Integrand size = 24, antiderivative size = 269 \[ \int \frac {\left (a+b x^2\right )^2}{x^{9/2} \left (c+d x^2\right )} \, dx=-\frac {2 a^2}{7 c x^{7/2}}-\frac {2 a (2 b c-a d)}{3 c^2 x^{3/2}}-\frac {(b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{11/4} \sqrt [4]{d}}+\frac {(b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{11/4} \sqrt [4]{d}}-\frac {(b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{11/4} \sqrt [4]{d}}+\frac {(b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{11/4} \sqrt [4]{d}} \]
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Time = 0.21 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {473, 464, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\left (a+b x^2\right )^2}{x^{9/2} \left (c+d x^2\right )} \, dx=-\frac {2 a^2}{7 c x^{7/2}}-\frac {(b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{11/4} \sqrt [4]{d}}+\frac {(b c-a d)^2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} c^{11/4} \sqrt [4]{d}}-\frac {(b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{11/4} \sqrt [4]{d}}+\frac {(b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{11/4} \sqrt [4]{d}}-\frac {2 a (2 b c-a d)}{3 c^2 x^{3/2}} \]
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Rule 210
Rule 217
Rule 335
Rule 464
Rule 473
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2}{7 c x^{7/2}}+\frac {2 \int \frac {\frac {7}{2} a (2 b c-a d)+\frac {7}{2} b^2 c x^2}{x^{5/2} \left (c+d x^2\right )} \, dx}{7 c} \\ & = -\frac {2 a^2}{7 c x^{7/2}}-\frac {2 a (2 b c-a d)}{3 c^2 x^{3/2}}+\frac {(b c-a d)^2 \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{c^2} \\ & = -\frac {2 a^2}{7 c x^{7/2}}-\frac {2 a (2 b c-a d)}{3 c^2 x^{3/2}}+\frac {\left (2 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{c^2} \\ & = -\frac {2 a^2}{7 c x^{7/2}}-\frac {2 a (2 b c-a d)}{3 c^2 x^{3/2}}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{c^{5/2}}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{c^{5/2}} \\ & = -\frac {2 a^2}{7 c x^{7/2}}-\frac {2 a (2 b c-a d)}{3 c^2 x^{3/2}}+\frac {(b c-a d)^2 \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 )}{2 c^{5/2} \sqrt {d}}+\frac {(b c-a d)^2 \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 )}{2 c^{5/2} \sqrt {d}}-\frac {(b c-a d)^2 \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 )}{2 \sqrt {2} c^{11/4} \sqrt [4]{d}}-\frac {(b c-a d)^2 \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 )}{2 \sqrt {2} c^{11/4} \sqrt [4]{d}} \\ & = -\frac {2 a^2}{7 c x^{7/2}}-\frac {2 a (2 b c-a d)}{3 c^2 x^{3/2}}-\frac {(b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{11/4} \sqrt [4]{d}}+\frac {(b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{11/4} \sqrt [4]{d}}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{11/4} \sqrt [4]{d}}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{11/4} \sqrt [4]{d}} \\ & = -\frac {2 a^2}{7 c x^{7/2}}-\frac {2 a (2 b c-a d)}{3 c^2 x^{3/2}}-\frac {(b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{11/4} \sqrt [4]{d}}+\frac {(b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{11/4} \sqrt [4]{d}}-\frac {(b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{11/4} \sqrt [4]{d}}+\frac {(b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{11/4} \sqrt [4]{d}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.59 \[ \int \frac {\left (a+b x^2\right )^2}{x^{9/2} \left (c+d x^2\right )} \, dx=\frac {\frac {4 a c^{3/4} \left (-3 a c-14 b c x^2+7 a d x^2\right )}{x^{7/2}}-\frac {21 \sqrt {2} (b c-a d)^2 \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{\sqrt [4]{d}}+\frac {21 \sqrt {2} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt [4]{d}}}{42 c^{11/4}} \]
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Time = 2.83 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.58
method | result | size |
derivativedivides | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{3}}-\frac {2 a^{2}}{7 c \,x^{\frac {7}{2}}}+\frac {2 a \left (a d -2 b c \right )}{3 c^{2} x^{\frac {3}{2}}}\) | \(156\) |
default | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{3}}-\frac {2 a^{2}}{7 c \,x^{\frac {7}{2}}}+\frac {2 a \left (a d -2 b c \right )}{3 c^{2} x^{\frac {3}{2}}}\) | \(156\) |
