3.8.72 \(\int \frac {1}{x (a+b x^2)^2 (c+d x^2)^{3/2}} \, dx\) [772]

Optimal. Leaf size=170 \[ \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 c^{3/2}}+\frac {b^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{5/2}} \]

[Out]

-arctanh((d*x^2+c)^(1/2)/c^(1/2))/a^2/c^(3/2)+1/2*b^(3/2)*(-5*a*d+2*b*c)*arctanh(b^(1/2)*(d*x^2+c)^(1/2)/(-a*d
+b*c)^(1/2))/a^2/(-a*d+b*c)^(5/2)+1/2*d*(2*a*d+b*c)/a/c/(-a*d+b*c)^2/(d*x^2+c)^(1/2)+1/2*b/a/(-a*d+b*c)/(b*x^2
+a)/(d*x^2+c)^(1/2)

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Rubi [A]
time = 0.15, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 105, 157, 162, 65, 214} \begin {gather*} \frac {b^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 c^{3/2}}+\frac {b}{2 a \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}+\frac {d (2 a d+b c)}{2 a c \sqrt {c+d x^2} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]

[Out]

(d*(b*c + 2*a*d))/(2*a*c*(b*c - a*d)^2*Sqrt[c + d*x^2]) + b/(2*a*(b*c - a*d)*(a + b*x^2)*Sqrt[c + d*x^2]) - Ar
cTanh[Sqrt[c + d*x^2]/Sqrt[c]]/(a^2*c^(3/2)) + (b^(3/2)*(2*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt
[b*c - a*d]])/(2*a^2*(b*c - a*d)^(5/2))

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 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 157

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

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 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (a+b x)^2 (c+d x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {b}{2 a (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {\text {Subst}\left (\int \frac {b c-a d+\frac {3 b d x}{2}}{x (a+b x) (c+d x)^{3/2}} \, dx,x,x^2\right )}{2 a (b c-a d)}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} (b c-a d)^2-\frac {1}{4} b d (b c+2 a d) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{a c (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a^2 c}-\frac {\left (b^2 (2 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^2 (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{a^2 c d}-\frac {\left (b^2 (2 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 a^2 d (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 \sqrt {c+d x^2}}+\frac {b}{2 a (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 c^{3/2}}+\frac {b^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.77, size = 157, normalized size = 0.92 \begin {gather*} \frac {\frac {a \left (2 a^2 d^2+2 a b d^2 x^2+b^2 c \left (c+d x^2\right )\right )}{c (b c-a d)^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {b^{3/2} (2 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{5/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{3/2}}}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]

[Out]

((a*(2*a^2*d^2 + 2*a*b*d^2*x^2 + b^2*c*(c + d*x^2)))/(c*(b*c - a*d)^2*(a + b*x^2)*Sqrt[c + d*x^2]) - (b^(3/2)*
(2*b*c - 5*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(5/2) - (2*ArcTanh[Sqrt[c
+ d*x^2]/Sqrt[c]])/c^(3/2))/(2*a^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1967\) vs. \(2(144)=288\).
time = 0.10, size = 1968, normalized size = 11.58

