Integrand size = 24, antiderivative size = 376 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {d \left (6 b^2 c^2-21 a b c d+11 a^2 d^2\right ) x^{3/2}}{6 a b^3}-\frac {d^2 (7 b c-11 a d) x^{7/2}}{14 a b^2}+\frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac {(b c-a d)^2 (b c+11 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}+\frac {(b c-a d)^2 (b c+11 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}+\frac {(b c-a d)^2 (b c+11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}-\frac {(b c-a d)^2 (b c+11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}} \]
[Out]
Time = 0.32 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {477, 479, 584, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^2 (11 a d+b c)}{4 \sqrt {2} a^{5/4} b^{15/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (11 a d+b c)}{4 \sqrt {2} a^{5/4} b^{15/4}}+\frac {(b c-a d)^2 (11 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}-\frac {(b c-a d)^2 (11 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}-\frac {d x^{3/2} \left (11 a^2 d^2-21 a b c d+6 b^2 c^2\right )}{6 a b^3}-\frac {d^2 x^{7/2} (7 b c-11 a d)}{14 a b^2}+\frac {x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )} \]
[In]
[Out]
Rule 210
Rule 303
Rule 477
Rule 479
Rule 584
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^2 \left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right ) \\ & = \frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \frac {x^2 \left (c+d x^4\right ) \left (-c (b c+3 a d)+d (7 b c-11 a d) x^4\right )}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a b} \\ & = \frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {d \left (6 b^2 c^2-21 a b c d+11 a^2 d^2\right ) x^2}{b^2}+\frac {d^2 (7 b c-11 a d) x^6}{b}-\frac {\left (b^3 c^3+9 a b^2 c^2 d-21 a^2 b c d^2+11 a^3 d^3\right ) x^2}{b^2 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a b} \\ & = -\frac {d \left (6 b^2 c^2-21 a b c d+11 a^2 d^2\right ) x^{3/2}}{6 a b^3}-\frac {d^2 (7 b c-11 a d) x^{7/2}}{14 a b^2}+\frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a b^3} \\ & = -\frac {d \left (6 b^2 c^2-21 a b c d+11 a^2 d^2\right ) x^{3/2}}{6 a b^3}-\frac {d^2 (7 b c-11 a d) x^{7/2}}{14 a b^2}+\frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a b^{7/2}}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a b^{7/2}} \\ & = -\frac {d \left (6 b^2 c^2-21 a b c d+11 a^2 d^2\right ) x^{3/2}}{6 a b^3}-\frac {d^2 (7 b c-11 a d) x^{7/2}}{14 a b^2}+\frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a b^4}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a b^4}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{5/4} b^{15/4}} \\ & = -\frac {d \left (6 b^2 c^2-21 a b c d+11 a^2 d^2\right ) x^{3/2}}{6 a b^3}-\frac {d^2 (7 b c-11 a d) x^{7/2}}{14 a b^2}+\frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}+\frac {(b c-a d)^2 (b c+11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}-\frac {(b c-a d)^2 (b c+11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}+\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}-\frac {\left ((b c-a d)^2 (b c+11 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}} \\ & = -\frac {d \left (6 b^2 c^2-21 a b c d+11 a^2 d^2\right ) x^{3/2}}{6 a b^3}-\frac {d^2 (7 b c-11 a d) x^{7/2}}{14 a b^2}+\frac {(b c-a d) x^{3/2} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac {(b c-a d)^2 (b c+11 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}+\frac {(b c-a d)^2 (b c+11 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{15/4}}+\frac {(b c-a d)^2 (b c+11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}}-\frac {(b c-a d)^2 (b c+11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{15/4}} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (21 b^3 c^3-77 a^3 d^3+a^2 b d^2 \left (147 c-44 d x^2\right )+3 a b^2 d \left (-21 c^2+28 c d x^2+4 d^2 x^4\right )\right )}{a+b x^2}-21 \sqrt {2} (b c-a d)^2 (b c+11 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-21 \sqrt {2} (b c-a d)^2 (b c+11 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{168 a^{5/4} b^{15/4}} \]
[In]
[Out]
Time = 2.78 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.53
method | result | size |
