Integrand size = 21, antiderivative size = 62 \[ \int \frac {x^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {x^{n/2} \sqrt {a+b x^n}}{b n}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{b^{3/2} n} \]
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Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {362, 294, 212} \[ \int \frac {x^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {x^{n/2} \sqrt {a+b x^n}}{b n}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{b^{3/2} n} \]
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Rule 212
Rule 294
Rule 362
Rubi steps \begin{align*} \text {integral}& = \frac {(2 a) \text {Subst}\left (\int \frac {x^2}{\left (1-b x^2\right )^2} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{n} \\ & = \frac {x^{n/2} \sqrt {a+b x^n}}{b n}-\frac {a \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{b n} \\ & = \frac {x^{n/2} \sqrt {a+b x^n}}{b n}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{b^{3/2} n} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.31 \[ \int \frac {x^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {\sqrt {b} x^{n/2} \left (a+b x^n\right )-a^{3/2} \sqrt {1+\frac {b x^n}{a}} \text {arcsinh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a}}\right )}{b^{3/2} n \sqrt {a+b x^n}} \]
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Time = 3.75 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\frac {{\mathrm e}^{\frac {n \ln \left (x \right )}{2}} \sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}}}{b n}-\frac {a \ln \left (\sqrt {b}\, {\mathrm e}^{\frac {n \ln \left (x \right )}{2}}+\sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}}\right )}{b^{\frac {3}{2}} n}\) | \(64\) |
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none
Time = 0.30 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.84 \[ \int \frac {x^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\left [\frac {2 \, \sqrt {b x^{n} + a} b x^{\frac {1}{2} \, n} + a \sqrt {b} \log \left (2 \, \sqrt {b x^{n} + a} \sqrt {b} x^{\frac {1}{2} \, n} - 2 \, b x^{n} - a\right )}{2 \, b^{2} n}, \frac {\sqrt {b x^{n} + a} b x^{\frac {1}{2} \, n} + a \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{\frac {1}{2} \, n}}{\sqrt {b x^{n} + a}}\right )}{b^{2} n}\right ] \]
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Time = 1.43 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.79 \[ \int \frac {x^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {\sqrt {a} x^{\frac {n}{2}} \sqrt {1 + \frac {b x^{n}}{a}}}{b n} - \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}} n} \]
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\[ \int \frac {x^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \]
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\[ \int \frac {x^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \]
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Timed out. \[ \int \frac {x^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int \frac {x^{\frac {3\,n}{2}-1}}{\sqrt {a+b\,x^n}} \,d x \]
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