Integrand size = 26, antiderivative size = 113 \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}-\frac {d \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}+\frac {d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 113, 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 = {462, 335, 338, 304, 211, 214} \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {d \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}+\frac {d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}} \]
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Rule 211
Rule 214
Rule 304
Rule 335
Rule 338
Rule 462
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}+\frac {d \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{3/4}} \, dx}{e^2} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}+\frac {(2 d) \text {Subst}\left (\int \frac {x^2}{\left (a+\frac {b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt {e x}\right )}{e^3} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}+\frac {(2 d) \text {Subst}\left (\int \frac {x^2}{1-\frac {b x^4}{e^2}} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{e^3} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}+\frac {d \text {Subst}\left (\int \frac {1}{e-\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{\sqrt {b} e}-\frac {d \text {Subst}\left (\int \frac {1}{e+\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{\sqrt {b} e} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}-\frac {d \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88 \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=\frac {x \left (-2 b^{3/4} c \sqrt [4]{a+b x^2}-a d \sqrt {x} \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )+a d \sqrt {x} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )\right )}{a b^{3/4} (e x)^{3/2}} \]
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\[\int \frac {d \,x^{2}+c}{\left (e x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}d x\]
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Timed out. \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 3.92 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.75 \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=\frac {\sqrt [4]{b} c \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {1}{4}\right )}{2 a e^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {d x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \]
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\[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=\int \frac {d\,x^2+c}{{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{3/4}} \,d x \]
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