Optimal. Leaf size=50 \[ \frac {2 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {b} x^{-n/2}}{\sqrt {c}}\right )}{b^{3/2} n}-\frac {2 x^{-n/2}}{b n} \]
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Rubi [A] time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1584, 345, 193, 321, 205} \[ \frac {2 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {b} x^{-n/2}}{\sqrt {c}}\right )}{b^{3/2} n}-\frac {2 x^{-n/2}}{b n} \]
Antiderivative was successfully verified.
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Rule 193
Rule 205
Rule 321
Rule 345
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{-1+\frac {n}{2}}}{b x^n+c x^{2 n}} \, dx &=\int \frac {x^{-1-\frac {n}{2}}}{b+c x^n} \, dx\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{b+\frac {c}{x^2}} \, dx,x,x^{-n/2}\right )}{n}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {x^2}{c+b x^2} \, dx,x,x^{-n/2}\right )}{n}\\ &=-\frac {2 x^{-n/2}}{b n}+\frac {(2 c) \operatorname {Subst}\left (\int \frac {1}{c+b x^2} \, dx,x,x^{-n/2}\right )}{b n}\\ &=-\frac {2 x^{-n/2}}{b n}+\frac {2 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {b} x^{-n/2}}{\sqrt {c}}\right )}{b^{3/2} n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 32, normalized size = 0.64 \[ -\frac {2 x^{-n/2} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {c x^n}{b}\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 151, normalized size = 3.02 \[ \left [\frac {x x^{\frac {1}{2} \, n - 1} \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} x^{n - 2} - 2 \, b x x^{\frac {1}{2} \, n - 1} \sqrt {-\frac {c}{b}} - b}{c x^{2} x^{n - 2} + b}\right ) - 2}{b n x x^{\frac {1}{2} \, n - 1}}, \frac {2 \, {\left (x x^{\frac {1}{2} \, n - 1} \sqrt {\frac {c}{b}} \arctan \left (\frac {b \sqrt {\frac {c}{b}}}{c x x^{\frac {1}{2} \, n - 1}}\right ) - 1\right )}}{b n x x^{\frac {1}{2} \, n - 1}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 38, normalized size = 0.76 \[ -\frac {2 \, {\left (\frac {c \arctan \left (\frac {c \sqrt {x^{n}}}{\sqrt {b c}}\right )}{\sqrt {b c} b} + \frac {1}{b \sqrt {x^{n}}}\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 79, normalized size = 1.58 \[ -\frac {2 x^{-\frac {n}{2}}}{b n}+\frac {\sqrt {-b c}\, \ln \left (x^{\frac {n}{2}}-\frac {\sqrt {-b c}}{c}\right )}{b^{2} n}-\frac {\sqrt {-b c}\, \ln \left (x^{\frac {n}{2}}+\frac {\sqrt {-b c}}{c}\right )}{b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -c \int \frac {x^{\frac {1}{2} \, n}}{b c x x^{n} + b^{2} x}\,{d x} - \frac {2}{b n x^{\frac {1}{2} \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^{\frac {n}{2}-1}}{b\,x^n+c\,x^{2\,n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.00, size = 36, normalized size = 0.72 \[ - \frac {2 \operatorname {atan}{\left (\frac {x^{\frac {n}{2}}}{\sqrt {\frac {b}{c}}} \right )}}{b n \sqrt {\frac {b}{c}}} - \frac {2 x^{- \frac {n}{2}}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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