Integrand size = 26, antiderivative size = 136 \[ \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx=\frac {d (e x)^{3/2} \sqrt [4]{a+b x^2}}{2 b e}-\frac {(4 b c-3 a d) \sqrt {e} \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{7/4}}+\frac {(4 b c-3 a d) \sqrt {e} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{7/4}} \]
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Time = 0.06 (sec) , antiderivative size = 136, 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 = {470, 335, 338, 304, 211, 214} \[ \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx=-\frac {\sqrt {e} (4 b c-3 a d) \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{7/4}}+\frac {\sqrt {e} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{7/4}}+\frac {d (e x)^{3/2} \sqrt [4]{a+b x^2}}{2 b e} \]
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Rule 211
Rule 214
Rule 304
Rule 335
Rule 338
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {d (e x)^{3/2} \sqrt [4]{a+b x^2}}{2 b e}-\frac {\left (-2 b c+\frac {3 a d}{2}\right ) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{3/4}} \, dx}{2 b} \\ & = \frac {d (e x)^{3/2} \sqrt [4]{a+b x^2}}{2 b e}+\frac {(4 b c-3 a 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 )}{2 b e} \\ & = \frac {d (e x)^{3/2} \sqrt [4]{a+b x^2}}{2 b e}+\frac {(4 b c-3 a 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 )}{2 b e} \\ & = \frac {d (e x)^{3/2} \sqrt [4]{a+b x^2}}{2 b e}+\frac {((4 b c-3 a d) e) \text {Subst}\left (\int \frac {1}{e-\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{4 b^{3/2}}-\frac {((4 b c-3 a d) e) \text {Subst}\left (\int \frac {1}{e+\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{4 b^{3/2}} \\ & = \frac {d (e x)^{3/2} \sqrt [4]{a+b x^2}}{2 b e}-\frac {(4 b c-3 a d) \sqrt {e} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{7/4}}+\frac {(4 b c-3 a d) \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{7/4}} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx=\frac {\sqrt {e x} \left (2 b^{3/4} d x^{3/2} \sqrt [4]{a+b x^2}+(-4 b c+3 a d) \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )+(4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )\right )}{4 b^{7/4} \sqrt {x}} \]
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\[\int \frac {\sqrt {e x}\, \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {3}{4}}}d x\]
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Timed out. \[ \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 3.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx=\frac {c \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} + \frac {d \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} \Gamma \left (\frac {11}{4}\right )} \]
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\[ \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \sqrt {e x}}{{\left (b x^{2} + a\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \sqrt {e x}}{{\left (b x^{2} + a\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx=\int \frac {\sqrt {e\,x}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{3/4}} \,d x \]
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