3.1225 \(\int \frac{a+b \tan ^{-1}(c x)}{x^4 (d+e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=423 \[ \frac{16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt{d+e x^2}}-\frac{b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right ) \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{3 d^4 \left (c^2 d-e\right )^{3/2}}-\frac{b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{b c \left (c^2 d+6 e\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 d^{7/2}}+\frac{16 b e^2}{3 c d^4 \sqrt{d+e x^2}}-\frac{b c e}{2 d^3 \sqrt{d+e x^2}}-\frac{b c}{6 d^2 x^2 \sqrt{d+e x^2}}+\frac{b c e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 d^{7/2}} \]

[Out]

-(b*c*e)/(2*d^3*Sqrt[d + e*x^2]) + (16*b*e^2)/(3*c*d^4*Sqrt[d + e*x^2]) - (b*c*(c^2*d + 6*e))/(3*d^3*Sqrt[d +
e*x^2]) + (b*(c^2*d - 2*e)*(c^4*d^2 + 8*c^2*d*e - 8*e^2))/(3*c*d^4*(c^2*d - e)*Sqrt[d + e*x^2]) - (b*c)/(6*d^2
*x^2*Sqrt[d + e*x^2]) - (a + b*ArcTan[c*x])/(3*d*x^3*(d + e*x^2)^(3/2)) + (2*e*(a + b*ArcTan[c*x]))/(d^2*x*(d
+ e*x^2)^(3/2)) + (8*e^2*x*(a + b*ArcTan[c*x]))/(3*d^3*(d + e*x^2)^(3/2)) + (16*e^2*x*(a + b*ArcTan[c*x]))/(3*
d^4*Sqrt[d + e*x^2]) + (b*c*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*d^(7/2)) + (b*c*(c^2*d + 6*e)*ArcTanh[Sqrt[
d + e*x^2]/Sqrt[d]])/(3*d^(7/2)) - (b*(c^2*d - 2*e)*(c^4*d^2 + 8*c^2*d*e - 8*e^2)*ArcTanh[(c*Sqrt[d + e*x^2])/
Sqrt[c^2*d - e]])/(3*d^4*(c^2*d - e)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 1.09543, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {271, 192, 191, 4976, 12, 6725, 266, 51, 63, 208, 261, 444} \[ \frac{16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt{d+e x^2}}-\frac{b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right ) \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{3 d^4 \left (c^2 d-e\right )^{3/2}}-\frac{b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{b c \left (c^2 d+6 e\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 d^{7/2}}+\frac{16 b e^2}{3 c d^4 \sqrt{d+e x^2}}-\frac{b c \sqrt{d+e x^2}}{2 d^3 x^2}+\frac{b c}{3 d^2 x^2 \sqrt{d+e x^2}}+\frac{b c e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 d^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x])/(x^4*(d + e*x^2)^(5/2)),x]

[Out]

(16*b*e^2)/(3*c*d^4*Sqrt[d + e*x^2]) - (b*c*(c^2*d + 6*e))/(3*d^3*Sqrt[d + e*x^2]) + (b*(c^2*d - 2*e)*(c^4*d^2
+ 8*c^2*d*e - 8*e^2))/(3*c*d^4*(c^2*d - e)*Sqrt[d + e*x^2]) + (b*c)/(3*d^2*x^2*Sqrt[d + e*x^2]) - (b*c*Sqrt[d
+ e*x^2])/(2*d^3*x^2) - (a + b*ArcTan[c*x])/(3*d*x^3*(d + e*x^2)^(3/2)) + (2*e*(a + b*ArcTan[c*x]))/(d^2*x*(d
+ e*x^2)^(3/2)) + (8*e^2*x*(a + b*ArcTan[c*x]))/(3*d^3*(d + e*x^2)^(3/2)) + (16*e^2*x*(a + b*ArcTan[c*x]))/(3
*d^4*Sqrt[d + e*x^2]) + (b*c*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*d^(7/2)) + (b*c*(c^2*d + 6*e)*ArcTanh[Sqrt
[d + e*x^2]/Sqrt[d]])/(3*d^(7/2)) - (b*(c^2*d - 2*e)*(c^4*d^2 + 8*c^2*d*e - 8*e^2)*ArcTanh[(c*Sqrt[d + e*x^2])
/Sqrt[c^2*d - e]])/(3*d^4*(c^2*d - e)^(3/2))

