Integrand size = 24, antiderivative size = 311 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=-\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}-\frac {c^{5/4} (b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{17/4}}-\frac {c^{5/4} (b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{17/4}} \]
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Time = 0.24 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {472, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=-\frac {c^{5/4} (b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} d^{17/4}}-\frac {c^{5/4} (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{17/4}}-\frac {2 c \sqrt {x} (b c-a d)^2}{d^4}+\frac {2 x^{5/2} (b c-a d)^2}{5 d^3}-\frac {2 b x^{9/2} (b c-2 a d)}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d} \]
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Rule 210
Rule 217
Rule 327
Rule 335
Rule 472
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b (b c-2 a d) x^{7/2}}{d^2}+\frac {b^2 x^{11/2}}{d}+\frac {\left (b^2 c^2-2 a b c d+a^2 d^2\right ) x^{7/2}}{d^2 \left (c+d x^2\right )}\right ) \, dx \\ & = -\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}+\frac {(b c-a d)^2 \int \frac {x^{7/2}}{c+d x^2} \, dx}{d^2} \\ & = \frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}-\frac {\left (c (b c-a d)^2\right ) \int \frac {x^{3/2}}{c+d x^2} \, dx}{d^3} \\ & = -\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}+\frac {\left (c^2 (b c-a d)^2\right ) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{d^4} \\ & = -\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}+\frac {\left (2 c^2 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^4} \\ & = -\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}+\frac {\left (c^{3/2} (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^4}+\frac {\left (c^{3/2} (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^4} \\ & = -\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}+\frac {\left (c^{3/2} (b c-a 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 )}{2 d^{9/2}}+\frac {\left (c^{3/2} (b c-a 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 )}{2 d^{9/2}}-\frac {\left (c^{5/4} (b c-a 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 )}{2 \sqrt {2} d^{17/4}}-\frac {\left (c^{5/4} (b c-a 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 )}{2 \sqrt {2} d^{17/4}} \\ & = -\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}-\frac {c^{5/4} (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} d^{17/4}}+\frac {c^{5/4} (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} d^{17/4}}+\frac {\left (c^{5/4} (b c-a 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 )}{\sqrt {2} d^{17/4}}-\frac {\left (c^{5/4} (b c-a 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 )}{\sqrt {2} d^{17/4}} \\ & = -\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}-\frac {c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{17/4}}-\frac {c^{5/4} (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} d^{17/4}}+\frac {c^{5/4} (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} d^{17/4}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.70 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {2 \sqrt {x} \left (117 a^2 d^2 \left (-5 c+d x^2\right )+26 a b d \left (45 c^2-9 c d x^2+5 d^2 x^4\right )+b^2 \left (-585 c^3+117 c^2 d x^2-65 c d^2 x^4+45 d^3 x^6\right )\right )}{585 d^4}-\frac {c^{5/4} (b c-a d)^2 \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{\sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} d^{17/4}} \]
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Time = 2.80 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(-\frac {2 \left (-\frac {b^{2} d^{3} x^{\frac {13}{2}}}{13}+\frac {\left (-\left (a d -b c \right ) b \,d^{2}-a b \,d^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {\left (-\left (a d -b c \right ) a \,d^{2}+b d \left (a c d -b \,c^{2}\right )\right ) x^{\frac {5}{2}}}{5}+\left (a d -b c \right ) \left (a c d -b \,c^{2}\right ) \sqrt {x}\right )}{d^{4}}+\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{4}}\) | \(230\) |
