Integrand size = 26, antiderivative size = 192 \[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {2 (b c-a d) (e x)^{11/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {7 (6 b c-11 a d) e^3 (e x)^{3/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac {(6 b c-11 a d) e (e x)^{7/2}}{15 a b^2 \sqrt [4]{a+b x^2}}+\frac {7 \sqrt {a} (6 b c-11 a d) e^4 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 b^{7/2} \sqrt [4]{a+b x^2}} \]
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Time = 0.07 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {468, 291, 290, 342, 202} \[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {7 \sqrt {a} e^4 \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} (6 b c-11 a d) E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 b^{7/2} \sqrt [4]{a+b x^2}}+\frac {7 e^3 (e x)^{3/2} (6 b c-11 a d)}{30 b^3 \sqrt [4]{a+b x^2}}-\frac {e (e x)^{7/2} (6 b c-11 a d)}{15 a b^2 \sqrt [4]{a+b x^2}}+\frac {2 (e x)^{11/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
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Rule 202
Rule 290
Rule 291
Rule 342
Rule 468
Rubi steps \begin{align*} \text {integral}& = \frac {2 (b c-a d) (e x)^{11/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {\left (2 \left (-3 b c+\frac {11 a d}{2}\right )\right ) \int \frac {(e x)^{9/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{5 a b} \\ & = \frac {2 (b c-a d) (e x)^{11/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(6 b c-11 a d) e (e x)^{7/2}}{15 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left (7 (6 b c-11 a d) e^2\right ) \int \frac {(e x)^{5/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{30 b^2} \\ & = \frac {2 (b c-a d) (e x)^{11/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {7 (6 b c-11 a d) e^3 (e x)^{3/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac {(6 b c-11 a d) e (e x)^{7/2}}{15 a b^2 \sqrt [4]{a+b x^2}}-\frac {\left (7 a (6 b c-11 a d) e^4\right ) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{5/4}} \, dx}{20 b^3} \\ & = \frac {2 (b c-a d) (e x)^{11/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {7 (6 b c-11 a d) e^3 (e x)^{3/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac {(6 b c-11 a d) e (e x)^{7/2}}{15 a b^2 \sqrt [4]{a+b x^2}}-\frac {\left (7 a (6 b c-11 a d) e^4 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x}\right ) \int \frac {1}{\left (1+\frac {a}{b x^2}\right )^{5/4} x^2} \, dx}{20 b^4 \sqrt [4]{a+b x^2}} \\ & = \frac {2 (b c-a d) (e x)^{11/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {7 (6 b c-11 a d) e^3 (e x)^{3/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac {(6 b c-11 a d) e (e x)^{7/2}}{15 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left (7 a (6 b c-11 a d) e^4 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{20 b^4 \sqrt [4]{a+b x^2}} \\ & = \frac {2 (b c-a d) (e x)^{11/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {7 (6 b c-11 a d) e^3 (e x)^{3/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac {(6 b c-11 a d) e (e x)^{7/2}}{15 a b^2 \sqrt [4]{a+b x^2}}+\frac {7 \sqrt {a} (6 b c-11 a d) e^4 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 b^{7/2} \sqrt [4]{a+b x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.60 \[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {e^3 (e x)^{3/2} \left (-77 a^2 d+a b \left (42 c-22 d x^2\right )+4 b^2 x^2 \left (3 c+d x^2\right )+7 (-6 b c+11 a d) \left (a+b x^2\right ) \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {9}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{12 b^3 \left (a+b x^2\right )^{5/4}} \]
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\[\int \frac {\left (e x \right )^{\frac {9}{2}} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {9}{4}}}d x\]
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\[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}} \,d x } \]
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\[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int \frac {{\left (e\,x\right )}^{9/2}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{9/4}} \,d x \]
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