Rule 271

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

Rule 192

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

Rule 191

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

Rule 4976

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 + c^
2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ((IGtQ[q, 0] &&  !(ILtQ[(m - 1)/2, 0] && GtQ[m +
2*q + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[q, 0] && GtQ[m + 2*q + 3, 0])) || (ILtQ[(m + 2*q + 1)/2, 0] &&
!ILtQ[(m - 1)/2, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 261

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

Rule 444

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

Rubi steps

\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{x^4 \left (d+e x^2\right )^{5/2}} \, dx &=-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt{d+e x^2}}-(b c) \int \frac{-d^3+6 d^2 e x^2+24 d e^2 x^4+16 e^3 x^6}{3 d^4 x^3 \left (1+c^2 x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx\\ &=-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt{d+e x^2}}-\frac{(b c) \int \frac{-d^3+6 d^2 e x^2+24 d e^2 x^4+16 e^3 x^6}{x^3 \left (1+c^2 x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx}{3 d^4}\\ &=-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt{d+e x^2}}-\frac{(b c) \int \left (-\frac{d^3}{x^3 \left (d+e x^2\right )^{3/2}}+\frac{d^2 \left (c^2 d+6 e\right )}{x \left (d+e x^2\right )^{3/2}}+\frac{16 e^3 x}{c^2 \left (d+e x^2\right )^{3/2}}+\frac{\left (c^2 d-2 e\right ) \left (-c^4 d^2-8 c^2 d e+8 e^2\right ) x}{c^2 \left (1+c^2 x^2\right ) \left (d+e x^2\right )^{3/2}}\right ) \, dx}{3 d^4}\\ &=-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt{d+e x^2}}+\frac{(b c) \int \frac{1}{x^3 \left (d+e x^2\right )^{3/2}} \, dx}{3 d}-\frac{\left (16 b e^3\right ) \int \frac{x}{\left (d+e x^2\right )^{3/2}} \, dx}{3 c d^4}-\frac{\left (b c \left (c^2 d+6 e\right )\right ) \int \frac{1}{x \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2}+\frac{\left (b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )\right ) \int \frac{x}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx}{3 c d^4}\\ &=\frac{16 b e^2}{3 c d^4 \sqrt{d+e x^2}}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt{d+e x^2}}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x^2 (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 d}-\frac{\left (b c \left (c^2 d+6 e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 d^2}+\frac{\left (b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+c^2 x\right ) (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 c d^4}\\ &=\frac{16 b e^2}{3 c d^4 \sqrt{d+e x^2}}-\frac{b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt{d+e x^2}}+\frac{b c}{3 d^2 x^2 \sqrt{d+e x^2}}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt{d+e x^2}}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{d+e x}} \, dx,x,x^2\right )}{2 d^2}-\frac{\left (b c \left (c^2 d+6 e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )}{6 d^3}+\frac{\left (b c \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+c^2 x\right ) \sqrt{d+e x}} \, dx,x,x^2\right )}{6 d^4 \left (c^2 d-e\right )}\\ &=\frac{16 b e^2}{3 c d^4 \sqrt{d+e x^2}}-\frac{b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt{d+e x^2}}+\frac{b c}{3 d^2 x^2 \sqrt{d+e x^2}}-\frac{b c \sqrt{d+e x^2}}{2 d^3 x^2}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt{d+e x^2}}-\frac{(b c e) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )}{4 d^3}-\frac{\left (b c \left (c^2 d+6 e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{3 d^3 e}+\frac{\left (b c \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{c^2 d}{e}+\frac{c^2 x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{3 d^4 \left (c^2 d-e\right ) e}\\ &=\frac{16 b e^2}{3 c d^4 \sqrt{d+e x^2}}-\frac{b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt{d+e x^2}}+\frac{b c}{3 d^2 x^2 \sqrt{d+e x^2}}-\frac{b c \sqrt{d+e x^2}}{2 d^3 x^2}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt{d+e x^2}}+\frac{b c \left (c^2 d+6 e\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 d^{7/2}}-\frac{b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right ) \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{3 d^4 \left (c^2 d-e\right )^{3/2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{2 d^3}\\ &=\frac{16 b e^2}{3 c d^4 \sqrt{d+e x^2}}-\frac{b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt{d+e x^2}}+\frac{b c}{3 d^2 x^2 \sqrt{d+e x^2}}-\frac{b c \sqrt{d+e x^2}}{2 d^3 x^2}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac{2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac{16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt{d+e x^2}}+\frac{b c e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 d^{7/2}}+\frac{b c \left (c^2 d+6 e\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 d^{7/2}}-\frac{b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right ) \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{3 d^4 \left (c^2 d-e\right )^{3/2}}\\ \end{align*}

