Integrand size = 24, antiderivative size = 325 \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^3}{13 a x^{13/2}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{5 a^3 x^{5/2}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}-\frac {\sqrt [4]{b} (b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} (b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}}-\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}} \]
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Time = 0.23 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {477, 472, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=-\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}}-\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{5 a^3 x^{5/2}}-\frac {2 c^3}{13 a x^{13/2}} \]
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Rule 210
Rule 303
Rule 472
Rule 477
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^{14} \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {c^3}{a x^{14}}+\frac {c^2 (-b c+3 a d)}{a^2 x^{10}}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x^6}+\frac {(-b c+a d)^3}{a^4 x^2}-\frac {b (-b c+a d)^3 x^2}{a^4 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 c^3}{13 a x^{13/2}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{5 a^3 x^{5/2}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}+\frac {\left (2 b (b c-a d)^3\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^4} \\ & = -\frac {2 c^3}{13 a x^{13/2}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{5 a^3 x^{5/2}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}-\frac {\left (\sqrt {b} (b c-a d)^3\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^4}+\frac {\left (\sqrt {b} (b c-a d)^3\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^4} \\ & = -\frac {2 c^3}{13 a x^{13/2}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{5 a^3 x^{5/2}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}+\frac {(b c-a d)^3 \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 )}{2 a^4}+\frac {(b c-a d)^3 \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 )}{2 a^4}+\frac {\left (\sqrt [4]{b} (b c-a d)^3\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 )}{2 \sqrt {2} a^{17/4}}+\frac {\left (\sqrt [4]{b} (b c-a d)^3\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 )}{2 \sqrt {2} a^{17/4}} \\ & = -\frac {2 c^3}{13 a x^{13/2}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{5 a^3 x^{5/2}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}+\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}}-\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}}+\frac {\left (\sqrt [4]{b} (b c-a d)^3\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 )}{\sqrt {2} a^{17/4}}-\frac {\left (\sqrt [4]{b} (b c-a d)^3\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 )}{\sqrt {2} a^{17/4}} \\ & = -\frac {2 c^3}{13 a x^{13/2}}+\frac {2 c^2 (b c-3 a d)}{9 a^2 x^{9/2}}-\frac {2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{5 a^3 x^{5/2}}+\frac {2 (b c-a d)^3}{a^4 \sqrt {x}}-\frac {\sqrt [4]{b} (b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} (b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}}-\frac {\sqrt [4]{b} (b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{17/4}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.73 \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=\frac {1170 b^3 c^3 x^6-234 a b^2 c^2 x^4 \left (c+15 d x^2\right )+26 a^2 b c x^2 \left (5 c^2+27 c d x^2+135 d^2 x^4\right )-6 a^3 \left (15 c^3+65 c^2 d x^2+117 c d^2 x^4+195 d^3 x^6\right )}{585 a^4 x^{13/2}}+\frac {\sqrt [4]{b} (-b c+a d)^3 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {2} a^{17/4}}+\frac {\sqrt [4]{b} (-b c+a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{17/4}} \]
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Time = 2.75 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2 c^{3}}{13 a \,x^{\frac {13}{2}}}-\frac {2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{a^{4} \sqrt {x}}-\frac {2 c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{9 a^{2} x^{\frac {9}{2}}}\) | \(249\) |
default | \(-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2 c^{3}}{13 a \,x^{\frac {13}{2}}}-\frac {2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{a^{4} \sqrt {x}}-\frac {2 c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{9 a^{2} x^{\frac {9}{2}}}\) | \(249\) |
risch | \(-\frac {2 \left (585 a^{3} d^{3} x^{6}-1755 a^{2} b c \,d^{2} x^{6}+1755 a \,b^{2} c^{2} d \,x^{6}-585 b^{3} c^{3} x^{6}+351 a^{3} c \,d^{2} x^{4}-351 a^{2} b \,c^{2} d \,x^{4}+117 a \,b^{2} c^{3} x^{4}+195 a^{3} c^{2} d \,x^{2}-65 a^{2} b \,c^{3} x^{2}+45 c^{3} a^{3}\right )}{585 a^{4} x^{\frac {13}{2}}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(269\) |
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 2059, normalized size of antiderivative = 6.34 \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=\text {Timed out} \]
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none
Time = 0.32 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.02 \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=\frac {{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, a^{4}} + \frac {2 \, {\left (585 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{6} - 45 \, a^{3} c^{3} - 117 \, {\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{4} + 65 \, {\left (a^{2} b c^{3} - 3 \, a^{3} c^{2} d\right )} x^{2}\right )}}{585 \, a^{4} x^{\frac {13}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (244) = 488\).
