Integrand size = 19, antiderivative size = 68 \[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=-\frac {2 x^{-3 n/2}}{3 a n}+\frac {2 b x^{-n/2}}{a^2 n}-\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{5/2} n} \]
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Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {369, 352, 199, 327, 211} \[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=-\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{5/2} n}+\frac {2 b x^{-n/2}}{a^2 n}-\frac {2 x^{-3 n/2}}{3 a n} \]
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Rule 199
Rule 211
Rule 327
Rule 352
Rule 369
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^{-3 n/2}}{3 a n}-\frac {b \int \frac {x^{-1-\frac {n}{2}}}{a+b x^n} \, dx}{a} \\ & = -\frac {2 x^{-3 n/2}}{3 a n}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{a+\frac {b}{x^2}} \, dx,x,x^{-n/2}\right )}{a n} \\ & = -\frac {2 x^{-3 n/2}}{3 a n}+\frac {(2 b) \text {Subst}\left (\int \frac {x^2}{b+a x^2} \, dx,x,x^{-n/2}\right )}{a n} \\ & = -\frac {2 x^{-3 n/2}}{3 a n}+\frac {2 b x^{-n/2}}{a^2 n}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,x^{-n/2}\right )}{a^2 n} \\ & = -\frac {2 x^{-3 n/2}}{3 a n}+\frac {2 b x^{-n/2}}{a^2 n}-\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{5/2} n} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.50 \[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=-\frac {2 x^{-3 n/2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\frac {b x^n}{a}\right )}{3 a n} \]
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Time = 3.80 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.43
method | result | size |
risch | \(\frac {2 b \,x^{-\frac {n}{2}}}{a^{2} n}-\frac {2 x^{-\frac {3 n}{2}}}{3 a n}+\frac {\sqrt {-a b}\, b \ln \left (x^{\frac {n}{2}}+\frac {\sqrt {-a b}}{b}\right )}{a^{3} n}-\frac {\sqrt {-a b}\, b \ln \left (x^{\frac {n}{2}}-\frac {\sqrt {-a b}}{b}\right )}{a^{3} n}\) | \(97\) |
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none
Time = 0.33 (sec) , antiderivative size = 251, normalized size of antiderivative = 3.69 \[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=\left [-\frac {2 \, a x x^{-\frac {3}{2} \, n - 1} - 3 \, b \sqrt {-\frac {b}{a}} \log \left (-\frac {2 \, a^{3} x^{\frac {5}{3}} x^{-\frac {5}{2} \, n - \frac {5}{3}} \sqrt {-\frac {b}{a}} - a^{3} x^{2} x^{-3 \, n - 2} - 2 \, a^{2} b x x^{-\frac {3}{2} \, n - 1} \sqrt {-\frac {b}{a}} + 2 \, a^{2} b x^{\frac {4}{3}} x^{-2 \, n - \frac {4}{3}} + 2 \, a b^{2} x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}} \sqrt {-\frac {b}{a}} - 2 \, a b^{2} x^{\frac {2}{3}} x^{-n - \frac {2}{3}} + b^{3}}{a^{3} x^{2} x^{-3 \, n - 2} + b^{3}}\right ) - 6 \, b x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}}}{3 \, a^{2} n}, -\frac {2 \, {\left (a x x^{-\frac {3}{2} \, n - 1} - 3 \, b \sqrt {\frac {b}{a}} \arctan \left (\frac {\sqrt {\frac {b}{a}}}{x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}}}\right ) - 3 \, b x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}}\right )}}{3 \, a^{2} n}\right ] \]
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Time = 1.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.82 \[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=- \frac {2 x^{- \frac {3 n}{2}}}{3 a n} + \frac {2 b x^{- \frac {n}{2}}}{a^{2} n} + \frac {2 b^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{a^{\frac {5}{2}} n} \]
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\[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=\int { \frac {x^{-\frac {3}{2} \, n - 1}}{b x^{n} + a} \,d x } \]
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\[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=\int { \frac {x^{-\frac {3}{2} \, n - 1}}{b x^{n} + a} \,d x } \]
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Timed out. \[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=\int \frac {1}{x^{\frac {3\,n}{2}+1}\,\left (a+b\,x^n\right )} \,d x \]
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