method	result	size
default	\(\text {Expression too large to display}\)	\(1968\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x^2+a)^2/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/4/a/(-a*b)^(1/2)*(1/(a*d-b*c)*b/(x+1/b*(-a*b)^(1/2))/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a
*b)^(1/2))-(a*d-b*c)/b)^(1/2)-3*d*(-a*b)^(1/2)/(a*d-b*c)*(-1/(a*d-b*c)*b/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^
(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-2*d*(-a*b)^(1/2)/(a*d-b*c)*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b
)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d
-b*c)/b)^(1/2)+1/(a*d-b*c)*b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2
*(-(a*d-b*c)/b)^(1/2)*(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x
+1/b*(-a*b)^(1/2))))+4*d/(a*d-b*c)*b*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b
)/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))-1/2/a^2*(-1/(a*d-b*c)*
b/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+2*d*(-a*b)^(1/2)/(a*d-b
*c)*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(
-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/(a*d-b*c)*b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*
d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1
/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2))))-1/2/a^2*(-1/(a*d-b*c)*b/(d*(x+1/b*(-a*b)^(1/2))^2-
2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-2*d*(-a*b)^(1/2)/(a*d-b*c)*(2*d*(x+1/b*(-a*b)^(1/2)
)-2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^
(1/2))-(a*d-b*c)/b)^(1/2)+1/(a*d-b*c)*b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*
b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b
)^(1/2))/(x+1/b*(-a*b)^(1/2))))+1/a^2*(1/c/(d*x^2+c)^(1/2)-1/c^(3/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x))-1/
4/a/(-a*b)^(1/2)*(1/(a*d-b*c)*b/(x-1/b*(-a*b)^(1/2))/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b
)^(1/2))-(a*d-b*c)/b)^(1/2)+3*d*(-a*b)^(1/2)/(a*d-b*c)*(-1/(a*d-b*c)*b/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1
/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+2*d*(-a*b)^(1/2)/(a*d-b*c)*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^
(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b
*c)/b)^(1/2)+1/(a*d-b*c)*b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(
-(a*d-b*c)/b)^(1/2)*(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1
/b*(-a*b)^(1/2))))+4*d/(a*d-b*c)*b*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/
(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))

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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(1/x/(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(3/2)*x), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (144) = 288\).
time = 3.55, size = 1992, normalized size = 11.72 \begin {gather*} \left [-\frac {{\left (2 \, a b^{2} c^{4} - 5 \, a^{2} b c^{3} d + {\left (2 \, b^{3} c^{3} d - 5 \, a b^{2} c^{2} d^{2}\right )} x^{4} + {\left (2 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 5 \, a^{2} b c^{2} d^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 4 \, {\left (a b^{2} c^{3} + 2 \, a^{3} c d^{2} + {\left (a b^{2} c^{2} d + 2 \, a^{2} b c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2} + {\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{4} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{2}\right )}}, \frac {8 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - {\left (2 \, a b^{2} c^{4} - 5 \, a^{2} b c^{3} d + {\left (2 \, b^{3} c^{3} d - 5 \, a b^{2} c^{2} d^{2}\right )} x^{4} + {\left (2 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 5 \, a^{2} b c^{2} d^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (a b^{2} c^{3} + 2 \, a^{3} c d^{2} + {\left (a b^{2} c^{2} d + 2 \, a^{2} b c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2} + {\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{4} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{2}\right )}}, -\frac {{\left (2 \, a b^{2} c^{4} - 5 \, a^{2} b c^{3} d + {\left (2 \, b^{3} c^{3} d - 5 \, a b^{2} c^{2} d^{2}\right )} x^{4} + {\left (2 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 5 \, a^{2} b c^{2} d^{2}\right )} x^{2}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) - 2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, {\left (a b^{2} c^{3} + 2 \, a^{3} c d^{2} + {\left (a b^{2} c^{2} d + 2 \, a^{2} b c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2} + {\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{4} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{2}\right )}}, -\frac {{\left (2 \, a b^{2} c^{4} - 5 \, a^{2} b c^{3} d + {\left (2 \, b^{3} c^{3} d - 5 \, a b^{2} c^{2} d^{2}\right )} x^{4} + {\left (2 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 5 \, a^{2} b c^{2} d^{2}\right )} x^{2}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) - 4 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - 2 \, {\left (a b^{2} c^{3} + 2 \, a^{3} c d^{2} + {\left (a b^{2} c^{2} d + 2 \, a^{2} b c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2} + {\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{4} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

