\(\int \frac {a+b \arctan (c x)}{x^4 (d+e x^2)^{5/2}} \, dx\) [1225]

Optimal result

Integrand size = 23, antiderivative size = 423 \[ \int \frac {a+b \arctan (c x)}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c e}{2 d^3 \sqrt {d+e x^2}}+\frac {16 b e^2}{3 c d^4 \sqrt {d+e x^2}}-\frac {b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt {d+e x^2}}-\frac {b c}{6 d^2 x^2 \sqrt {d+e x^2}}-\frac {a+b \arctan (c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e (a+b \arctan (c x))}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x (a+b \arctan (c x))}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x (a+b \arctan (c x))}{3 d^4 \sqrt {d+e x^2}}+\frac {b c e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 d^{7/2}}+\frac {b c \left (c^2 d+6 e\right ) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 d^{7/2}}-\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right ) \text {arctanh}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^4 \left (c^2 d-e\right )^{3/2}} \]

[Out]

1/3*(-a-b*arctan(c*x))/d/x^3/(e*x^2+d)^(3/2)+2*e*(a+b*arctan(c*x))/d^2/x/(e*x^2+d)^(3/2)+8/3*e^2*x*(a+b*arctan
(c*x))/d^3/(e*x^2+d)^(3/2)+1/2*b*c*e*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(7/2)+1/3*b*c*(c^2*d+6*e)*arctanh((e*x
^2+d)^(1/2)/d^(1/2))/d^(7/2)-1/3*b*(c^2*d-2*e)*(c^4*d^2+8*c^2*d*e-8*e^2)*arctanh(c*(e*x^2+d)^(1/2)/(c^2*d-e)^(
1/2))/d^4/(c^2*d-e)^(3/2)-1/2*b*c*e/d^3/(e*x^2+d)^(1/2)+16/3*b*e^2/c/d^4/(e*x^2+d)^(1/2)-1/3*b*c*(c^2*d+6*e)/d
^3/(e*x^2+d)^(1/2)+1/3*b*(c^2*d-2*e)*(c^4*d^2+8*c^2*d*e-8*e^2)/c/d^4/(c^2*d-e)/(e*x^2+d)^(1/2)-1/6*b*c/d^2/x^2
/(e*x^2+d)^(1/2)+16/3*e^2*x*(a+b*arctan(c*x))/d^4/(e*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {277, 198, 197, 5096, 12, 6857, 272, 44, 53, 65, 214, 267, 455} \[ \int \frac {a+b \arctan (c x)}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\frac {16 e^2 x (a+b \arctan (c x))}{3 d^4 \sqrt {d+e x^2}}+\frac {8 e^2 x (a+b \arctan (c x))}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e (a+b \arctan (c x))}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac {a+b \arctan (c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {b c \left (c^2 d+6 e\right ) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 d^{7/2}}-\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right ) \text {arctanh}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^4 \left (c^2 d-e\right )^{3/2}}+\frac {b c e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 d^{7/2}}-\frac {b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {16 b e^2}{3 c d^4 \sqrt {d+e x^2}}-\frac {b c e}{2 d^3 \sqrt {d+e x^2}}-\frac {b c}{6 d^2 x^2 \sqrt {d+e x^2}} \]

[In]

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

[Out]

-1/2*(b*c*e)/(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[Sqr
t[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 12

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

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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 65

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 197

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 198

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 214

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

Rule 267

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 272

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 277

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 455

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]

