Integrand size = 20, antiderivative size = 20 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{5/2}} \, dx=-\frac {2 a}{3 x^{3/2}}+b \text {Int}\left (\frac {\csc \left (c+d \sqrt {x}\right )}{x^{5/2}},x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{5/2}} \, dx=\int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{5/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{x^{5/2}}+\frac {b \csc \left (c+d \sqrt {x}\right )}{x^{5/2}}\right ) \, dx \\ & = -\frac {2 a}{3 x^{3/2}}+b \int \frac {\csc \left (c+d \sqrt {x}\right )}{x^{5/2}} \, dx \\ \end{align*}
Not integrable
Time = 28.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{5/2}} \, dx=\int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{5/2}} \, dx \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80
\[\int \frac {a +b \csc \left (c +d \sqrt {x}\right )}{x^{\frac {5}{2}}}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{5/2}} \, dx=\int { \frac {b \csc \left (d \sqrt {x} + c\right ) + a}{x^{\frac {5}{2}}} \,d x } \]
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Not integrable
Time = 6.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{5/2}} \, dx=\int \frac {a + b \csc {\left (c + d \sqrt {x} \right )}}{x^{\frac {5}{2}}}\, dx \]
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Not integrable
Time = 0.61 (sec) , antiderivative size = 115, normalized size of antiderivative = 5.75 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{5/2}} \, dx=\int { \frac {b \csc \left (d \sqrt {x} + c\right ) + a}{x^{\frac {5}{2}}} \,d x } \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{5/2}} \, dx=\int { \frac {b \csc \left (d \sqrt {x} + c\right ) + a}{x^{\frac {5}{2}}} \,d x } \]
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Not integrable
Time = 18.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{5/2}} \, dx=\int \frac {a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}}{x^{5/2}} \,d x \]
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