[-1/8*((2*a*b^2*c^4 - 5*a^2*b*c^3*d + (2*b^3*c^3*d - 5*a*b^2*c^2*d^2)*x^4 + (2*b^3*c^4 - 3*a*b^2*c^3*d - 5*a^2
*b*c^2*d^2)*x^2)*sqrt(b/(b*c - a*d))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b
*d^2)*x^2 - 4*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*c - a*d)))
/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*
d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c
) + 2*c)/x^2) - 4*(a*b^2*c^3 + 2*a^3*c*d^2 + (a*b^2*c^2*d + 2*a^2*b*c*d^2)*x^2)*sqrt(d*x^2 + c))/(a^3*b^2*c^5
- 2*a^4*b*c^4*d + a^5*c^3*d^2 + (a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^4 + (a^2*b^3*c^5 - a^3*b
^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x^2), 1/8*(8*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a
*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*sqrt(-c)*arctan(sqrt(-c)/sq
rt(d*x^2 + c)) - (2*a*b^2*c^4 - 5*a^2*b*c^3*d + (2*b^3*c^3*d - 5*a*b^2*c^2*d^2)*x^4 + (2*b^3*c^4 - 3*a*b^2*c^3
*d - 5*a^2*b*c^2*d^2)*x^2)*sqrt(b/(b*c - a*d))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c
*d - 3*a*b*d^2)*x^2 - 4*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*
c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(a*b^2*c^3 + 2*a^3*c*d^2 + (a*b^2*c^2*d + 2*a^2*b*c*d^2)*x^2)*sqrt
(d*x^2 + c))/(a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2 + (a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*
x^4 + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x^2), -1/4*((2*a*b^2*c^4 - 5*a^2*b*c^3*d + (
2*b^3*c^3*d - 5*a*b^2*c^2*d^2)*x^4 + (2*b^3*c^4 - 3*a*b^2*c^3*d - 5*a^2*b*c^2*d^2)*x^2)*sqrt(-b/(b*c - a*d))*a
rctan(1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-b/(b*c - a*d))/(b*d*x^2 + b*c)) - 2*(a*b^2*c^3 - 2*a^2
*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^
3*d^3)*x^2)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) - 2*(a*b^2*c^3 + 2*a^3*c*d^2 + (a*b^2*
c^2*d + 2*a^2*b*c*d^2)*x^2)*sqrt(d*x^2 + c))/(a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2 + (a^2*b^3*c^4*d - 2*a
^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^4 + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x^2), -1/4*(
(2*a*b^2*c^4 - 5*a^2*b*c^3*d + (2*b^3*c^3*d - 5*a*b^2*c^2*d^2)*x^4 + (2*b^3*c^4 - 3*a*b^2*c^3*d - 5*a^2*b*c^2*
d^2)*x^2)*sqrt(-b/(b*c - a*d))*arctan(1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-b/(b*c - a*d))/(b*d*x^
2 + b*c)) - 4*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3
- a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^2 + c)) - 2*(a*b^2*c^3 + 2*a^3*c
*d^2 + (a*b^2*c^2*d + 2*a^2*b*c*d^2)*x^2)*sqrt(d*x^2 + c))/(a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2 + (a^2*b
^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^4 + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3
)*x^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x**2+a)**2/(d*x**2+c)**(3/2),x)

[Out]

Integral(1/(x*(a + b*x**2)**2*(c + d*x**2)**(3/2)), x)

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Giac [A]
time = 0.56, size = 225, normalized size = 1.32 \begin {gather*} -\frac {{\left (2 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {{\left (d x^{2} + c\right )} b^{2} c d + 2 \, {\left (d x^{2} + c\right )} a b d^{2} - 2 \, a b c d^{2} + 2 \, a^{2} d^{3}}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} {\left ({\left (d x^{2} + c\right )}^{\frac {3}{2}} b - \sqrt {d x^{2} + c} b c + \sqrt {d x^{2} + c} a d\right )}} + \frac {\arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

