Optimal. Leaf size=377 \[ -\frac {20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3}{24 a^2 c^3 (b c-a d)^3 x^3}+\frac {20 b^4 c^4-24 a b^3 c^3 d-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4}{8 a^3 c^4 (b c-a d)^3 x}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x^3 \left (c+d x^2\right )}+\frac {b^{9/2} (5 b c-11 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} (b c-a d)^4}+\frac {d^{7/2} \left (99 b^2 c^2-110 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} (b c-a d)^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.46, antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {483, 593, 597,
536, 211} \begin {gather*} \frac {b^{9/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (5 b c-11 a d)}{2 a^{7/2} (b c-a d)^4}+\frac {d^{7/2} \left (35 a^2 d^2-110 a b c d+99 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} (b c-a d)^4}+\frac {d \left (-7 a^2 d^2+15 a b c d+4 b^2 c^2\right )}{8 a c^2 x^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac {-35 a^3 d^3+75 a^2 b c d^2-24 a b^2 c^2 d+20 b^3 c^3}{24 a^2 c^3 x^3 (b c-a d)^3}+\frac {-35 a^4 d^4+75 a^3 b c d^3-24 a^2 b^2 c^2 d^2-24 a b^3 c^3 d+20 b^4 c^4}{8 a^3 c^4 x (b c-a d)^3}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac {d (a d+2 b c)}{4 a c x^3 \left (c+d x^2\right )^2 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 483
Rule 536
Rule 593
Rule 597
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\int \frac {-5 b c+2 a d-9 b d x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx}{2 a (b c-a d)}\\ &=\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\int \frac {-2 \left (10 b^2 c^2-8 a b c d+7 a^2 d^2\right )-14 b d (2 b c+a d) x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{8 a c (b c-a d)^2}\\ &=\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x^3 \left (c+d x^2\right )}-\frac {\int \frac {-2 \left (20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3\right )-10 b d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right ) x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{16 a c^2 (b c-a d)^3}\\ &=-\frac {20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3}{24 a^2 c^3 (b c-a d)^3 x^3}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x^3 \left (c+d x^2\right )}+\frac {\int \frac {-6 \left (20 b^4 c^4-24 a b^3 c^3 d-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4\right )-6 b d \left (20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3\right ) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{48 a^2 c^3 (b c-a d)^3}\\ &=-\frac {20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3}{24 a^2 c^3 (b c-a d)^3 x^3}+\frac {20 b^4 c^4-24 a b^3 c^3 d-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4}{8 a^3 c^4 (b c-a d)^3 x}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x^3 \left (c+d x^2\right )}-\frac {\int \frac {-6 \left (20 b^5 c^5-24 a b^4 c^4 d-24 a^2 b^3 c^3 d^2-24 a^3 b^2 c^2 d^3+75 a^4 b c d^4-35 a^5 d^5\right )-6 b d \left (20 b^4 c^4-24 a b^3 c^3 d-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{48 a^3 c^4 (b c-a d)^3}\\ &=-\frac {20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3}{24 a^2 c^3 (b c-a d)^3 x^3}+\frac {20 b^4 c^4-24 a b^3 c^3 d-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4}{8 a^3 c^4 (b c-a d)^3 x}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x^3 \left (c+d x^2\right )}+\frac {\left (b^5 (5 b c-11 a d)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^3 (b c-a d)^4}+\frac {\left (d^4 \left (99 b^2 c^2-110 a b c d+35 a^2 d^2\right )\right ) \int \frac {1}{c+d x^2} \, dx}{8 c^4 (b c-a d)^4}\\ &=-\frac {20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3}{24 a^2 c^3 (b c-a d)^3 x^3}+\frac {20 b^4 c^4-24 a b^3 c^3 d-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4}{8 a^3 c^4 (b c-a d)^3 x}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x^3 \left (c+d x^2\right )}+\frac {b^{9/2} (5 b c-11 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} (b c-a d)^4}+\frac {d^{7/2} \left (99 b^2 c^2-110 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} (b c-a d)^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.32, size = 230, normalized size = 0.61 \begin {gather*} \frac {1}{24} \left (-\frac {8}{a^2 c^3 x^3}+\frac {48 b c+72 a d}{a^3 c^4 x}-\frac {12 b^5 x}{a^3 (-b c+a d)^3 \left (a+b x^2\right )}+\frac {6 d^4 x}{c^3 (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {3 d^4 (19 b c-11 a d) x}{c^4 (b c-a d)^3 \left (c+d x^2\right )}+\frac {12 b^{9/2} (5 b c-11 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2} (b c-a d)^4}+\frac {3 d^{7/2} \left (99 b^2 c^2-110 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{9/2} (b c-a d)^4}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.22, size = 222, normalized size = 0.59
method | result | size |
default | \(-\frac {b^{5} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (11 a d -5 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3} \left (a d -b c \right )^{4}}+\frac {d^{4} \left (\frac {\left (\frac {11}{8} a^{2} d^{3}-\frac {15}{4} a b c \,d^{2}+\frac {19}{8} b^{2} c^{2} d \right ) x^{3}+\frac {c \left (13 a^{2} d^{2}-34 a b c d +21 b^{2} c^{2}\right ) x}{8}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (35 a^{2} d^{2}-110 a b c d +99 b^{2} c^{2}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}}\right )}{c^{4} \left (a d -b c \right )^{4}}-\frac {1}{3 a^{2} c^{3} x^{3}}-\frac {-3 a d -2 b c}{a^{3} c^{4} x}\) | \(222\) |
risch | \(\text {Expression too large to display}\) | \(2662\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 738 vs.
