Integrand size = 26, antiderivative size = 221 \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {(4 b c-9 a d) e^{7/2} \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}+\frac {(4 b c-9 a d) e^{7/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}} \]
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Time = 0.10 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {468, 291, 294, 335, 246, 218, 214, 211} \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {e^{7/2} (4 b c-9 a d) \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}+\frac {e^{7/2} (4 b c-9 a d) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}-\frac {e^3 \sqrt {e x} (4 b c-9 a d)}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {e (e x)^{5/2} (4 b c-9 a d)}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {2 (e x)^{9/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
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Rule 211
Rule 214
Rule 218
Rule 246
Rule 291
Rule 294
Rule 335
Rule 468
Rubi steps \begin{align*} \text {integral}& = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {\left (2 \left (-2 b c+\frac {9 a d}{2}\right )\right ) \int \frac {(e x)^{7/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{5 a b} \\ & = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^2\right ) \int \frac {(e x)^{3/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{4 b^2} \\ & = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^4\right ) \int \frac {1}{\sqrt {e x} \sqrt [4]{a+b x^2}} \, dx}{4 b^3} \\ & = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 b^3} \\ & = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^3\right ) \text {Subst}\left (\int \frac {1}{1-\frac {b x^4}{e^2}} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{2 b^3} \\ & = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-9 a d) e^4\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{4 b^3}+\frac {\left ((4 b c-9 a d) e^4\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{4 b^3} \\ & = \frac {2 (b c-a d) (e x)^{9/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(4 b c-9 a d) e^3 \sqrt {e x}}{2 b^3 \sqrt [4]{a+b x^2}}-\frac {(4 b c-9 a d) e (e x)^{5/2}}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac {(4 b c-9 a d) e^{7/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}+\frac {(4 b c-9 a d) e^{7/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}} \\ \end{align*}
Time = 1.90 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.68 \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {(e x)^{7/2} \left (\frac {2 \sqrt [4]{b} \sqrt {x} \left (45 a^2 d+b^2 x^2 \left (-24 c+5 d x^2\right )+a b \left (-20 c+54 d x^2\right )\right )}{\left (a+b x^2\right )^{5/4}}+5 (4 b c-9 a d) \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )+5 (4 b c-9 a d) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )\right )}{20 b^{13/4} x^{7/2}} \]
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\[\int \frac {\left (e x \right )^{\frac {7}{2}} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {9}{4}}}d x\]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 904, normalized size of antiderivative = 4.09 \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {4 \, {\left (5 \, b^{2} d e^{3} x^{4} - 6 \, {\left (4 \, b^{2} c - 9 \, a b d\right )} e^{3} x^{2} - 5 \, {\left (4 \, a b c - 9 \, a^{2} d\right )} e^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} + 5 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (4 \, b c - 9 \, a d\right )} \sqrt {e x} e^{3} + {\left (b^{4} x^{2} + a b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) - 5 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (4 \, b c - 9 \, a d\right )} \sqrt {e x} e^{3} - {\left (b^{4} x^{2} + a b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) - 5 \, {\left (-i \, b^{5} x^{4} - 2 i \, a b^{4} x^{2} - i \, a^{2} b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (4 \, b c - 9 \, a d\right )} \sqrt {e x} e^{3} + {\left (i \, b^{4} x^{2} + i \, a b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) - 5 \, {\left (i \, b^{5} x^{4} + 2 i \, a b^{4} x^{2} + i \, a^{2} b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (4 \, b c - 9 \, a d\right )} \sqrt {e x} e^{3} + {\left (-i \, b^{4} x^{2} - i \, a b^{3}\right )} \left (\frac {{\left (256 \, b^{4} c^{4} - 2304 \, a b^{3} c^{3} d + 7776 \, a^{2} b^{2} c^{2} d^{2} - 11664 \, a^{3} b c d^{3} + 6561 \, a^{4} d^{4}\right )} e^{14}}{b^{13}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right )}{40 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \]
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Timed out. \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}} \,d x } \]
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\[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{9/4}} \,d x \]
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