3.12.82 \(\int \frac {\sqrt {d+e x^2} (a+b \text {ArcTan}(c x))}{x^6} \, dx\) [1182]

Optimal. Leaf size=224 \[ \frac {b c \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{120 d x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4}-\frac {\left (d+e x^2\right )^{3/2} (a+b \text {ArcTan}(c x))}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} (a+b \text {ArcTan}(c x))}{15 d^2 x^3}-\frac {b c \left (24 c^4 d^2-20 c^2 d e-15 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{120 d^{3/2}}+\frac {b \left (c^2 d-e\right )^{3/2} \left (3 c^2 d+2 e\right ) \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{15 d^2} \]

[Out]

-1/5*(e*x^2+d)^(3/2)*(a+b*arctan(c*x))/d/x^5+2/15*e*(e*x^2+d)^(3/2)*(a+b*arctan(c*x))/d^2/x^3+1/30*b*c*(3*c^2*
d-e)*e*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(3/2)+1/40*b*c*e^2*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(3/2)-1/15*b*c
*(c^2*d-e)*(3*c^2*d+2*e)*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(3/2)+1/15*b*(c^2*d-e)^(3/2)*(3*c^2*d+2*e)*arctanh
(c*(e*x^2+d)^(1/2)/(c^2*d-e)^(1/2))/d^2-1/20*b*c*(e*x^2+d)^(1/2)/x^4+1/30*b*c*(3*c^2*d-e)*(e*x^2+d)^(1/2)/d/x^
2-1/40*b*c*e*(e*x^2+d)^(1/2)/d/x^2

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Rubi [A]
time = 0.24, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {277, 270, 5096, 12, 587, 154, 162, 65, 214} \begin {gather*} \frac {2 e \left (d+e x^2\right )^{3/2} (a+b \text {ArcTan}(c x))}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} (a+b \text {ArcTan}(c x))}{5 d x^5}+\frac {b \left (3 c^2 d+2 e\right ) \left (c^2 d-e\right )^{3/2} \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{15 d^2}+\frac {b c \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{120 d x^2}-\frac {b c \left (24 c^4 d^2-20 c^2 d e-15 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{120 d^{3/2}}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^6,x]

[Out]

(b*c*(12*c^2*d - e)*Sqrt[d + e*x^2])/(120*d*x^2) - (b*c*(d + e*x^2)^(3/2))/(20*d*x^4) - ((d + e*x^2)^(3/2)*(a
+ b*ArcTan[c*x]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(15*d^2*x^3) - (b*c*(24*c^4*d^2 - 20
*c^2*d*e - 15*e^2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(120*d^(3/2)) + (b*(c^2*d - e)^(3/2)*(3*c^2*d + 2*e)*ArcT
anh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(15*d^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 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 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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 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 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 587

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n],
x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[Simplify[(m + 1)/n]]

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]))

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^6} \, dx &=-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-(b c) \int \frac {\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{15 d^2 x^5 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c) \int \frac {\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{x^5 \left (1+c^2 x^2\right )} \, dx}{15 d^2}\\ &=-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c) \text {Subst}\left (\int \frac {(d+e x)^{3/2} (-3 d+2 e x)}{x^3 \left (1+c^2 x\right )} \, dx,x,x^2\right )}{30 d^2}\\ &=-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c) \text {Subst}\left (\int \frac {\sqrt {d+e x} \left (\frac {1}{2} d \left (12 c^2 d-e\right )+\frac {1}{2} e \left (3 c^2 d+8 e\right ) x\right )}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )}{60 d^2}\\ &=\frac {b c \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{120 d x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c) \text {Subst}\left (\int \frac {-\frac {1}{4} d \left (24 c^4 d^2-20 c^2 d e-15 e^2\right )-\frac {1}{4} e \left (12 c^4 d^2-7 c^2 d e-16 e^2\right ) x}{x \left (1+c^2 x\right ) \sqrt {d+e x}} \, dx,x,x^2\right )}{60 d^2}\\ &=\frac {b c \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{120 d x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (b c \left (c^2 d-e\right )^2 \left (3 c^2 d+2 e\right )\right ) \text {Subst}\left (\int \frac {1}{\left (1+c^2 x\right ) \sqrt {d+e x}} \, dx,x,x^2\right )}{30 d^2}+\frac {\left (b c \left (24 c^4 d^2-20 c^2 d e-15 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{240 d}\\ &=\frac {b c \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{120 d x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (b c \left (c^2 d-e\right )^2 \left (3 c^2 d+2 e\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 )}{15 d^2 e}+\frac {\left (b c \left (24 c^4 d^2-20 c^2 d e-15 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{120 d e}\\ &=\frac {b c \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{120 d x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 d x^4}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {b c \left (24 c^4 d^2-20 c^2 d e-15 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{120 d^{3/2}}+\frac {b \left (c^2 d-e\right )^{3/2} \left (3 c^2 d+2 e\right ) \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{15 d^2}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.38, size = 413, normalized size = 1.84 \begin {gather*} \frac {-\sqrt {d+e x^2} \left (8 a \left (3 d^2+d e x^2-2 e^2 x^4\right )+b c d x \left (7 e x^2+d \left (6-12 c^2 x^2\right )\right )\right )-8 b \sqrt {d+e x^2} \left (3 d^2+d e x^2-2 e^2 x^4\right ) \text {ArcTan}(c x)+b c \sqrt {d} \left (24 c^4 d^2-20 c^2 d e-15 e^2\right ) x^5 \log (x)-b c \sqrt {d} \left (24 c^4 d^2-20 c^2 d e-15 e^2\right ) x^5 \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )+4 b \left (c^2 d-e\right )^{3/2} \left (3 c^2 d+2 e\right ) x^5 \log \left (-\frac {60 c d^2 \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{5/2} \left (3 c^2 d+2 e\right ) (i+c x)}\right )+4 b \left (c^2 d-e\right )^{3/2} \left (3 c^2 d+2 e\right ) x^5 \log \left (-\frac {60 c d^2 \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{5/2} \left (3 c^2 d+2 e\right ) (-i+c x)}\right )}{120 d^2 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^6,x]