Mathematica [C]  time = 2.13869, size = 510, normalized size = 1.21 \[ -\frac{\frac{2 a \left (-6 d^2 e x^2+d^3-24 d e^2 x^4-16 e^3 x^6\right )}{x^3 \left (d+e x^2\right )^{3/2}}+\frac{b \left (6 c^4 d^2 e+c^6 d^3-24 c^2 d e^2+16 e^3\right ) \log \left (\frac{12 c d^4 \sqrt{c^2 d-e} \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right )}{b (c x+i) \left (6 c^4 d^2 e+c^6 d^3-24 c^2 d e^2+16 e^3\right )}\right )}{\left (c^2 d-e\right )^{3/2}}+\frac{b \left (6 c^4 d^2 e+c^6 d^3-24 c^2 d e^2+16 e^3\right ) \log \left (\frac{12 c d^4 \sqrt{c^2 d-e} \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right )}{b (c x-i) \left (6 c^4 d^2 e+c^6 d^3-24 c^2 d e^2+16 e^3\right )}\right )}{\left (c^2 d-e\right )^{3/2}}+\frac{b c d \left (c^2 d \left (d+e x^2\right )+e \left (e x^2-d\right )\right )}{x^2 \left (c^2 d-e\right ) \sqrt{d+e x^2}}-b c \sqrt{d} \left (2 c^2 d+15 e\right ) \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )+b c \sqrt{d} \log (x) \left (2 c^2 d+15 e\right )+\frac{2 b \tan ^{-1}(c x) \left (-6 d^2 e x^2+d^3-24 d e^2 x^4-16 e^3 x^6\right )}{x^3 \left (d+e x^2\right )^{3/2}}}{6 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTan[c*x])/(x^4*(d + e*x^2)^(5/2)),x]

[Out]

-((2*a*(d^3 - 6*d^2*e*x^2 - 24*d*e^2*x^4 - 16*e^3*x^6))/(x^3*(d + e*x^2)^(3/2)) + (b*c*d*(e*(-d + e*x^2) + c^2
*d*(d + e*x^2)))/((c^2*d - e)*x^2*Sqrt[d + e*x^2]) + (2*b*(d^3 - 6*d^2*e*x^2 - 24*d*e^2*x^4 - 16*e^3*x^6)*ArcT
an[c*x])/(x^3*(d + e*x^2)^(3/2)) + b*c*Sqrt[d]*(2*c^2*d + 15*e)*Log[x] - b*c*Sqrt[d]*(2*c^2*d + 15*e)*Log[d +
Sqrt[d]*Sqrt[d + e*x^2]] + (b*(c^6*d^3 + 6*c^4*d^2*e - 24*c^2*d*e^2 + 16*e^3)*Log[(12*c*d^4*Sqrt[c^2*d - e]*(c
*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^6*d^3 + 6*c^4*d^2*e - 24*c^2*d*e^2 + 16*e^3)*(I + c*x))])
/(c^2*d - e)^(3/2) + (b*(c^6*d^3 + 6*c^4*d^2*e - 24*c^2*d*e^2 + 16*e^3)*Log[(12*c*d^4*Sqrt[c^2*d - e]*(c*d + I
*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^6*d^3 + 6*c^4*d^2*e - 24*c^2*d*e^2 + 16*e^3)*(-I + c*x))])/(c^2
*d - e)^(3/2))/(6*d^4)

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Maple [F]  time = 0.601, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{{x}^{4}} \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x))/x^4/(e*x^2+d)^(5/2),x)

[Out]

int((a+b*arctan(c*x))/x^4/(e*x^2+d)^(5/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/x^4/(e*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 20.367, size = 7200, normalized size = 17.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/x^4/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