\(2 (347) = 694\).
time = 0.55, size = 738, normalized size = 1.96 \begin {gather*} \frac {{\left (5 \, b^{6} c - 11 \, a b^{5} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )} \sqrt {a b}} + \frac {{\left (99 \, b^{2} c^{2} d^{4} - 110 \, a b c d^{5} + 35 \, a^{2} d^{6}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} \sqrt {c d}} - \frac {8 \, a^{2} b^{3} c^{6} - 24 \, a^{3} b^{2} c^{5} d + 24 \, a^{4} b c^{4} d^{2} - 8 \, a^{5} c^{3} d^{3} - 3 \, {\left (20 \, b^{5} c^{4} d^{2} - 24 \, a b^{4} c^{3} d^{3} - 24 \, a^{2} b^{3} c^{2} d^{4} + 75 \, a^{3} b^{2} c d^{5} - 35 \, a^{4} b d^{6}\right )} x^{8} - {\left (120 \, b^{5} c^{5} d - 104 \, a b^{4} c^{4} d^{2} - 192 \, a^{2} b^{3} c^{3} d^{3} + 303 \, a^{3} b^{2} c^{2} d^{4} + 50 \, a^{4} b c d^{5} - 105 \, a^{5} d^{6}\right )} x^{6} - {\left (60 \, b^{5} c^{6} + 8 \, a b^{4} c^{5} d - 176 \, a^{2} b^{3} c^{4} d^{2} + 319 \, a^{4} b c^{2} d^{4} - 175 \, a^{5} c d^{5}\right )} x^{4} - 8 \, {\left (5 \, a b^{4} c^{6} - 8 \, a^{2} b^{3} c^{5} d - 6 \, a^{3} b^{2} c^{4} d^{2} + 16 \, a^{4} b c^{3} d^{3} - 7 \, a^{5} c^{2} d^{4}\right )} x^{2}}{24 \, {\left ({\left (a^{3} b^{4} c^{7} d^{2} - 3 \, a^{4} b^{3} c^{6} d^{3} + 3 \, a^{5} b^{2} c^{5} d^{4} - a^{6} b c^{4} d^{5}\right )} x^{9} + {\left (2 \, a^{3} b^{4} c^{8} d - 5 \, a^{4} b^{3} c^{7} d^{2} + 3 \, a^{5} b^{2} c^{6} d^{3} + a^{6} b c^{5} d^{4} - a^{7} c^{4} d^{5}\right )} x^{7} + {\left (a^{3} b^{4} c^{9} - a^{4} b^{3} c^{8} d - 3 \, a^{5} b^{2} c^{7} d^{2} + 5 \, a^{6} b c^{6} d^{3} - 2 \, a^{7} c^{5} d^{4}\right )} x^{5} + {\left (a^{4} b^{3} c^{9} - 3 \, a^{5} b^{2} c^{8} d + 3 \, a^{6} b c^{7} d^{2} - a^{7} c^{6} d^{3}\right )} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1033 vs.
\(2 (347) = 694\).
