Integrand size = 26, antiderivative size = 103 \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx=-\frac {2 c}{a e \sqrt {e x} \sqrt [4]{a+b x^2}}+\frac {2 (2 b c-a d) \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{a^{3/2} \sqrt {b} e^2 \sqrt [4]{a+b x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {464, 290, 342, 202} \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx=\frac {2 \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} (2 b c-a d) E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{a^{3/2} \sqrt {b} e^2 \sqrt [4]{a+b x^2}}-\frac {2 c}{a e \sqrt {e x} \sqrt [4]{a+b x^2}} \]
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Rule 202
Rule 290
Rule 342
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c}{a e \sqrt {e x} \sqrt [4]{a+b x^2}}-\frac {(2 b c-a d) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{5/4}} \, dx}{a e^2} \\ & = -\frac {2 c}{a e \sqrt {e x} \sqrt [4]{a+b x^2}}-\frac {\left ((2 b c-a d) \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}{a b e^2 \sqrt [4]{a+b x^2}} \\ & = -\frac {2 c}{a e \sqrt {e x} \sqrt [4]{a+b x^2}}+\frac {\left ((2 b c-a d) \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 )}{a b e^2 \sqrt [4]{a+b x^2}} \\ & = -\frac {2 c}{a e \sqrt {e x} \sqrt [4]{a+b x^2}}+\frac {2 (2 b c-a d) \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 )}{a^{3/2} \sqrt {b} e^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 = 10.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.75 \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx=\frac {x \left (-6 a c+2 (-2 b c+a d) x^2 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{3 a^2 (e x)^{3/2} \sqrt [4]{a+b x^2}} \]
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\[\int \frac {d \,x^{2}+c}{\left (e x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {5}{4}}}d x\]
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\[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
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Result contains complex when optimal does not.
Time = 15.39 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.80 \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx=- \frac {d {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{b^{\frac {5}{4}} e^{\frac {3}{2}} x} + \frac {c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{4}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]
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\[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx=\int \frac {d\,x^2+c}{{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{5/4}} \,d x \]
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