-1/2*(2*b^3*c - 5*a*b^2*d)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/((a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^
2)*sqrt(-b^2*c + a*b*d)) + 1/2*((d*x^2 + c)*b^2*c*d + 2*(d*x^2 + c)*a*b*d^2 - 2*a*b*c*d^2 + 2*a^2*d^3)/((a*b^2
*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*((d*x^2 + c)^(3/2)*b - sqrt(d*x^2 + c)*b*c + sqrt(d*x^2 + c)*a*d)) + arctan(
sqrt(d*x^2 + c)/sqrt(-c))/(a^2*sqrt(-c)*c)

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Mupad [B]
time = 1.90, size = 2500, normalized size = 14.71 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x)

[Out]

atanh((240*a^3*b^11*c^11*d^4*(c + d*x^2)^(1/2))/((c^3)^(1/2)*(64*a^12*b^2*c*d^13 - 240*a^3*b^11*c^10*d^4 + 208
0*a^4*b^10*c^9*d^5 - 7760*a^5*b^9*c^8*d^6 + 16384*a^6*b^8*c^7*d^7 - 21584*a^7*b^7*c^6*d^8 + 18400*a^8*b^6*c^5*
d^9 - 10160*a^9*b^5*c^4*d^10 + 3520*a^10*b^4*c^3*d^11 - 704*a^11*b^3*c^2*d^12)) - (2080*a^4*b^10*c^10*d^5*(c +
d*x^2)^(1/2))/((c^3)^(1/2)*(64*a^12*b^2*c*d^13 - 240*a^3*b^11*c^10*d^4 + 2080*a^4*b^10*c^9*d^5 - 7760*a^5*b^9
*c^8*d^6 + 16384*a^6*b^8*c^7*d^7 - 21584*a^7*b^7*c^6*d^8 + 18400*a^8*b^6*c^5*d^9 - 10160*a^9*b^5*c^4*d^10 + 35
20*a^10*b^4*c^3*d^11 - 704*a^11*b^3*c^2*d^12)) + (7760*a^5*b^9*c^9*d^6*(c + d*x^2)^(1/2))/((c^3)^(1/2)*(64*a^1
2*b^2*c*d^13 - 240*a^3*b^11*c^10*d^4 + 2080*a^4*b^10*c^9*d^5 - 7760*a^5*b^9*c^8*d^6 + 16384*a^6*b^8*c^7*d^7 -
21584*a^7*b^7*c^6*d^8 + 18400*a^8*b^6*c^5*d^9 - 10160*a^9*b^5*c^4*d^10 + 3520*a^10*b^4*c^3*d^11 - 704*a^11*b^3
*c^2*d^12)) - (16384*a^6*b^8*c^8*d^7*(c + d*x^2)^(1/2))/((c^3)^(1/2)*(64*a^12*b^2*c*d^13 - 240*a^3*b^11*c^10*d
^4 + 2080*a^4*b^10*c^9*d^5 - 7760*a^5*b^9*c^8*d^6 + 16384*a^6*b^8*c^7*d^7 - 21584*a^7*b^7*c^6*d^8 + 18400*a^8*
b^6*c^5*d^9 - 10160*a^9*b^5*c^4*d^10 + 3520*a^10*b^4*c^3*d^11 - 704*a^11*b^3*c^2*d^12)) + (21584*a^7*b^7*c^7*d
^8*(c + d*x^2)^(1/2))/((c^3)^(1/2)*(64*a^12*b^2*c*d^13 - 240*a^3*b^11*c^10*d^4 + 2080*a^4*b^10*c^9*d^5 - 7760*