[-1/12*(((b*c^6*d^3*e^2 + 6*b*c^4*d^2*e^3 - 24*b*c^2*d*e^4 + 16*b*e^5)*x^7 + 2*(b*c^6*d^4*e + 6*b*c^4*d^3*e^2
- 24*b*c^2*d^2*e^3 + 16*b*d*e^4)*x^5 + (b*c^6*d^5 + 6*b*c^4*d^4*e - 24*b*c^2*d^3*e^2 + 16*b*d^2*e^3)*x^3)*sqrt
(c^2*d - e)*log((c^4*e^2*x^4 + 8*c^4*d^2 - 8*c^2*d*e + 2*(4*c^4*d*e - 3*c^2*e^2)*x^2 + 4*(c^3*e*x^2 + 2*c^3*d
- c*e)*sqrt(c^2*d - e)*sqrt(e*x^2 + d) + e^2)/(c^4*x^4 + 2*c^2*x^2 + 1)) - ((2*b*c^7*d^3*e^2 + 11*b*c^5*d^2*e^
3 - 28*b*c^3*d*e^4 + 15*b*c*e^5)*x^7 + 2*(2*b*c^7*d^4*e + 11*b*c^5*d^3*e^2 - 28*b*c^3*d^2*e^3 + 15*b*c*d*e^4)*
x^5 + (2*b*c^7*d^5 + 11*b*c^5*d^4*e - 28*b*c^3*d^3*e^2 + 15*b*c*d^2*e^3)*x^3)*sqrt(d)*log(-(e*x^2 + 2*sqrt(e*x
^2 + d)*sqrt(d) + 2*d)/x^2) + 2*(2*a*c^4*d^5 - 4*a*c^2*d^4*e - 32*(a*c^4*d^2*e^3 - 2*a*c^2*d*e^4 + a*e^5)*x^6
+ 2*a*d^3*e^2 + (b*c^5*d^3*e^2 - b*c*d*e^4)*x^5 - 48*(a*c^4*d^3*e^2 - 2*a*c^2*d^2*e^3 + a*d*e^4)*x^4 + 2*(b*c^
5*d^4*e - b*c^3*d^3*e^2)*x^3 - 12*(a*c^4*d^4*e - 2*a*c^2*d^3*e^2 + a*d^2*e^3)*x^2 + (b*c^5*d^5 - 2*b*c^3*d^4*e
+ b*c*d^3*e^2)*x + 2*(b*c^4*d^5 - 2*b*c^2*d^4*e - 16*(b*c^4*d^2*e^3 - 2*b*c^2*d*e^4 + b*e^5)*x^6 + b*d^3*e^2
- 24*(b*c^4*d^3*e^2 - 2*b*c^2*d^2*e^3 + b*d*e^4)*x^4 - 6*(b*c^4*d^4*e - 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x^2)*arct
an(c*x))*sqrt(e*x^2 + d))/((c^4*d^6*e^2 - 2*c^2*d^5*e^3 + d^4*e^4)*x^7 + 2*(c^4*d^7*e - 2*c^2*d^6*e^2 + d^5*e^
3)*x^5 + (c^4*d^8 - 2*c^2*d^7*e + d^6*e^2)*x^3), -1/12*(2*((b*c^6*d^3*e^2 + 6*b*c^4*d^2*e^3 - 24*b*c^2*d*e^4 +
16*b*e^5)*x^7 + 2*(b*c^6*d^4*e + 6*b*c^4*d^3*e^2 - 24*b*c^2*d^2*e^3 + 16*b*d*e^4)*x^5 + (b*c^6*d^5 + 6*b*c^4*
d^4*e - 24*b*c^2*d^3*e^2 + 16*b*d^2*e^3)*x^3)*sqrt(-c^2*d + e)*arctan(-1/2*(c^2*e*x^2 + 2*c^2*d - e)*sqrt(-c^2
*d + e)*sqrt(e*x^2 + d)/(c^3*d^2 - c*d*e + (c^3*d*e - c*e^2)*x^2)) - ((2*b*c^7*d^3*e^2 + 11*b*c^5*d^2*e^3 - 28
*b*c^3*d*e^4 + 15*b*c*e^5)*x^7 + 2*(2*b*c^7*d^4*e + 11*b*c^5*d^3*e^2 - 28*b*c^3*d^2*e^3 + 15*b*c*d*e^4)*x^5 +