Time = 0.30 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.65 \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \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 )}{2 \, a^{5} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \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 )}{2 \, a^{5} b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \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 )}{4 \, a^{5} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \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 )}{4 \, a^{5} b^{2}} + \frac {2 \, {\left (585 \, b^{3} c^{3} x^{6} - 1755 \, a b^{2} c^{2} d x^{6} + 1755 \, a^{2} b c d^{2} x^{6} - 585 \, a^{3} d^{3} x^{6} - 117 \, a b^{2} c^{3} x^{4} + 351 \, a^{2} b c^{2} d x^{4} - 351 \, a^{3} c d^{2} x^{4} + 65 \, a^{2} b c^{3} x^{2} - 195 \, a^{3} c^{2} d x^{2} - 45 \, a^{3} c^{3}\right )}}{585 \, a^{4} x^{\frac {13}{2}}} \]
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Time = 5.16 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.97 \[ \int \frac {\left (c+d x^2\right )^3}{x^{15/2} \left (a+b x^2\right )} \, dx=\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (16\,a^{19}\,b^4\,d^6-96\,a^{18}\,b^5\,c\,d^5+240\,a^{17}\,b^6\,c^2\,d^4-320\,a^{16}\,b^7\,c^3\,d^3+240\,a^{15}\,b^8\,c^4\,d^2-96\,a^{14}\,b^9\,c^5\,d+16\,a^{13}\,b^{10}\,c^6\right )}{a^{17/4}\,\left (-16\,a^{18}\,b^4\,d^9+144\,a^{17}\,b^5\,c\,d^8-576\,a^{16}\,b^6\,c^2\,d^7+1344\,a^{15}\,b^7\,c^3\,d^6-2016\,a^{14}\,b^8\,c^4\,d^5+2016\,a^{13}\,b^9\,c^5\,d^4-1344\,a^{12}\,b^{10}\,c^6\,d^3+576\,a^{11}\,b^{11}\,c^7\,d^2-144\,a^{10}\,b^{12}\,c^8\,d+16\,a^9\,b^{13}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^{17/4}}-\frac {\frac {2\,c^3}{13\,a}+\frac {2\,x^6\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{a^4}+\frac {2\,c^2\,x^2\,\left (3\,a\,d-b\,c\right )}{9\,a^2}+\frac {2\,c\,x^4\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{5\,a^3}}{x^{13/2}}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (16\,a^{19}\,b^4\,d^6-96\,a^{18}\,b^5\,c\,d^5+240\,a^{17}\,b^6\,c^2\,d^4-320\,a^{16}\,b^7\,c^3\,d^3+240\,a^{15}\,b^8\,c^4\,d^2-96\,a^{14}\,b^9\,c^5\,d+16\,a^{13}\,b^{10}\,c^6\right )}{a^{17/4}\,\left (-16\,a^{18}\,b^4\,d^9+144\,a^{17}\,b^5\,c\,d^8-576\,a^{16}\,b^6\,c^2\,d^7+1344\,a^{15}\,b^7\,c^3\,d^6-2016\,a^{14}\,b^8\,c^4\,d^5+2016\,a^{13}\,b^9\,c^5\,d^4-1344\,a^{12}\,b^{10}\,c^6\,d^3+576\,a^{11}\,b^{11}\,c^7\,d^2-144\,a^{10}\,b^{12}\,c^8\,d+16\,a^9\,b^{13}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^{17/4}} \]
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