Integrand size = 24, antiderivative size = 376 \[ \int \frac {\left (c+d x^2\right )^3}{x^{7/2} \left (a+b x^2\right )^2} \, dx=-\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt {x}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac {3 (b c-a d)^2 (3 b c+a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} b^{7/4}}+\frac {3 (b c-a d)^2 (3 b c+a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} b^{7/4}}+\frac {3 (b c-a d)^2 (3 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} b^{7/4}}-\frac {3 (b c-a d)^2 (3 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} b^{7/4}} \]
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Time = 0.33 (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} \[ \int \frac {\left (c+d x^2\right )^3}{x^{7/2} \left (a+b x^2\right )^2} \, dx=-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^2 (a d+3 b c)}{4 \sqrt {2} a^{13/4} b^{7/4}}+\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (a d+3 b c)}{4 \sqrt {2} a^{13/4} b^{7/4}}+\frac {3 (b c-a d)^2 (a d+3 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} b^{7/4}}-\frac {3 (b c-a d)^2 (a d+3 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} b^{7/4}}-\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (2 a^2 d^2-15 a b c d+9 b^2 c^2\right )}{2 a^3 b \sqrt {x}}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{5/2} \left (a+b x^2\right )} \]
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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 {\left (c+d x^4\right )^3}{x^6 \left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right ) \\ & = \frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \frac {\left (c+d x^4\right ) \left (-c (9 b c-5 a d)-d (b c+3 a d) x^4\right )}{x^6 \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right )}{2 a b} \\ & = \frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {c^2 (-9 b c+5 a d)}{a x^6}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{a^2 x^2}-\frac {3 (-b c+a d)^2 (3 b c+a d) x^2}{a^2 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a b} \\ & = -\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt {x}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac {\left (3 (b c-a d)^2 (3 b c+a d)\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^3 b} \\ & = -\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt {x}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac {\left (3 (b c-a d)^2 (3 b c+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^3 b^{3/2}}+\frac {\left (3 (b c-a d)^2 (3 b c+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^3 b^{3/2}} \\ & = -\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt {x}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac {\left (3 (b c-a d)^2 (3 b c+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^3 b^2}+\frac {\left (3 (b c-a d)^2 (3 b c+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^3 b^2}+\frac {\left (3 (b c-a d)^2 (3 b c+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^{13/4} b^{7/4}}+\frac {\left (3 (b c-a d)^2 (3 b c+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^{13/4} b^{7/4}} \\ & = -\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt {x}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac {3 (b c-a d)^2 (3 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} b^{7/4}}-\frac {3 (b c-a d)^2 (3 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} b^{7/4}}+\frac {\left (3 (b c-a d)^2 (3 b c+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^{13/4} b^{7/4}}-\frac {\left (3 (b c-a d)^2 (3 b c+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^{13/4} b^{7/4}} \\ & = -\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt {x}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac {3 (b c-a d)^2 (3 b c+a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} b^{7/4}}+\frac {3 (b c-a d)^2 (3 b c+a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} b^{7/4}}+\frac {3 (b c-a d)^2 (3 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} b^{7/4}}-\frac {3 (b c-a d)^2 (3 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} b^{7/4}} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.63 \[ \int \frac {\left (c+d x^2\right )^3}{x^{7/2} \left (a+b x^2\right )^2} \, dx=\frac {-\frac {4 \sqrt [4]{a} b^{3/4} \left (-45 b^3 c^3 x^4+5 a^3 d^3 x^4+3 a b^2 c^2 x^2 \left (-12 c+25 d x^2\right )+a^2 b c \left (4 c^2+60 c d x^2-15 d^2 x^4\right )\right )}{x^{5/2} \left (a+b x^2\right )}-15 \sqrt {2} (b c-a d)^2 (3 b c+a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-15 \sqrt {2} (b c-a d)^2 (3 b c+a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{40 a^{13/4} b^{7/4}} \]
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Time = 2.70 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.52