risch | \(-\frac {2 x^{\frac {3}{2}} d^{2} \left (-3 b d \,x^{2}+14 a d -21 b c \right )}{21 b^{3}}+\frac {\left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (-\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 a \left (b \,x^{2}+a \right )}+\frac {\left (11 a d +b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{b^{3}}\) | \(198\) |
derivativedivides | \(-\frac {2 d^{2} \left (-\frac {b d \,x^{\frac {7}{2}}}{7}+\frac {\left (2 a d -3 b c \right ) x^{\frac {3}{2}}}{3}\right )}{b^{3}}+\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{\frac {3}{2}}}{2 a \left (b \,x^{2}+a \right )}+\frac {\left (11 a^{3} d^{3}-21 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{b^{3}}\) | \(235\) |
default | \(-\frac {2 d^{2} \left (-\frac {b d \,x^{\frac {7}{2}}}{7}+\frac {\left (2 a d -3 b c \right ) x^{\frac {3}{2}}}{3}\right )}{b^{3}}+\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{\frac {3}{2}}}{2 a \left (b \,x^{2}+a \right )}+\frac {\left (11 a^{3} d^{3}-21 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{b^{3}}\) | \(235\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 2091, normalized size of antiderivative = 5.56 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{\frac {3}{2}}}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {2 \, {\left (3 \, b d^{3} x^{\frac {7}{2}} + 7 \, {\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} x^{\frac {3}{2}}\right )}}{21 \, b^{3}} + \frac {{\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 21 \, a^{2} b c d^{2} + 11 \, a^{3} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a b^{3}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {b^{3} c^{3} x^{\frac {3}{2}} - 3 \, a b^{2} c^{2} d x^{\frac {3}{2}} + 3 \, a^{2} b c d^{2} x^{\frac {3}{2}} - a^{3} d^{3} x^{\frac {3}{2}}}{2 \, {\left (b x^{2} + a\right )} a b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} + 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 21 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 11 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{6}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} + 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 21 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 11 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{6}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} + 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 21 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 11 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{6}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} + 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 21 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 11 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{6}} + \frac {2 \, {\left (3 \, b^{12} d^{3} x^{\frac {7}{2}} + 21 \, b^{12} c d^{2} x^{\frac {3}{2}} - 14 \, a b^{11} d^{3} x^{\frac {3}{2}}\right )}}{21 \, b^{14}} \]
[In]
[Out]
Time = 5.39 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {2\,d^3\,x^{7/2}}{7\,b^2}-x^{3/2}\,\left (\frac {4\,a\,d^3}{3\,b^3}-\frac {2\,c\,d^2}{b^2}\right )-\frac {x^{3/2}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,a\,\left (b^4\,x^2+a\,b^3\right )}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (11\,a\,d+b\,c\right )\,\left (121\,a^6\,d^6-462\,a^5\,b\,c\,d^5+639\,a^4\,b^2\,c^2\,d^4-356\,a^3\,b^3\,c^3\,d^3+39\,a^2\,b^4\,c^4\,d^2+18\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{{\left (-a\right )}^{1/4}\,\left (1331\,a^9\,d^9-7623\,a^8\,b\,c\,d^8+17820\,a^7\,b^2\,c^2\,d^7-21372\,a^6\,b^3\,c^3\,d^6+13194\,a^5\,b^4\,c^4\,d^5-3186\,a^4\,b^5\,c^5\,d^4-372\,a^3\,b^6\,c^6\,d^3+180\,a^2\,b^7\,c^7\,d^2+27\,a\,b^8\,c^8\,d+b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (11\,a\,d+b\,c\right )}{4\,{\left (-a\right )}^{5/4}\,b^{15/4}}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (11\,a\,d+b\,c\right )\,\left (121\,a^6\,d^6-462\,a^5\,b\,c\,d^5+639\,a^4\,b^2\,c^2\,d^4-356\,a^3\,b^3\,c^3\,d^3+39\,a^2\,b^4\,c^4\,d^2+18\,a\,b^5\,c^5\,d+b^6\,c^6\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\left (1331\,a^9\,d^9-7623\,a^8\,b\,c\,d^8+17820\,a^7\,b^2\,c^2\,d^7-21372\,a^6\,b^3\,c^3\,d^6+13194\,a^5\,b^4\,c^4\,d^5-3186\,a^4\,b^5\,c^5\,d^4-372\,a^3\,b^6\,c^6\,d^3+180\,a^2\,b^7\,c^7\,d^2+27\,a\,b^8\,c^8\,d+b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (11\,a\,d+b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{5/4}\,b^{15/4}} \]
[In]
[Out]