a^5*b^9*c^8*d^6 + 16384*a^6*b^8*c^7*d^7 - 21584*a^7*b^7*c^6*d^8 + 18400*a^8*b^6*c^5*d^9 - 10160*a^9*b^5*c^4*d^
10 + 3520*a^10*b^4*c^3*d^11 - 704*a^11*b^3*c^2*d^12)) - (18400*a^8*b^6*c^6*d^9*(c + d*x^2)^(1/2))/((c^3)^(1/2)
*(64*a^12*b^2*c*d^13 - 240*a^3*b^11*c^10*d^4 + 2080*a^4*b^10*c^9*d^5 - 7760*a^5*b^9*c^8*d^6 + 16384*a^6*b^8*c^
7*d^7 - 21584*a^7*b^7*c^6*d^8 + 18400*a^8*b^6*c^5*d^9 - 10160*a^9*b^5*c^4*d^10 + 3520*a^10*b^4*c^3*d^11 - 704*
a^11*b^3*c^2*d^12)) + (10160*a^9*b^5*c^5*d^10*(c + d*x^2)^(1/2))/((c^3)^(1/2)*(64*a^12*b^2*c*d^13 - 240*a^3*b^
11*c^10*d^4 + 2080*a^4*b^10*c^9*d^5 - 7760*a^5*b^9*c^8*d^6 + 16384*a^6*b^8*c^7*d^7 - 21584*a^7*b^7*c^6*d^8 + 1
8400*a^8*b^6*c^5*d^9 - 10160*a^9*b^5*c^4*d^10 + 3520*a^10*b^4*c^3*d^11 - 704*a^11*b^3*c^2*d^12)) - (3520*a^10*
b^4*c^4*d^11*(c + d*x^2)^(1/2))/((c^3)^(1/2)*(64*a^12*b^2*c*d^13 - 240*a^3*b^11*c^10*d^4 + 2080*a^4*b^10*c^9*d
^5 - 7760*a^5*b^9*c^8*d^6 + 16384*a^6*b^8*c^7*d^7 - 21584*a^7*b^7*c^6*d^8 + 18400*a^8*b^6*c^5*d^9 - 10160*a^9*
b^5*c^4*d^10 + 3520*a^10*b^4*c^3*d^11 - 704*a^11*b^3*c^2*d^12)) + (704*a^11*b^3*c^3*d^12*(c + d*x^2)^(1/2))/((
c^3)^(1/2)*(64*a^12*b^2*c*d^13 - 240*a^3*b^11*c^10*d^4 + 2080*a^4*b^10*c^9*d^5 - 7760*a^5*b^9*c^8*d^6 + 16384*
a^6*b^8*c^7*d^7 - 21584*a^7*b^7*c^6*d^8 + 18400*a^8*b^6*c^5*d^9 - 10160*a^9*b^5*c^4*d^10 + 3520*a^10*b^4*c^3*d
^11 - 704*a^11*b^3*c^2*d^12)) - (64*a^12*b^2*c^2*d^13*(c + d*x^2)^(1/2))/((c^3)^(1/2)*(64*a^12*b^2*c*d^13 - 24
0*a^3*b^11*c^10*d^4 + 2080*a^4*b^10*c^9*d^5 - 7760*a^5*b^9*c^8*d^6 + 16384*a^6*b^8*c^7*d^7 - 21584*a^7*b^7*c^6
*d^8 + 18400*a^8*b^6*c^5*d^9 - 10160*a^9*b^5*c^4*d^10 + 3520*a^10*b^4*c^3*d^11 - 704*a^11*b^3*c^2*d^12)))/(a^2
*(c^3)^(1/2)) - (d^2/(b*c^2 - a*c*d) + (d*(c + d*x^2)*(b^2*c + 2*a*b*d))/(2*a*(b*c^2 - a*c*d)*(a*d - b*c)))/(b
*(c + d*x^2)^(3/2) + (c + d*x^2)^(1/2)*(a*d - b*c)) - (atan((((-b^3*(a*d - b*c)^5)^(1/2)*(5*a*d - 2*b*c)*((c +