Rule 5096

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 6857

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arctan (c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e (a+b \arctan (c x))}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x (a+b \arctan (c x))}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x (a+b \arctan (c x))}{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 \arctan (c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e (a+b \arctan (c x))}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x (a+b \arctan (c x))}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x (a+b \arctan (c x))}{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 \arctan (c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e (a+b \arctan (c x))}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x (a+b \arctan (c x))}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x (a+b \arctan (c x))}{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 \arctan (c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e (a+b \arctan (c x))}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x (a+b \arctan (c x))}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x (a+b \arctan (c x))}{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 \arctan (c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e (a+b \arctan (c x))}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x (a+b \arctan (c x))}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x (a+b \arctan (c x))}{3 d^4 \sqrt {d+e x^2}}+\frac {(b c) \text {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 ) \text {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 ) \text {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}{6 d^2 x^2 \sqrt {d+e x^2}}-\frac {a+b \arctan (c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e (a+b \arctan (c x))}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x (a+b \arctan (c x))}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x (a+b \arctan (c x))}{3 d^4 \sqrt {d+e x^2}}-\frac {(b c e) \text {Subst}\left (\int \frac {1}{x (d+e x)^{3/2}} \, dx,x,x^2\right )}{4 d^2}-\frac {\left (b c \left (c^2 d+6 e\right )\right ) \text {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 ) \text {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 {b c e}{2 d^3 \sqrt {d+e x^2}}+\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}{6 d^2 x^2 \sqrt {d+e x^2}}-\frac {a+b \arctan (c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e (a+b \arctan (c x))}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x (a+b \arctan (c x))}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x (a+b \arctan (c x))}{3 d^4 \sqrt {d+e x^2}}-\frac {(b c e) \text {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 ) \text {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 ) \text {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 {b c e}{2 d^3 \sqrt {d+e x^2}}+\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}{6 d^2 x^2 \sqrt {d+e x^2}}-\frac {a+b \arctan (c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e (a+b \arctan (c x))}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x (a+b \arctan (c x))}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x (a+b \arctan (c x))}{3 d^4 \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+6 e\right ) \text {arctanh}\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 ) \text {arctanh}\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) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{2 d^3} \\ & = -\frac {b c e}{2 d^3 \sqrt {d+e x^2}}+\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}{6 d^2 x^2 \sqrt {d+e x^2}}-\frac {a+b \arctan (c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e (a+b \arctan (c x))}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x (a+b \arctan (c x))}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x (a+b \arctan (c x))}{3 d^4 \sqrt {d+e x^2}}+\frac {b c e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 d^{7/2}}+\frac {b c \left (c^2 d+6 e\right ) \text {arctanh}\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 ) \text {arctanh}\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] (verified)

Result contains complex when optimal does not.

Time = 1.31 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.21 \[ \int \frac {a+b \arctan (c x)}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=-\frac {\frac {2 a \left (d^3-6 d^2 e x^2-24 d e^2 x^4-16 e^3 x^6\right )}{x^3 \left (d+e x^2\right )^{3/2}}+\frac {b c d \left (e \left (-d+e x^2\right )+c^2 d \left (d+e x^2\right )\right )}{\left (c^2 d-e\right ) x^2 \sqrt {d+e x^2}}+\frac {2 b \left (d^3-6 d^2 e x^2-24 d e^2 x^4-16 e^3 x^6\right ) \arctan (c x)}{x^3 \left (d+e x^2\right )^{3/2}}+b c \sqrt {d} \left (2 c^2 d+15 e\right ) \log (x)-b c \sqrt {d} \left (2 c^2 d+15 e\right ) \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )+\frac {b \left (c^6 d^3+6 c^4 d^2 e-24 c^2 d e^2+16 e^3\right ) \log \left (\frac {12 c d^4 \sqrt {c^2 d-e} \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^6 d^3+6 c^4 d^2 e-24 c^2 d e^2+16 e^3\right ) (i+c x)}\right )}{\left (c^2 d-e\right )^{3/2}}+\frac {b \left (c^6 d^3+6 c^4 d^2 e-24 c^2 d e^2+16 e^3\right ) \log \left (\frac {12 c d^4 \sqrt {c^2 d-e} \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^6 d^3+6 c^4 d^2 e-24 c^2 d e^2+16 e^3\right ) (-i+c x)}\right )}{\left (c^2 d-e\right )^{3/2}}}{6 d^4} \]

[In]

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

[Out]

-1/6*((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)*
ArcTan[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))/d^4

Maple [F]

\[\int \frac {a +b \arctan \left (c x \right )}{x^{4} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 837 vs. \(2 (365) = 730\).

Time = 1.77 (sec) , antiderivative size = 3460, normalized size of antiderivative = 8.18 \[ \int \frac {a+b \arctan (c x)}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \arctan (c x)}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arctan (c x)}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {a+b \arctan (c x)}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arctan (c x)}{x^4 \left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^4\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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