Integrand size = 23, antiderivative size = 53 \[ \int \frac {\sqrt {c x}}{\sqrt {a x^3+b x^n}} \, dx=\frac {2 \sqrt {c x} \text {arctanh}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{\sqrt {a} (3-n) \sqrt {x}} \]
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Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2056, 2054, 212} \[ \int \frac {\sqrt {c x}}{\sqrt {a x^3+b x^n}} \, dx=\frac {2 \sqrt {c x} \text {arctanh}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{\sqrt {a} (3-n) \sqrt {x}} \]
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Rule 212
Rule 2054
Rule 2056
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {a x^3+b x^n}} \, dx}{\sqrt {x}} \\ & = \frac {\left (2 \sqrt {c x}\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{(3-n) \sqrt {x}} \\ & = \frac {2 \sqrt {c x} \tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{\sqrt {a} (3-n) \sqrt {x}} \\ \end{align*}
Time = 1.50 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.68 \[ \int \frac {\sqrt {c x}}{\sqrt {a x^3+b x^n}} \, dx=-\frac {2 \sqrt {b} x^{\frac {1}{2} (-1+n)} \sqrt {c x} \sqrt {1+\frac {a x^{3-n}}{b}} \text {arcsinh}\left (\frac {\sqrt {a} x^{\frac {3}{2}-\frac {n}{2}}}{\sqrt {b}}\right )}{\sqrt {a} (-3+n) \sqrt {a x^3+b x^n}} \]
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\[\int \frac {\sqrt {c x}}{\sqrt {a \,x^{3}+b \,x^{n}}}d x\]
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Exception generated. \[ \int \frac {\sqrt {c x}}{\sqrt {a x^3+b x^n}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\sqrt {c x}}{\sqrt {a x^3+b x^n}} \, dx=\int \frac {\sqrt {c x}}{\sqrt {a x^{3} + b x^{n}}}\, dx \]
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\[ \int \frac {\sqrt {c x}}{\sqrt {a x^3+b x^n}} \, dx=\int { \frac {\sqrt {c x}}{\sqrt {a x^{3} + b x^{n}}} \,d x } \]
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\[ \int \frac {\sqrt {c x}}{\sqrt {a x^3+b x^n}} \, dx=\int { \frac {\sqrt {c x}}{\sqrt {a x^{3} + b x^{n}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {c x}}{\sqrt {a x^3+b x^n}} \, dx=\int \frac {\sqrt {c\,x}}{\sqrt {b\,x^n+a\,x^3}} \,d x \]
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