Optimal. Leaf size=37 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x^{2-n}+b x^2}}\right )}{\sqrt {b} n} \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1979, 2008, 206} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x^{2-n}+b x^2}}\right )}{\sqrt {b} n} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 1979
Rule 2008
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x^{2-n} \left (a+b x^n\right )}} \, dx &=\int \frac {1}{\sqrt {b x^2+a x^{2-n}}} \, dx\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+a x^{2-n}}}\right )}{n}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a x^{2-n}}}\right )}{\sqrt {b} n}\\ \end {align*}
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Mathematica [B] time = 0.05, size = 76, normalized size = 2.05 \begin {gather*} \frac {2 \sqrt {a} x^{1-\frac {n}{2}} \sqrt {\frac {b x^n}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a}}\right )}{\sqrt {b} n \sqrt {x^{2-n} \left (a+b x^n\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x^{2-n} \left (a+b x^n\right )}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.43, size = 102, normalized size = 2.76 \begin {gather*} \left [\frac {\log \left (\frac {2 \, b x x^{n} + a x + 2 \, \sqrt {b} x^{n} \sqrt {\frac {b x^{2} x^{n} + a x^{2}}{x^{n}}}}{x}\right )}{\sqrt {b} n}, -\frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {b x^{2} x^{n} + a x^{2}}{x^{n}}}}{b x}\right )}{b n}\right ] \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {{\left (b x^{n} + a\right )} x^{-n + 2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.73, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (b \,x^{n}+a \right ) x^{-n +2}}}\, dx \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {{\left (b x^{n} + a\right )} x^{-n + 2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {x^{2-n}\,\left (a+b\,x^n\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x^{2 - n} \left (a + b x^{n}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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