Integrand size = 24, antiderivative size = 375 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}} \]
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Time = 0.31 (sec) , antiderivative size = 375, 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, 470, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} d^{17/4}}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}+\frac {\sqrt {x} (13 b c-5 a d) (b c-a d)}{2 d^4}-\frac {x^{5/2} (13 b c-5 a d) (b c-a d)}{10 c d^3}+\frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {2 b^2 x^{9/2}}{9 d^2} \]
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Rule 210
Rule 217
Rule 327
Rule 335
Rule 470
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}}{2 c d^2 \left (c+d x^2\right )}-\frac {\int \frac {x^{7/2} \left (\frac {1}{2} (3 b c-5 a d) (3 b c-a d)-2 b^2 c d x^2\right )}{c+d x^2} \, dx}{2 c d^2} \\ & = \frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {((13 b c-5 a d) (b c-a d)) \int \frac {x^{7/2}}{c+d x^2} \, dx}{4 c d^2} \\ & = -\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((13 b c-5 a d) (b c-a d)) \int \frac {x^{3/2}}{c+d x^2} \, dx}{4 d^3} \\ & = \frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(c (13 b c-5 a d) (b c-a d)) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{4 d^4} \\ & = \frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(c (13 b c-5 a d) (b c-a d)) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 d^4} \\ & = \frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {\left (\sqrt {c} (13 b c-5 a d) (b c-a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 d^4}-\frac {\left (\sqrt {c} (13 b c-5 a d) (b c-a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 d^4} \\ & = \frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {\left (\sqrt {c} (13 b c-5 a d) (b c-a d)\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 )}{8 d^{9/2}}-\frac {\left (\sqrt {c} (13 b c-5 a d) (b c-a d)\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 )}{8 d^{9/2}}+\frac {\left (\sqrt [4]{c} (13 b c-5 a d) (b c-a d)\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 )}{8 \sqrt {2} d^{17/4}}+\frac {\left (\sqrt [4]{c} (13 b c-5 a d) (b c-a d)\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 )}{8 \sqrt {2} d^{17/4}} \\ & = \frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\left (\sqrt [4]{c} (13 b c-5 a d) (b c-a d)\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 )}{4 \sqrt {2} d^{17/4}}+\frac {\left (\sqrt [4]{c} (13 b c-5 a d) (b c-a d)\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 )}{4 \sqrt {2} d^{17/4}} \\ & = \frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}} \\ \end{align*}
Time = 0.75 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.68 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{d} \sqrt {x} \left (45 a^2 d^2 \left (5 c+4 d x^2\right )+18 a b d \left (-45 c^2-36 c d x^2+4 d^2 x^4\right )+b^2 \left (585 c^3+468 c^2 d x^2-52 c d^2 x^4+20 d^3 x^6\right )\right )}{c+d x^2}+45 \sqrt {2} \sqrt [4]{c} \left (13 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-45 \sqrt {2} \sqrt [4]{c} \left (13 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{360 d^{17/4}} \]
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Time = 2.81 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.58
method | result | size |
risch | \(\frac {2 \left (5 b^{2} d^{2} x^{4}+18 x^{2} a b \,d^{2}-18 x^{2} b^{2} c d +45 a^{2} d^{2}-180 a b c d +135 b^{2} c^{2}\right ) \sqrt {x}}{45 d^{4}}-\frac {c \left (2 a d -2 b c \right ) \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (5 a d -13 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{d^{4}}\) | \(216\) |
derivativedivides | \(\frac {\frac {2 b^{2} d^{2} x^{\frac {9}{2}}}{9}+\frac {4 a b \,d^{2} x^{\frac {5}{2}}}{5}-\frac {4 b^{2} c d \,x^{\frac {5}{2}}}{5}+2 a^{2} d^{2} \sqrt {x}-8 a b c d \sqrt {x}+6 b^{2} c^{2} \sqrt {x}}{d^{4}}-\frac {2 c \left (\frac {\left (-\frac {1}{4} a^{2} d^{2}+\frac {1}{2} a b c d -\frac {1}{4} b^{2} c^{2}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (5 a^{2} d^{2}-18 a b c d +13 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{d^{4}}\) | \(240\) |
default | \(\frac {\frac {2 b^{2} d^{2} x^{\frac {9}{2}}}{9}+\frac {4 a b \,d^{2} x^{\frac {5}{2}}}{5}-\frac {4 b^{2} c d \,x^{\frac {5}{2}}}{5}+2 a^{2} d^{2} \sqrt {x}-8 a b c d \sqrt {x}+6 b^{2} c^{2} \sqrt {x}}{d^{4}}-\frac {2 c \left (\frac {\left (-\frac {1}{4} a^{2} d^{2}+\frac {1}{2} a b c d -\frac {1}{4} b^{2} c^{2}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (5 a^{2} d^{2}-18 a b c d +13 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{d^{4}}\) | \(240\) |
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 1248, normalized size of antiderivative = 3.33 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.01 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {x}}{2 \, {\left (d^{5} x^{2} + c d^{4}\right )}} - \frac {{\left (\frac {2 \, \sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}\right )} c}{16 \, d^{4}} + \frac {2 \, {\left (5 \, b^{2} d^{2} x^{\frac {9}{2}} - 18 \, {\left (b^{2} c d - a b d^{2}\right )} x^{\frac {5}{2}} + 45 \, {\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right )}}{45 \, d^{4}} \]
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Time = 0.31 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.17 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=-\frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, d^{5}} - \frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, d^{5}} - \frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, d^{5}} + \frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, d^{5}} + \frac {b^{2} c^{3} \sqrt {x} - 2 \, a b c^{2} d \sqrt {x} + a^{2} c d^{2} \sqrt {x}}{2 \, {\left (d x^{2} + c\right )} d^{4}} + \frac {2 \, {\left (5 \, b^{2} d^{16} x^{\frac {9}{2}} - 18 \, b^{2} c d^{15} x^{\frac {5}{2}} + 18 \, a b d^{16} x^{\frac {5}{2}} + 135 \, b^{2} c^{2} d^{14} \sqrt {x} - 180 \, a b c d^{15} \sqrt {x} + 45 \, a^{2} d^{16} \sqrt {x}\right )}}{45 \, d^{18}} \]
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Time = 4.94 (sec) , antiderivative size = 1367, normalized size of antiderivative = 3.65 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
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