[Out]

(-(Sqrt[d + e*x^2]*(8*a*(3*d^2 + d*e*x^2 - 2*e^2*x^4) + b*c*d*x*(7*e*x^2 + d*(6 - 12*c^2*x^2)))) - 8*b*Sqrt[d
+ e*x^2]*(3*d^2 + d*e*x^2 - 2*e^2*x^4)*ArcTan[c*x] + b*c*Sqrt[d]*(24*c^4*d^2 - 20*c^2*d*e - 15*e^2)*x^5*Log[x]
- b*c*Sqrt[d]*(24*c^4*d^2 - 20*c^2*d*e - 15*e^2)*x^5*Log[d + Sqrt[d]*Sqrt[d + e*x^2]] + 4*b*(c^2*d - e)^(3/2)
*(3*c^2*d + 2*e)*x^5*Log[(-60*c*d^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(5/2)*(3*c
^2*d + 2*e)*(I + c*x))] + 4*b*(c^2*d - e)^(3/2)*(3*c^2*d + 2*e)*x^5*Log[(-60*c*d^2*(c*d + I*e*x + Sqrt[c^2*d -
e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(5/2)*(3*c^2*d + 2*e)*(-I + c*x))])/(120*d^2*x^5)

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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {e \,x^{2}+d}\, \left (a +b \arctan \left (c x \right )\right )}{x^{6}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x^6,x)

[Out]

int((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x^6,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x^6,x, algorithm="maxima")

[Out]

1/15*a*(2*(x^2*e + d)^(3/2)*e/(d^2*x^3) - 3*(x^2*e + d)^(3/2)/(d*x^5)) + b*integrate(sqrt(x^2*e + d)*arctan(c*
x)/x^6, x)