(2*b*c^7*d^5 + 11*b*c^5*d^4*e - 28*b*c^3*d^3*e^2 + 15*b*c*d^2*e^3)*x^3)*sqrt(d)*log(-(e*x^2 + 2*sqrt(e*x^2 + d
)*sqrt(d) + 2*d)/x^2) + 2*(2*a*c^4*d^5 - 4*a*c^2*d^4*e - 32*(a*c^4*d^2*e^3 - 2*a*c^2*d*e^4 + a*e^5)*x^6 + 2*a*
d^3*e^2 + (b*c^5*d^3*e^2 - b*c*d*e^4)*x^5 - 48*(a*c^4*d^3*e^2 - 2*a*c^2*d^2*e^3 + a*d*e^4)*x^4 + 2*(b*c^5*d^4*
e - b*c^3*d^3*e^2)*x^3 - 12*(a*c^4*d^4*e - 2*a*c^2*d^3*e^2 + a*d^2*e^3)*x^2 + (b*c^5*d^5 - 2*b*c^3*d^4*e + b*c
*d^3*e^2)*x + 2*(b*c^4*d^5 - 2*b*c^2*d^4*e - 16*(b*c^4*d^2*e^3 - 2*b*c^2*d*e^4 + b*e^5)*x^6 + b*d^3*e^2 - 24*(
b*c^4*d^3*e^2 - 2*b*c^2*d^2*e^3 + b*d*e^4)*x^4 - 6*(b*c^4*d^4*e - 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x^2)*arctan(c*x
))*sqrt(e*x^2 + d))/((c^4*d^6*e^2 - 2*c^2*d^5*e^3 + d^4*e^4)*x^7 + 2*(c^4*d^7*e - 2*c^2*d^6*e^2 + d^5*e^3)*x^5
+ (c^4*d^8 - 2*c^2*d^7*e + d^6*e^2)*x^3), -1/12*(2*((2*b*c^7*d^3*e^2 + 11*b*c^5*d^2*e^3 - 28*b*c^3*d*e^4 + 15
*b*c*e^5)*x^7 + 2*(2*b*c^7*d^4*e + 11*b*c^5*d^3*e^2 - 28*b*c^3*d^2*e^3 + 15*b*c*d*e^4)*x^5 + (2*b*c^7*d^5 + 11
*b*c^5*d^4*e - 28*b*c^3*d^3*e^2 + 15*b*c*d^2*e^3)*x^3)*sqrt(-d)*arctan(sqrt(-d)/sqrt(e*x^2 + d)) + ((b*c^6*d^3
*e^2 + 6*b*c^4*d^2*e^3 - 24*b*c^2*d*e^4 + 16*b*e^5)*x^7 + 2*(b*c^6*d^4*e + 6*b*c^4*d^3*e^2 - 24*b*c^2*d^2*e^3
+ 16*b*d*e^4)*x^5 + (b*c^6*d^5 + 6*b*c^4*d^4*e - 24*b*c^2*d^3*e^2 + 16*b*d^2*e^3)*x^3)*sqrt(c^2*d - e)*log((c^
4*e^2*x^4 + 8*c^4*d^2 - 8*c^2*d*e + 2*(4*c^4*d*e - 3*c^2*e^2)*x^2 + 4*(c^3*e*x^2 + 2*c^3*d - c*e)*sqrt(c^2*d -
e)*sqrt(e*x^2 + d) + e^2)/(c^4*x^4 + 2*c^2*x^2 + 1)) + 2*(2*a*c^4*d^5 - 4*a*c^2*d^4*e - 32*(a*c^4*d^2*e^3 - 2
*a*c^2*d*e^4 + a*e^5)*x^6 + 2*a*d^3*e^2 + (b*c^5*d^3*e^2 - b*c*d*e^4)*x^5 - 48*(a*c^4*d^3*e^2 - 2*a*c^2*d^2*e^
3 + a*d*e^4)*x^4 + 2*(b*c^5*d^4*e - b*c^3*d^3*e^2)*x^3 - 12*(a*c^4*d^4*e - 2*a*c^2*d^3*e^2 + a*d^2*e^3)*x^2 +
(b*c^5*d^5 - 2*b*c^3*d^4*e + b*c*d^3*e^2)*x + 2*(b*c^4*d^5 - 2*b*c^2*d^4*e - 16*(b*c^4*d^2*e^3 - 2*b*c^2*d*e^4