method | result | size |
risch | \(-\frac {2 c^{2} \left (15 a d \,x^{2}-10 c b \,x^{2}+a c \right )}{5 a^{3} x^{\frac {5}{2}}}+\frac {\left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (-\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 b \left (b \,x^{2}+a \right )}+\frac {3 \left (a d +3 b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3}}\) | \(197\) |
derivativedivides | \(-\frac {2 c^{3}}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 c^{2} \left (3 a d -2 b c \right )}{a^{3} \sqrt {x}}+\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{\frac {3}{2}}}{2 b \left (b \,x^{2}+a \right )}+\frac {3 \left (a^{3} d^{3}+a^{2} b c \,d^{2}-5 a \,b^{2} c^{2} d +3 b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a^{3}}\) | \(232\) |
default | \(-\frac {2 c^{3}}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 c^{2} \left (3 a d -2 b c \right )}{a^{3} \sqrt {x}}+\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{\frac {3}{2}}}{2 b \left (b \,x^{2}+a \right )}+\frac {3 \left (a^{3} d^{3}+a^{2} b c \,d^{2}-5 a \,b^{2} c^{2} d +3 b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a^{3}}\) | \(232\) |
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 2110, normalized size of antiderivative = 5.61 \[ \int \frac {\left (c+d x^2\right )^3}{x^{7/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{x^{7/2} \left (a+b x^2\right )^2} \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.84 \[ \int \frac {\left (c+d x^2\right )^3}{x^{7/2} \left (a+b x^2\right )^2} \, dx=-\frac {4 \, a^{2} b c^{3} - 5 \, {\left (9 \, b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{4} - 12 \, {\left (3 \, a b^{2} c^{3} - 5 \, a^{2} b c^{2} d\right )} x^{2}}{10 \, {\left (a^{3} b^{2} x^{\frac {9}{2}} + a^{4} b x^{\frac {5}{2}}\right )}} + \frac {3 \, {\left (3 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a^{3} b} \]
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Time = 0.30 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.34 \[ \int \frac {\left (c+d x^2\right )^3}{x^{7/2} \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^{3} b} + \frac {2 \, {\left (10 \, b c^{3} x^{2} - 15 \, a c^{2} d x^{2} - a c^{3}\right )}}{5 \, a^{3} x^{\frac {5}{2}}} + \frac {3 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + \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^{4} b^{4}} + \frac {3 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + \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^{4} b^{4}} - \frac {3 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + \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^{4} b^{4}} + \frac {3 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + \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^{4} b^{4}} \]
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Time = 0.26 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.74 \[ \int \frac {\left (c+d x^2\right )^3}{x^{7/2} \left (a+b x^2\right )^2} \, dx=\frac {3\,\mathrm {atan}\left (\frac {3\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+3\,b\,c\right )\,\left (288\,a^{16}\,b^5\,d^6+576\,a^{15}\,b^6\,c\,d^5-2592\,a^{14}\,b^7\,c^2\,d^4-1152\,a^{13}\,b^8\,c^3\,d^3+8928\,a^{12}\,b^9\,c^4\,d^2-8640\,a^{11}\,b^{10}\,c^5\,d+2592\,a^{10}\,b^{11}\,c^6\right )}{4\,{\left (-a\right )}^{13/4}\,b^{7/4}\,\left (216\,a^{16}\,b^3\,d^9+648\,a^{15}\,b^4\,c\,d^8-2592\,a^{14}\,b^5\,c^2\,d^7-4320\,a^{13}\,b^6\,c^3\,d^6+16848\,a^{12}\,b^7\,c^4\,d^5-1296\,a^{11}\,b^8\,c^5\,d^4-40608\,a^{10}\,b^9\,c^6\,d^3+54432\,a^9\,b^{10}\,c^7\,d^2-29160\,a^8\,b^{11}\,c^8\,d+5832\,a^7\,b^{12}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+3\,b\,c\right )}{4\,{\left (-a\right )}^{13/4}\,b^{7/4}}-\frac {\frac {2\,c^3}{5\,a}+\frac {x^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+15\,a\,b^2\,c^2\,d-9\,b^3\,c^3\right )}{2\,a^3\,b}+\frac {6\,c^2\,x^2\,\left (5\,a\,d-3\,b\,c\right )}{5\,a^2}}{a\,x^{5/2}+b\,x^{9/2}}-\frac {3\,\mathrm {atanh}\left (\frac {3\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+3\,b\,c\right )\,\left (288\,a^{16}\,b^5\,d^6+576\,a^{15}\,b^6\,c\,d^5-2592\,a^{14}\,b^7\,c^2\,d^4-1152\,a^{13}\,b^8\,c^3\,d^3+8928\,a^{12}\,b^9\,c^4\,d^2-8640\,a^{11}\,b^{10}\,c^5\,d+2592\,a^{10}\,b^{11}\,c^6\right )}{4\,{\left (-a\right )}^{13/4}\,b^{7/4}\,\left (216\,a^{16}\,b^3\,d^9+648\,a^{15}\,b^4\,c\,d^8-2592\,a^{14}\,b^5\,c^2\,d^7-4320\,a^{13}\,b^6\,c^3\,d^6+16848\,a^{12}\,b^7\,c^4\,d^5-1296\,a^{11}\,b^8\,c^5\,d^4-40608\,a^{10}\,b^9\,c^6\,d^3+54432\,a^9\,b^{10}\,c^7\,d^2-29160\,a^8\,b^{11}\,c^8\,d+5832\,a^7\,b^{12}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+3\,b\,c\right )}{4\,{\left (-a\right )}^{13/4}\,b^{7/4}} \]
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