Integrand size = 26, antiderivative size = 107 \[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}+\frac {2 \sqrt {b} (2 b c-3 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 a^{3/2} e^4 \left (a+b x^2\right )^{3/4}} \]
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Time = 0.06 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {464, 335, 243, 342, 281, 237} \[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=\frac {2 \sqrt {b} (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (2 b c-3 a d) \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 a^{3/2} e^4 \left (a+b x^2\right )^{3/4}}-\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}} \]
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Rule 237
Rule 243
Rule 281
Rule 335
Rule 342
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}-\frac {(2 b c-3 a d) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{3/4}} \, dx}{3 a e^2} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}-\frac {(2 (2 b c-3 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 )}{3 a e^3} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}-\frac {\left (2 (2 b c-3 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 )}{3 a e^3 \left (a+b x^2\right )^{3/4}} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}+\frac {\left (2 (2 b c-3 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 )}{3 a e^3 \left (a+b x^2\right )^{3/4}} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}+\frac {\left ((2 b c-3 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 )}{3 a e^3 \left (a+b x^2\right )^{3/4}} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}+\frac {2 \sqrt {b} (2 b c-3 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 )}{3 a^{3/2} e^4 \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 = 10.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.79 \[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=\frac {x \left (-2 c \left (a+b x^2\right )+2 (-2 b c+3 a d) x^2 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{3 a (e x)^{5/2} \left (a+b x^2\right )^{3/4}} \]
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\[\int \frac {d \,x^{2}+c}{\left (e x \right )^{\frac {5}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}d x\]
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\[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
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Result contains complex when optimal does not.
Time = 9.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.77 \[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=- \frac {d {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{b^{\frac {3}{4}} e^{\frac {5}{2}} x} + \frac {c \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \]
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\[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=\int \frac {d\,x^2+c}{{\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^{3/4}} \,d x \]
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