+ b*e^5)*x^6 + b*d^3*e^2 - 24*(b*c^4*d^3*e^2 - 2*b*c^2*d^2*e^3 + b*d*e^4)*x^4 - 6*(b*c^4*d^4*e - 2*b*c^2*d^3*
e^2 + b*d^2*e^3)*x^2)*arctan(c*x))*sqrt(e*x^2 + d))/((c^4*d^6*e^2 - 2*c^2*d^5*e^3 + d^4*e^4)*x^7 + 2*(c^4*d^7*
e - 2*c^2*d^6*e^2 + d^5*e^3)*x^5 + (c^4*d^8 - 2*c^2*d^7*e + d^6*e^2)*x^3), -1/6*(((b*c^6*d^3*e^2 + 6*b*c^4*d^2
*e^3 - 24*b*c^2*d*e^4 + 16*b*e^5)*x^7 + 2*(b*c^6*d^4*e + 6*b*c^4*d^3*e^2 - 24*b*c^2*d^2*e^3 + 16*b*d*e^4)*x^5
+ (b*c^6*d^5 + 6*b*c^4*d^4*e - 24*b*c^2*d^3*e^2 + 16*b*d^2*e^3)*x^3)*sqrt(-c^2*d + e)*arctan(-1/2*(c^2*e*x^2 +
2*c^2*d - e)*sqrt(-c^2*d + e)*sqrt(e*x^2 + d)/(c^3*d^2 - c*d*e + (c^3*d*e - c*e^2)*x^2)) + ((2*b*c^7*d^3*e^2
+ 11*b*c^5*d^2*e^3 - 28*b*c^3*d*e^4 + 15*b*c*e^5)*x^7 + 2*(2*b*c^7*d^4*e + 11*b*c^5*d^3*e^2 - 28*b*c^3*d^2*e^3
+ 15*b*c*d*e^4)*x^5 + (2*b*c^7*d^5 + 11*b*c^5*d^4*e - 28*b*c^3*d^3*e^2 + 15*b*c*d^2*e^3)*x^3)*sqrt(-d)*arctan
(sqrt(-d)/sqrt(e*x^2 + d)) + (2*a*c^4*d^5 - 4*a*c^2*d^4*e - 32*(a*c^4*d^2*e^3 - 2*a*c^2*d*e^4 + a*e^5)*x^6 + 2
*a*d^3*e^2 + (b*c^5*d^3*e^2 - b*c*d*e^4)*x^5 - 48*(a*c^4*d^3*e^2 - 2*a*c^2*d^2*e^3 + a*d*e^4)*x^4 + 2*(b*c^5*d
^4*e - b*c^3*d^3*e^2)*x^3 - 12*(a*c^4*d^4*e - 2*a*c^2*d^3*e^2 + a*d^2*e^3)*x^2 + (b*c^5*d^5 - 2*b*c^3*d^4*e +
b*c*d^3*e^2)*x + 2*(b*c^4*d^5 - 2*b*c^2*d^4*e - 16*(b*c^4*d^2*e^3 - 2*b*c^2*d*e^4 + b*e^5)*x^6 + b*d^3*e^2 - 2
4*(b*c^4*d^3*e^2 - 2*b*c^2*d^2*e^3 + b*d*e^4)*x^4 - 6*(b*c^4*d^4*e - 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x^2)*arctan(
c*x))*sqrt(e*x^2 + d))/((c^4*d^6*e^2 - 2*c^2*d^5*e^3 + d^4*e^4)*x^7 + 2*(c^4*d^7*e - 2*c^2*d^6*e^2 + d^5*e^3)*
x^5 + (c^4*d^8 - 2*c^2*d^7*e + d^6*e^2)*x^3)]

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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((a+b*atan(c*x))/x**4/(e*x**2+d)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac{5}{2}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/x^4/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arctan(c*x) + a)/((e*x^2 + d)^(5/2)*x^4), x)