default | \(-\frac {2 \left (-\frac {b^{2} d^{3} x^{\frac {13}{2}}}{13}+\frac {\left (-\left (a d -b c \right ) b \,d^{2}-a b \,d^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {\left (-\left (a d -b c \right ) a \,d^{2}+b d \left (a c d -b \,c^{2}\right )\right ) x^{\frac {5}{2}}}{5}+\left (a d -b c \right ) \left (a c d -b \,c^{2}\right ) \sqrt {x}\right )}{d^{4}}+\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{4}}\) | \(230\) |
risch | \(-\frac {2 \left (-45 b^{2} d^{3} x^{6}-130 a b \,d^{3} x^{4}+65 b^{2} c \,d^{2} x^{4}-117 a^{2} d^{3} x^{2}+234 a b c \,d^{2} x^{2}-117 b^{2} c^{2} d \,x^{2}+585 c \,a^{2} d^{2}-1170 a b \,c^{2} d +585 b^{2} c^{3}\right ) \sqrt {x}}{585 d^{4}}+\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{4}}\) | \(230\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 1193, normalized size of antiderivative = 3.84 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\text {Too large to display} \]
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Time = 82.27 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.80 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\begin {cases} \tilde {\infty } \left (\frac {2 a^{2} x^{\frac {5}{2}}}{5} + \frac {4 a b x^{\frac {9}{2}}}{9} + \frac {2 b^{2} x^{\frac {13}{2}}}{13}\right ) & \text {for}\: c = 0 \wedge d = 0 \\\frac {\frac {2 a^{2} x^{\frac {9}{2}}}{9} + \frac {4 a b x^{\frac {13}{2}}}{13} + \frac {2 b^{2} x^{\frac {17}{2}}}{17}}{c} & \text {for}\: d = 0 \\\frac {\frac {2 a^{2} x^{\frac {5}{2}}}{5} + \frac {4 a b x^{\frac {9}{2}}}{9} + \frac {2 b^{2} x^{\frac {13}{2}}}{13}}{d} & \text {for}\: c = 0 \\- \frac {2 a^{2} c \sqrt {x}}{d^{2}} - \frac {a^{2} c \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2}} + \frac {a^{2} c \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2}} + \frac {a^{2} c \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{2}} + \frac {2 a^{2} x^{\frac {5}{2}}}{5 d} + \frac {4 a b c^{2} \sqrt {x}}{d^{3}} + \frac {a b c^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{d^{3}} - \frac {a b c^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{d^{3}} - \frac {2 a b c^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{3}} - \frac {4 a b c x^{\frac {5}{2}}}{5 d^{2}} + \frac {4 a b x^{\frac {9}{2}}}{9 d} - \frac {2 b^{2} c^{3} \sqrt {x}}{d^{4}} - \frac {b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{4}} + \frac {b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{4}} + \frac {b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{4}} + \frac {2 b^{2} c^{2} x^{\frac {5}{2}}}{5 d^{3}} - \frac {2 b^{2} c x^{\frac {9}{2}}}{9 d^{2}} + \frac {2 b^{2} x^{\frac {13}{2}}}{13 d} & \text {otherwise} \end {cases} \]
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Time = 0.34 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.16 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {{\left (\frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}\right )} c^{2}}{4 \, d^{4}} + \frac {2 \, {\left (45 \, b^{2} d^{3} x^{\frac {13}{2}} - 65 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{\frac {9}{2}} + 117 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {5}{2}} - 585 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {x}\right )}}{585 \, d^{4}} \]
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Time = 0.32 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.40 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} c d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, d^{5}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} c d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, d^{5}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} c d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, d^{5}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} c d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, d^{5}} + \frac {2 \, {\left (45 \, b^{2} d^{12} x^{\frac {13}{2}} - 65 \, b^{2} c d^{11} x^{\frac {9}{2}} + 130 \, a b d^{12} x^{\frac {9}{2}} + 117 \, b^{2} c^{2} d^{10} x^{\frac {5}{2}} - 234 \, a b c d^{11} x^{\frac {5}{2}} + 117 \, a^{2} d^{12} x^{\frac {5}{2}} - 585 \, b^{2} c^{3} d^{9} \sqrt {x} + 1170 \, a b c^{2} d^{10} \sqrt {x} - 585 \, a^{2} c d^{11} \sqrt {x}\right )}}{585 \, d^{13}} \]
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Time = 0.43 (sec) , antiderivative size = 1202, normalized size of antiderivative = 3.86 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\text {Too large to display} \]
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