Optimal. Leaf size=68 \[ -\frac {2 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x^{-n/2}}{\sqrt {c}}\right )}{b^{5/2} n}+\frac {2 c x^{-n/2}}{b^2 n}-\frac {2 x^{-3 n/2}}{3 b n} \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1584, 362, 345, 193, 321, 205} \[ -\frac {2 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x^{-n/2}}{\sqrt {c}}\right )}{b^{5/2} n}+\frac {2 c x^{-n/2}}{b^2 n}-\frac {2 x^{-3 n/2}}{3 b n} \]
Antiderivative was successfully verified.
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Rule 193
Rule 205
Rule 321
Rule 345
Rule 362
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 {3 n}{2}}}{b+c x^n} \, dx\\ &=-\frac {2 x^{-3 n/2}}{3 b n}-\frac {c \int \frac {x^{-1-\frac {n}{2}}}{b+c x^n} \, dx}{b}\\ &=-\frac {2 x^{-3 n/2}}{3 b n}+\frac {(2 c) \operatorname {Subst}\left (\int \frac {1}{b+\frac {c}{x^2}} \, dx,x,x^{-n/2}\right )}{b n}\\ &=-\frac {2 x^{-3 n/2}}{3 b n}+\frac {(2 c) \operatorname {Subst}\left (\int \frac {x^2}{c+b x^2} \, dx,x,x^{-n/2}\right )}{b n}\\ &=-\frac {2 x^{-3 n/2}}{3 b n}+\frac {2 c x^{-n/2}}{b^2 n}-\frac {\left (2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+b x^2} \, dx,x,x^{-n/2}\right )}{b^2 n}\\ &=-\frac {2 x^{-3 n/2}}{3 b n}+\frac {2 c x^{-n/2}}{b^2 n}-\frac {2 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x^{-n/2}}{\sqrt {c}}\right )}{b^{5/2} n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.50 \[ -\frac {2 x^{-3 n/2} \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {c x^n}{b}\right )}{3 b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 161, normalized size = 2.37 \[ \left [-\frac {2 \, b x^{3} x^{-\frac {3}{2} \, n - 3} - 6 \, c x x^{-\frac {1}{2} \, n - 1} - 3 \, c \sqrt {-\frac {c}{b}} \log \left (\frac {b x^{2} x^{-n - 2} - 2 \, b x x^{-\frac {1}{2} \, n - 1} \sqrt {-\frac {c}{b}} - c}{b x^{2} x^{-n - 2} + c}\right )}{3 \, b^{2} n}, -\frac {2 \, {\left (b x^{3} x^{-\frac {3}{2} \, n - 3} - 3 \, c x x^{-\frac {1}{2} \, n - 1} - 3 \, c \sqrt {\frac {c}{b}} \arctan \left (\frac {\sqrt {\frac {c}{b}}}{x x^{-\frac {1}{2} \, n - 1}}\right )\right )}}{3 \, b^{2} n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{-\frac {1}{2} \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 97, normalized size = 1.43 \[ -\frac {2 x^{-\frac {3 n}{2}}}{3 b n}+\frac {2 c \,x^{-\frac {n}{2}}}{b^{2} n}-\frac {\sqrt {-b c}\, c \ln \left (x^{\frac {n}{2}}-\frac {\sqrt {-b c}}{c}\right )}{b^{3} n}+\frac {\sqrt {-b c}\, c \ln \left (x^{\frac {n}{2}}+\frac {\sqrt {-b c}}{c}\right )}{b^{3} n} \]
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 \[ c^{2} \int \frac {x^{\frac {1}{2} \, n}}{b^{2} c x x^{n} + b^{3} x}\,{d x} + \frac {2 \, {\left (3 \, c x^{n} - b\right )}}{3 \, b^{2} n x^{\frac {3}{2} \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{\frac {n}{2}+1}\,\left (b\,x^n+c\,x^{2\,n}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.92, size = 58, normalized size = 0.85 \[ - \frac {2 x^{- \frac {3 n}{2}}}{3 b n} + \frac {2 c x^{- \frac {n}{2}}}{b^{2} n} - \frac {2 c^{2} \operatorname {atan}{\left (\frac {x^{- \frac {n}{2}}}{\sqrt {\frac {c}{b}}} \right )}}{b^{3} n \sqrt {\frac {c}{b}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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