Integrand size = 26, antiderivative size = 181 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{7/4}} \, dx=-\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (10 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{3/4}}+\frac {4 (10 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^3 e^3 (e x)^{3/2}}-\frac {8 b^{3/2} (10 b c-7 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{21 a^{7/2} e^6 \left (a+b x^2\right )^{3/4}} \]
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Time = 0.09 (sec) , antiderivative size = 181, 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 = {464, 296, 331, 335, 243, 342, 281, 237} \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{7/4}} \, dx=-\frac {8 b^{3/2} (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (10 b c-7 a d) \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{21 a^{7/2} e^6 \left (a+b x^2\right )^{3/4}}+\frac {4 \sqrt [4]{a+b x^2} (10 b c-7 a d)}{21 a^3 e^3 (e x)^{3/2}}-\frac {2 (10 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{3/4}}-\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{3/4}} \]
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Rule 237
Rule 243
Rule 281
Rule 296
Rule 331
Rule 335
Rule 342
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{3/4}}-\frac {(10 b c-7 a d) \int \frac {1}{(e x)^{5/2} \left (a+b x^2\right )^{7/4}} \, dx}{7 a e^2} \\ & = -\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (10 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{3/4}}-\frac {(2 (10 b c-7 a d)) \int \frac {1}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx}{7 a^2 e^2} \\ & = -\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (10 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{3/4}}+\frac {4 (10 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^3 e^3 (e x)^{3/2}}+\frac {(4 b (10 b c-7 a d)) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{3/4}} \, dx}{21 a^3 e^4} \\ & = -\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (10 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{3/4}}+\frac {4 (10 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^3 e^3 (e x)^{3/2}}+\frac {(8 b (10 b c-7 a d)) \text {Subst}\left (\int \frac {1}{\left (a+\frac {b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt {e x}\right )}{21 a^3 e^5} \\ & = -\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (10 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{3/4}}+\frac {4 (10 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^3 e^3 (e x)^{3/2}}+\frac {\left (8 b (10 b c-7 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a e^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt {e x}\right )}{21 a^3 e^5 \left (a+b x^2\right )^{3/4}} \\ & = -\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (10 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{3/4}}+\frac {4 (10 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^3 e^3 (e x)^{3/2}}-\frac {\left (8 b (10 b c-7 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a e^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {e x}}\right )}{21 a^3 e^5 \left (a+b x^2\right )^{3/4}} \\ & = -\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (10 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{3/4}}+\frac {4 (10 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^3 e^3 (e x)^{3/2}}-\frac {\left (4 b (10 b c-7 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a e^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{e x}\right )}{21 a^3 e^5 \left (a+b x^2\right )^{3/4}} \\ & = -\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (10 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{3/4}}+\frac {4 (10 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^3 e^3 (e x)^{3/2}}-\frac {8 b^{3/2} (10 b c-7 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{21 a^{7/2} e^6 \left (a+b x^2\right )^{3/4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.45 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{7/4}} \, dx=-\frac {2 \sqrt {e x} \left (3 a c+(-10 b c+7 a d) x^2 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {7}{4},\frac {1}{4},-\frac {b x^2}{a}\right )\right )}{21 a^2 e^5 x^4 \left (a+b x^2\right )^{3/4}} \]
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\[\int \frac {d \,x^{2}+c}{\left (e x \right )^{\frac {9}{2}} \left (b \,x^{2}+a \right )^{\frac {7}{4}}}d x\]
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\[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (e x\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{7/4}} \, dx=\text {Timed out} \]
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\[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (e x\right )^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (e x\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{7/4}} \, dx=\int \frac {d\,x^2+c}{{\left (e\,x\right )}^{9/2}\,{\left (b\,x^2+a\right )}^{7/4}} \,d x \]
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