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Fricas [A]
time = 2.82, size = 1244, normalized size = 5.55 \begin {gather*} \left [-\frac {4 \, {\left (3 \, b c^{4} d^{2} x^{5} - b c^{2} d x^{5} e - 2 \, b x^{5} e^{2}\right )} \sqrt {c^{2} d - e} \log \left (\frac {8 \, c^{4} d^{2} - 4 \, {\left (2 \, c^{3} d + {\left (c^{3} x^{2} - c\right )} e\right )} \sqrt {c^{2} d - e} \sqrt {x^{2} e + d} + {\left (c^{4} x^{4} - 6 \, c^{2} x^{2} + 1\right )} e^{2} + 8 \, {\left (c^{4} d x^{2} - c^{2} d\right )} e}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) + {\left (24 \, b c^{5} d^{2} x^{5} - 20 \, b c^{3} d x^{5} e - 15 \, b c x^{5} e^{2}\right )} \sqrt {d} \log \left (-\frac {x^{2} e + 2 \, \sqrt {x^{2} e + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - 2 \, {\left (12 \, b c^{3} d^{2} x^{3} + 16 \, a x^{4} e^{2} - 6 \, b c d^{2} x - 24 \, a d^{2} + 8 \, {\left (2 \, b x^{4} e^{2} - b d x^{2} e - 3 \, b d^{2}\right )} \arctan \left (c x\right ) - {\left (7 \, b c d x^{3} + 8 \, a d x^{2}\right )} e\right )} \sqrt {x^{2} e + d}}{240 \, d^{2} x^{5}}, \frac {8 \, {\left (3 \, b c^{4} d^{2} x^{5} - b c^{2} d x^{5} e - 2 \, b x^{5} e^{2}\right )} \sqrt {-c^{2} d + e} \arctan \left (-\frac {{\left (2 \, c^{2} d + {\left (c^{2} x^{2} - 1\right )} e\right )} \sqrt {-c^{2} d + e} \sqrt {x^{2} e + d}}{2 \, {\left (c^{3} d^{2} - c x^{2} e^{2} + {\left (c^{3} d x^{2} - c d\right )} e\right )}}\right ) - {\left (24 \, b c^{5} d^{2} x^{5} - 20 \, b c^{3} d x^{5} e - 15 \, b c x^{5} e^{2}\right )} \sqrt {d} \log \left (-\frac {x^{2} e + 2 \, \sqrt {x^{2} e + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (12 \, b c^{3} d^{2} x^{3} + 16 \, a x^{4} e^{2} - 6 \, b c d^{2} x - 24 \, a d^{2} + 8 \, {\left (2 \, b x^{4} e^{2} - b d x^{2} e - 3 \, b d^{2}\right )} \arctan \left (c x\right ) - {\left (7 \, b c d x^{3} + 8 \, a d x^{2}\right )} e\right )} \sqrt {x^{2} e + d}}{240 \, d^{2} x^{5}}, \frac {{\left (24 \, b c^{5} d^{2} x^{5} - 20 \, b c^{3} d x^{5} e - 15 \, b c x^{5} e^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) - 2 \, {\left (3 \, b c^{4} d^{2} x^{5} - b c^{2} d x^{5} e - 2 \, b x^{5} e^{2}\right )} \sqrt {c^{2} d - e} \log \left (\frac {8 \, c^{4} d^{2} - 4 \, {\left (2 \, c^{3} d + {\left (c^{3} x^{2} - c\right )} e\right )} \sqrt {c^{2} d - e} \sqrt {x^{2} e + d} + {\left (c^{4} x^{4} - 6 \, c^{2} x^{2} + 1\right )} e^{2} + 8 \, {\left (c^{4} d x^{2} - c^{2} d\right )} e}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) + {\left (12 \, b c^{3} d^{2} x^{3} + 16 \, a x^{4} e^{2} - 6 \, b c d^{2} x - 24 \, a d^{2} + 8 \, {\left (2 \, b x^{4} e^{2} - b d x^{2} e - 3 \, b d^{2}\right )} \arctan \left (c x\right ) - {\left (7 \, b c d x^{3} + 8 \, a d x^{2}\right )} e\right )} \sqrt {x^{2} e + d}}{120 \, d^{2} x^{5}}, \frac {4 \, {\left (3 \, b c^{4} d^{2} x^{5} - b c^{2} d x^{5} e - 2 \, b x^{5} e^{2}\right )} \sqrt {-c^{2} d + e} \arctan \left (-\frac {{\left (2 \, c^{2} d + {\left (c^{2} x^{2} - 1\right )} e\right )} \sqrt {-c^{2} d + e} \sqrt {x^{2} e + d}}{2 \, {\left (c^{3} d^{2} - c x^{2} e^{2} + {\left (c^{3} d x^{2} - c d\right )} e\right )}}\right ) + {\left (24 \, b c^{5} d^{2} x^{5} - 20 \, b c^{3} d x^{5} e - 15 \, b c x^{5} e^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) + {\left (12 \, b c^{3} d^{2} x^{3} + 16 \, a x^{4} e^{2} - 6 \, b c d^{2} x - 24 \, a d^{2} + 8 \, {\left (2 \, b x^{4} e^{2} - b d x^{2} e - 3 \, b d^{2}\right )} \arctan \left (c x\right ) - {\left (7 \, b c d x^{3} + 8 \, a d x^{2}\right )} e\right )} \sqrt {x^{2} e + d}}{120 \, d^{2} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x^6,x, algorithm="fricas")

[Out]