time = 15.13, size = 4225, normalized size = 11.21 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.72, size = 367, normalized size = 0.97 \begin {gather*} \frac {b^{5} x}{2 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} {\left (b x^{2} + a\right )}} + \frac {{\left (5 \, b^{6} c - 11 \, a b^{5} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )} \sqrt {a b}} + \frac {{\left (99 \, b^{2} c^{2} d^{4} - 110 \, a b c d^{5} + 35 \, a^{2} d^{6}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} \sqrt {c d}} + \frac {19 \, b c d^{5} x^{3} - 11 \, a d^{6} x^{3} + 21 \, b c^{2} d^{4} x - 13 \, a c d^{5} x}{8 \, {\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}\right )} {\left (d x^{2} + c\right )}^{2}} + \frac {6 \, b c x^{2} + 9 \, a d x^{2} - a c}{3 \, a^{3} c^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.23, size = 1161, normalized size = 3.08 \begin {gather*} \frac {\frac {x^2\,\left (7\,a\,d+5\,b\,c\right )}{3\,a^2\,c^2}-\frac {1}{3\,a\,c}+\frac {x^8\,\left (35\,a^4\,b\,d^6-75\,a^3\,b^2\,c\,d^5+24\,a^2\,b^3\,c^2\,d^4+24\,a\,b^4\,c^3\,d^3-20\,b^5\,c^4\,d^2\right )}{8\,a^3\,c^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {x^4\,\left (-175\,a^5\,d^5+319\,a^4\,b\,c\,d^4-176\,a^2\,b^3\,c^3\,d^2+8\,a\,b^4\,c^4\,d+60\,b^5\,c^5\right )}{24\,a^3\,c^3\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,x^6\,\left (105\,a^5\,d^5-50\,a^4\,b\,c\,d^4-303\,a^3\,b^2\,c^2\,d^3+192\,a^2\,b^3\,c^3\,d^2+104\,a\,b^4\,c^4\,d-120\,b^5\,c^5\right )}{24\,a^3\,c^4\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x^5\,\left (b\,c^2+2\,a\,d\,c\right )+x^7\,\left (a\,d^2+2\,b\,c\,d\right )+a\,c^2\,x^3+b\,d^2\,x^9}+\frac {\mathrm {atan}\left (\frac {b^3\,c^{11}\,x\,{\left (-a^7\,b^9\right )}^{3/2}\,400{}\mathrm {i}+a^{18}\,b\,d^{11}\,x\,\sqrt {-a^7\,b^9}\,1225{}\mathrm {i}+a^{14}\,b^5\,c^4\,d^7\,x\,\sqrt {-a^7\,b^9}\,9801{}\mathrm {i}-a^{15}\,b^4\,c^3\,d^8\,x\,\sqrt {-a^7\,b^9}\,21780{}\mathrm {i}+a^{16}\,b^3\,c^2\,d^9\,x\,\sqrt {-a^7\,b^9}\,19030{}\mathrm {i}-a\,b^2\,c^{10}\,d\,x\,{\left (-a^7\,b^9\right )}^{3/2}\,1760{}\mathrm {i}+a^2\,b\,c^9\,d^2\,x\,{\left (-a^7\,b^9\right )}^{3/2}\,1936{}\mathrm {i}-a^{17}\,b^2\,c\,d^{10}\,x\,\sqrt {-a^7\,b^9}\,7700{}\mathrm {i}}{-1225\,a^{22}\,b^5\,d^{11}+7700\,a^{21}\,b^6\,c\,d^{10}-19030\,a^{20}\,b^7\,c^2\,d^9+21780\,a^{19}\,b^8\,c^3\,d^8-9801\,a^{18}\,b^9\,c^4\,d^7+1936\,a^{13}\,b^{14}\,c^9\,d^2-1760\,a^{12}\,b^{15}\,c^{10}\,d+400\,a^{11}\,b^{16}\,c^{11}}\right )\,\left (11\,a\,d-5\,b\,c\right )\,\sqrt {-a^7\,b^9}\,1{}\mathrm {i}}{2\,\left (a^{11}\,d^4-4\,a^{10}\,b\,c\,d^3+6\,a^9\,b^2\,c^2\,d^2-4\,a^8\,b^3\,c^3\,d+a^7\,b^4\,c^4\right )}-\frac {\mathrm {atan}\left (\frac {a^{11}\,d^5\,x\,{\left (-c^9\,d^7\right )}^{3/2}\,1225{}\mathrm {i}+b^{11}\,c^{20}\,d\,x\,\sqrt {-c^9\,d^7}\,400{}\mathrm {i}-a^8\,b^3\,c^3\,d^2\,x\,{\left (-c^9\,d^7\right )}^{3/2}\,21780{}\mathrm {i}+a^9\,b^2\,c^2\,d^3\,x\,{\left (-c^9\,d^7\right )}^{3/2}\,19030{}\mathrm {i}+a^2\,b^9\,c^{18}\,d^3\,x\,\sqrt {-c^9\,d^7}\,1936{}\mathrm {i}-a^{10}\,b\,c\,d^4\,x\,{\left (-c^9\,d^7\right )}^{3/2}\,7700{}\mathrm {i}+a^7\,b^4\,c^4\,d\,x\,{\left (-c^9\,d^7\right )}^{3/2}\,9801{}\mathrm {i}-a\,b^{10}\,c^{19}\,d^2\,x\,\sqrt {-c^9\,d^7}\,1760{}\mathrm {i}}{1225\,a^{11}\,c^{14}\,d^{15}-7700\,a^{10}\,b\,c^{15}\,d^{14}+19030\,a^9\,b^2\,c^{16}\,d^{13}-21780\,a^8\,b^3\,c^{17}\,d^{12}+9801\,a^7\,b^4\,c^{18}\,d^{11}-1936\,a^2\,b^9\,c^{23}\,d^6+1760\,a\,b^{10}\,c^{24}\,d^5-400\,b^{11}\,c^{25}\,d^4}\right )\,\sqrt {-c^9\,d^7}\,\left (35\,a^2\,d^2-110\,a\,b\,c\,d+99\,b^2\,c^2\right )\,1{}\mathrm {i}}{8\,\left (a^4\,c^9\,d^4-4\,a^3\,b\,c^{10}\,d^3+6\,a^2\,b^2\,c^{11}\,d^2-4\,a\,b^3\,c^{12}\,d+b^4\,c^{13}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________