d*x^2)^(1/2)*(128*a^3*b^13*c^13*d^2 - 1344*a^4*b^12*c^12*d^3 + 6160*a^5*b^11*c^11*d^4 - 16160*a^6*b^10*c^10*d
^5 + 26800*a^7*b^9*c^9*d^6 - 29312*a^8*b^8*c^8*d^7 + 21424*a^9*b^7*c^7*d^8 - 10400*a^10*b^6*c^6*d^9 + 3280*a^1
1*b^5*c^5*d^10 - 640*a^12*b^4*c^4*d^11 + 64*a^13*b^3*c^3*d^12) + ((-b^3*(a*d - b*c)^5)^(1/2)*(5*a*d - 2*b*c)*(
64*a^6*b^12*c^14*d^3 - 896*a^7*b^11*c^13*d^4 + 4992*a^8*b^10*c^12*d^5 - 15360*a^9*b^9*c^11*d^6 + 29568*a^10*b^
8*c^10*d^7 - 37632*a^11*b^7*c^9*d^8 + 32256*a^12*b^6*c^8*d^9 - 18432*a^13*b^5*c^7*d^10 + 6720*a^14*b^4*c^6*d^1
1 - 1408*a^15*b^3*c^5*d^12 + 128*a^16*b^2*c^4*d^13 - ((-b^3*(a*d - b*c)^5)^(1/2)*(c + d*x^2)^(1/2)*(5*a*d - 2*
b*c)*(512*a^7*b^13*c^16*d^2 - 5376*a^8*b^12*c^15*d^3 + 25600*a^9*b^11*c^14*d^4 - 72960*a^10*b^10*c^13*d^5 + 13
8240*a^11*b^9*c^12*d^6 - 182784*a^12*b^8*c^11*d^7 + 172032*a^13*b^7*c^10*d^8 - 115200*a^14*b^6*c^9*d^9 + 53760
*a^15*b^5*c^8*d^10 - 16640*a^16*b^4*c^7*d^11 + 3072*a^17*b^3*c^6*d^12 - 256*a^18*b^2*c^5*d^13))/(4*(a^7*d^5 -
a^2*b^5*c^5 + 5*a^3*b^4*c^4*d - 10*a^4*b^3*c^3*d^2 + 10*a^5*b^2*c^2*d^3 - 5*a^6*b*c*d^4))))/(4*(a^7*d^5 - a^2*
b^5*c^5 + 5*a^3*b^4*c^4*d - 10*a^4*b^3*c^3*d^2 + 10*a^5*b^2*c^2*d^3 - 5*a^6*b*c*d^4)))*1i)/(4*(a^7*d^5 - a^2*b
^5*c^5 + 5*a^3*b^4*c^4*d - 10*a^4*b^3*c^3*d^2 + 10*a^5*b^2*c^2*d^3 - 5*a^6*b*c*d^4)) + ((-b^3*(a*d - b*c)^5)^(
1/2)*(5*a*d - 2*b*c)*((c + d*x^2)^(1/2)*(128*a^3*b^13*c^13*d^2 - 1344*a^4*b^12*c^12*d^3 + 6160*a^5*b^11*c^11*d
^4 - 16160*a^6*b^10*c^10*d^5 + 26800*a^7*b^9*c^9*d^6 - 29312*a^8*b^8*c^8*d^7 + 21424*a^9*b^7*c^7*d^8 - 10400*a
^10*b^6*c^6*d^9 + 3280*a^11*b^5*c^5*d^10 - 640*a^12*b^4*c^4*d^11 + 64*a^13*b^3*c^3*d^12) - ((-b^3*(a*d - b*c)^
5)^(1/2)*(5*a*d - 2*b*c)*(64*a^6*b^12*c^14*d^3 - 896*a^7*b^11*c^13*d^4 + 4992*a^8*b^10*c^12*d^5 - 15360*a^9*b^
9*c^11*d^6 + 29568*a^10*b^8*c^10*d^7 - 37632*a^...

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