1.91 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.03 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \left (-6-6 \cot ^2\left (\frac {1}{2} \arctan (a x)\right )-2 a^2 x^2 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+28 a x \arctan (a x) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+4 a^3 x^3 \arctan (a x) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-3 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+60 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1-e^{i \arctan (a x)}\right )-60 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1+e^{i \arctan (a x)}\right )+4 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (-a x+\sqrt {1+a^2 x^2}\right )+48 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )-\sin \left (\frac {1}{2} \arctan (a x)\right )\right )-48 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )+\sin \left (\frac {1}{2} \arctan (a x)\right )\right )+60 i \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-60 i \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+3 \arctan (a x) \csc \left (\frac {1}{2} \arctan (a x)\right ) \sec \left (\frac {1}{2} \arctan (a x)\right )\right ) \tan \left (\frac {1}{2} \arctan (a x)\right )}{24 \sqrt {1+a^2 x^2}} \]

[In]

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

[Out]

(a^2*c^2*Sqrt[c + a^2*c*x^2]*(-6 - 6*Cot[ArcTan[a*x]/2]^2 - 2*a^2*x^2*Csc[ArcTan[a*x]/2]^2 + 28*a*x*ArcTan[a*x
]*Csc[ArcTan[a*x]/2]^2 + 4*a^3*x^3*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 - 3*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Csc[Arc
Tan[a*x]/2]^2 + 60*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 - E^(I*ArcTan[a*x])] - 60*ArcTan[a*x]*Cot[ArcTan[a*x]/
2]*Log[1 + E^(I*ArcTan[a*x])] + 4*Cot[ArcTan[a*x]/2]*Log[-(a*x) + Sqrt[1 + a^2*x^2]] + 48*Cot[ArcTan[a*x]/2]*L
og[Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]] - 48*Cot[ArcTan[a*x]/2]*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2
]] + (60*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, -E^(I*ArcTan[a*x])] - (60*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, E^(I*ArcT
an[a*x])] + 3*ArcTan[a*x]*Csc[ArcTan[a*x]/2]*Sec[ArcTan[a*x]/2])*Tan[ArcTan[a*x]/2])/(24*Sqrt[1 + a^2*x^2])

Maple [A] (verified)

Time = 1.93 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.70

method	result	size
default	\(-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (-2 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{4} x^{4}+\sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+15 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) \arctan \left (a x \right ) a^{2} x^{2}-14 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}-15 i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) a^{2} x^{2}-26 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a^{2} x^{2}-15 i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a^{2} x^{2}+3 \sqrt {a^{2} x^{2}+1}\, a x +3 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\right ) c^{2}}{6 \sqrt {a^{2} x^{2}+1}\, x^{2}}\)	\(256\)

[In]

int((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/6/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)*(-2*arctan(a*x)*(a^2*x^2+1)^(1/2)*a^4*x^4+(a^2*x^2+1)^(1/2)*a
^3*x^3+15*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)*arctan(a*x)*a^2*x^2-14*arctan(a*x)*(a^2*x^2+1)^(1/2)*a^2*x^2-15*I*
dilog((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)*a^2*x^2-26*I*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*a^2*x^2-15*I*dilog((1+I*
a*x)/(a^2*x^2+1)^(1/2))*a^2*x^2+3*(a^2*x^2+1)^(1/2)*a*x+3*arctan(a*x)*(a^2*x^2+1)^(1/2))*c^2/x^2

Fricas [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )}{x^{3}} \,d x } \]

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x^3,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*sqrt(a^2*c*x^2 + c)*arctan(a*x)/x^3, x)

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}{\left (a x \right )}}{x^{3}}\, dx \]

[In]

integrate((a**2*c*x**2+c)**(5/2)*atan(a*x)/x**3,x)

[Out]

Integral((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)/x**3, x)

Maxima [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )}{x^{3}} \,d x } \]

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x^3,x, algorithm="maxima")

[Out]

1/3*(a^4*c^2*x^2 + a^2*c^2)*sqrt(a^2*x^2 + 1)*sqrt(c)*arctan(a*x) - 1/6*(a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*(a^3*
c^2*x*cos(1/2*arctan2(4*a*x, -a^2*x^2 + 3)) + 2*a^2*c^2*sin(1/2*arctan2(4*a*x, -a^2*x^2 + 3)))*sqrt(c) + 1/12*
(a^2*c^2*arctan2((a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*sin(1/2*arctan2(4*a*x, a^2*x^2 - 3)) + 2, a*x + (a^4*x^4 + 1
0*a^2*x^2 + 9)^(1/4)*cos(1/2*arctan2(4*a*x, a^2*x^2 - 3))) + a^2*c^2*arctan2((a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*
sin(1/2*arctan2(4*a*x, a^2*x^2 - 3)) - 2, -a*x + (a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*cos(1/2*arctan2(4*a*x, a^2*x
^2 - 3))) + 24*a^2*c^2*integrate(sqrt(a^2*x^2 + 1)*arctan(a*x)/x, x) + 12*c^2*integrate(sqrt(a^2*x^2 + 1)*arct
an(a*x)/x^3, x))*sqrt(c)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x^3} \,d x \]

[In]

int((atan(a*x)*(c + a^2*c*x^2)^(5/2))/x^3,x)

[Out]

int((atan(a*x)*(c + a^2*c*x^2)^(5/2))/x^3, x)