[-1/240*(4*(3*b*c^4*d^2*x^5 - b*c^2*d*x^5*e - 2*b*x^5*e^2)*sqrt(c^2*d - e)*log((8*c^4*d^2 - 4*(2*c^3*d + (c^3*
x^2 - c)*e)*sqrt(c^2*d - e)*sqrt(x^2*e + d) + (c^4*x^4 - 6*c^2*x^2 + 1)*e^2 + 8*(c^4*d*x^2 - c^2*d)*e)/(c^4*x^
4 + 2*c^2*x^2 + 1)) + (24*b*c^5*d^2*x^5 - 20*b*c^3*d*x^5*e - 15*b*c*x^5*e^2)*sqrt(d)*log(-(x^2*e + 2*sqrt(x^2*
e + d)*sqrt(d) + 2*d)/x^2) - 2*(12*b*c^3*d^2*x^3 + 16*a*x^4*e^2 - 6*b*c*d^2*x - 24*a*d^2 + 8*(2*b*x^4*e^2 - b*
d*x^2*e - 3*b*d^2)*arctan(c*x) - (7*b*c*d*x^3 + 8*a*d*x^2)*e)*sqrt(x^2*e + d))/(d^2*x^5), 1/240*(8*(3*b*c^4*d^
2*x^5 - b*c^2*d*x^5*e - 2*b*x^5*e^2)*sqrt(-c^2*d + e)*arctan(-1/2*(2*c^2*d + (c^2*x^2 - 1)*e)*sqrt(-c^2*d + e)
*sqrt(x^2*e + d)/(c^3*d^2 - c*x^2*e^2 + (c^3*d*x^2 - c*d)*e)) - (24*b*c^5*d^2*x^5 - 20*b*c^3*d*x^5*e - 15*b*c*
x^5*e^2)*sqrt(d)*log(-(x^2*e + 2*sqrt(x^2*e + d)*sqrt(d) + 2*d)/x^2) + 2*(12*b*c^3*d^2*x^3 + 16*a*x^4*e^2 - 6*
b*c*d^2*x - 24*a*d^2 + 8*(2*b*x^4*e^2 - b*d*x^2*e - 3*b*d^2)*arctan(c*x) - (7*b*c*d*x^3 + 8*a*d*x^2)*e)*sqrt(x
^2*e + d))/(d^2*x^5), 1/120*((24*b*c^5*d^2*x^5 - 20*b*c^3*d*x^5*e - 15*b*c*x^5*e^2)*sqrt(-d)*arctan(sqrt(-d)/s
qrt(x^2*e + d)) - 2*(3*b*c^4*d^2*x^5 - b*c^2*d*x^5*e - 2*b*x^5*e^2)*sqrt(c^2*d - e)*log((8*c^4*d^2 - 4*(2*c^3*
d + (c^3*x^2 - c)*e)*sqrt(c^2*d - e)*sqrt(x^2*e + d) + (c^4*x^4 - 6*c^2*x^2 + 1)*e^2 + 8*(c^4*d*x^2 - c^2*d)*e
)/(c^4*x^4 + 2*c^2*x^2 + 1)) + (12*b*c^3*d^2*x^3 + 16*a*x^4*e^2 - 6*b*c*d^2*x - 24*a*d^2 + 8*(2*b*x^4*e^2 - b*
d*x^2*e - 3*b*d^2)*arctan(c*x) - (7*b*c*d*x^3 + 8*a*d*x^2)*e)*sqrt(x^2*e + d))/(d^2*x^5), 1/120*(4*(3*b*c^4*d^
2*x^5 - b*c^2*d*x^5*e - 2*b*x^5*e^2)*sqrt(-c^2*d + e)*arctan(-1/2*(2*c^2*d + (c^2*x^2 - 1)*e)*sqrt(-c^2*d + e)
*sqrt(x^2*e + d)/(c^3*d^2 - c*x^2*e^2 + (c^3*d*x^2 - c*d)*e)) + (24*b*c^5*d^2*x^5 - 20*b*c^3*d*x^5*e - 15*b*c*
x^5*e^2)*sqrt(-d)*arctan(sqrt(-d)/sqrt(x^2*e + d)) + (12*b*c^3*d^2*x^3 + 16*a*x^4*e^2 - 6*b*c*d^2*x - 24*a*d^2
+ 8*(2*b*x^4*e^2 - b*d*x^2*e - 3*b*d^2)*arctan(c*x) - (7*b*c*d*x^3 + 8*a*d*x^2)*e)*sqrt(x^2*e + d))/(d^2*x^5)
]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(1/2)*(a+b*atan(c*x))/x**6,x)

[Out]

Integral((a + b*atan(c*x))*sqrt(d + e*x**2)/x**6, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x^6,x, algorithm="giac")

[Out]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\sqrt {e\,x^2+d}}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*atan(c*x))*(d + e*x^2)^(1/2))/x^6,x)

[Out]

int(((a + b*atan(c*x))*(d + e*x^2)^(1/2))/x^6, x)

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