Integrand size = 24, antiderivative size = 440 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt {x}}{16 c d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}} \]
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Time = 0.27 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {474, 468, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=-\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}-\frac {\sqrt {x} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right )}{16 c d^4}+\frac {x^{5/2} \left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right )}{80 c^2 d^3}-\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (5 a^2 d^2-90 a b c d+117 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}-\frac {x^{9/2} (b c-a d) (17 b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {x^{9/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]
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Rule 210
Rule 217
Rule 327
Rule 335
Rule 468
Rule 474
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\int \frac {x^{7/2} \left (\frac {1}{2} \left (-8 a^2 d^2+9 (b c-a d)^2\right )-4 b^2 c d x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c d^2} \\ & = \frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \int \frac {x^{7/2}}{c+d x^2} \, dx}{32 c^2 d^2} \\ & = \frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \int \frac {x^{3/2}}{c+d x^2} \, dx}{32 c d^3} \\ & = -\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt {x}}{16 c d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{32 d^4} \\ & = -\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt {x}}{16 c d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 d^4} \\ & = -\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt {x}}{16 c d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {c} d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {c} d^4} \\ & = -\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt {x}}{16 c d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {c} d^{9/2}}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {c} d^{9/2}}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{3/4} d^{17/4}} \\ & = -\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt {x}}{16 c d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}} \\ & = -\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \sqrt {x}}{16 c d^4}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) x^{5/2}}{80 c^2 d^3}+\frac {(b c-a d)^2 x^{9/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (17 b c-a d) x^{9/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} d^{17/4}}-\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}}+\frac {\left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} d^{17/4}} \\ \end{align*}
Time = 0.84 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.58 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {\frac {4 \sqrt [4]{d} \sqrt {x} \left (-5 a^2 d^2 \left (5 c+9 d x^2\right )+10 a b d \left (45 c^2+81 c d x^2+32 d^2 x^4\right )-b^2 \left (585 c^3+1053 c^2 d x^2+416 c d^2 x^4-32 d^3 x^6\right )\right )}{\left (c+d x^2\right )^2}-\frac {5 \sqrt {2} \left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{3/4}}+\frac {5 \sqrt {2} \left (117 b^2 c^2-90 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{3/4}}}{320 d^{17/4}} \]
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Time = 2.85 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {2 b \left (b d \,x^{2}+10 a d -15 b c \right ) \sqrt {x}}{5 d^{4}}+\frac {\frac {2 \left (-\frac {9}{32} a^{2} d^{3}+\frac {17}{16} a b c \,d^{2}-\frac {25}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}-\frac {c \left (5 a^{2} d^{2}-26 a b c d +21 b^{2} c^{2}\right ) \sqrt {x}}{16}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}-90 a b c d +117 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c}}{d^{4}}\) | \(230\) |
derivativedivides | \(\frac {2 b \left (\frac {b \,x^{\frac {5}{2}} d}{5}+2 a d \sqrt {x}-3 b c \sqrt {x}\right )}{d^{4}}+\frac {\frac {2 \left (\left (-\frac {9}{32} a^{2} d^{3}+\frac {17}{16} a b c \,d^{2}-\frac {25}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}-\frac {c \left (5 a^{2} d^{2}-26 a b c d +21 b^{2} c^{2}\right ) \sqrt {x}}{32}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}-90 a b c d +117 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c}}{d^{4}}\) | \(234\) |
default | \(\frac {2 b \left (\frac {b \,x^{\frac {5}{2}} d}{5}+2 a d \sqrt {x}-3 b c \sqrt {x}\right )}{d^{4}}+\frac {\frac {2 \left (\left (-\frac {9}{32} a^{2} d^{3}+\frac {17}{16} a b c \,d^{2}-\frac {25}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}-\frac {c \left (5 a^{2} d^{2}-26 a b c d +21 b^{2} c^{2}\right ) \sqrt {x}}{32}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}-90 a b c d +117 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c}}{d^{4}}\) | \(234\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1307, normalized size of antiderivative = 2.97 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.88 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=-\frac {{\left (25 \, b^{2} c^{2} d - 34 \, a b c d^{2} + 9 \, a^{2} d^{3}\right )} x^{\frac {5}{2}} + {\left (21 \, b^{2} c^{3} - 26 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} \sqrt {x}}{16 \, {\left (d^{6} x^{4} + 2 \, c d^{5} x^{2} + c^{2} d^{4}\right )}} + \frac {2 \, {\left (b^{2} d x^{\frac {5}{2}} - 5 \, {\left (3 \, b^{2} c - 2 \, a b d\right )} \sqrt {x}\right )}}{5 \, d^{4}} + \frac {\frac {2 \, \sqrt {2} {\left (117 \, b^{2} c^{2} - 90 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (117 \, b^{2} c^{2} - 90 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (117 \, b^{2} c^{2} - 90 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (117 \, b^{2} c^{2} - 90 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, d^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.02 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {\sqrt {2} {\left (117 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c d^{5}} + \frac {\sqrt {2} {\left (117 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c d^{5}} + \frac {\sqrt {2} {\left (117 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c d^{5}} - \frac {\sqrt {2} {\left (117 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c d^{5}} - \frac {25 \, b^{2} c^{2} d x^{\frac {5}{2}} - 34 \, a b c d^{2} x^{\frac {5}{2}} + 9 \, a^{2} d^{3} x^{\frac {5}{2}} + 21 \, b^{2} c^{3} \sqrt {x} - 26 \, a b c^{2} d \sqrt {x} + 5 \, a^{2} c d^{2} \sqrt {x}}{16 \, {\left (d x^{2} + c\right )}^{2} d^{4}} + \frac {2 \, {\left (b^{2} d^{12} x^{\frac {5}{2}} - 15 \, b^{2} c d^{11} \sqrt {x} + 10 \, a b d^{12} \sqrt {x}\right )}}{5 \, d^{15}} \]
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Time = 5.64 (sec) , antiderivative size = 1426, normalized size of antiderivative = 3.24 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
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