risch | \(-\frac {2 \left (-7 a d \,x^{2}+14 c b \,x^{2}+3 a c \right ) a}{21 c^{2} x^{\frac {7}{2}}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{3}}\) | \(156\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1117, normalized size of antiderivative = 4.15 \[ \int \frac {\left (a+b x^2\right )^2}{x^{9/2} \left (c+d x^2\right )} \, dx=\frac {21 \, c^{2} x^{4} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{11} d}\right )^{\frac {1}{4}} \log \left (c^{3} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{11} d}\right )^{\frac {1}{4}} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) + 21 i \, c^{2} x^{4} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{11} d}\right )^{\frac {1}{4}} \log \left (i \, c^{3} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{11} d}\right )^{\frac {1}{4}} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) - 21 i \, c^{2} x^{4} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{11} d}\right )^{\frac {1}{4}} \log \left (-i \, c^{3} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{11} d}\right )^{\frac {1}{4}} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) - 21 \, c^{2} x^{4} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{11} d}\right )^{\frac {1}{4}} \log \left (-c^{3} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{11} d}\right )^{\frac {1}{4}} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) - 4 \, {\left (3 \, a^{2} c + 7 \, {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {x}}{42 \, c^{2} x^{4}} \]
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Time = 63.90 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b x^2\right )^2}{x^{9/2} \left (c+d x^2\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 a^{2}}{11 x^{\frac {11}{2}}} - \frac {4 a b}{7 x^{\frac {7}{2}}} - \frac {2 b^{2}}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: c = 0 \wedge d = 0 \\\frac {- \frac {2 a^{2}}{11 x^{\frac {11}{2}}} - \frac {4 a b}{7 x^{\frac {7}{2}}} - \frac {2 b^{2}}{3 x^{\frac {3}{2}}}}{d} & \text {for}\: c = 0 \\\frac {- \frac {2 a^{2}}{7 x^{\frac {7}{2}}} - \frac {4 a b}{3 x^{\frac {3}{2}}} + 2 b^{2} \sqrt {x}}{c} & \text {for}\: d = 0 \\- \frac {2 a^{2}}{7 c x^{\frac {7}{2}}} + \frac {2 a^{2} d}{3 c^{2} x^{\frac {3}{2}}} - \frac {a^{2} d^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 c^{3}} + \frac {a^{2} d^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 c^{3}} + \frac {a^{2} d^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{c^{3}} - \frac {4 a b}{3 c x^{\frac {3}{2}}} + \frac {a b d \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{c^{2}} - \frac {a b d \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{c^{2}} - \frac {2 a b d \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{c^{2}} - \frac {b^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 c} + \frac {b^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 c} + \frac {b^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{c} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^2}{x^{9/2} \left (c+d x^2\right )} \, dx=\frac {\frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{4 \, c^{2}} - \frac {2 \, {\left (3 \, a^{2} c + 7 \, {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )}}{21 \, c^{2} x^{\frac {7}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b x^2\right )^2}{x^{9/2} \left (c+d x^2\right )} \, dx=\frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c^{3} d} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c^{3} d} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, c^{3} d} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, c^{3} d} - \frac {2 \, {\left (14 \, a b c x^{2} - 7 \, a^{2} d x^{2} + 3 \, a^{2} c\right )}}{21 \, c^{2} x^{\frac {7}{2}}} \]
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Time = 4.97 (sec) , antiderivative size = 1209, normalized size of antiderivative = 4.49 \[ \int \frac {\left (a+b x^2\right )^2}{x^{9/2} \left (c+d x^2\right )} \, dx=\text {Too large to display} \]
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