Optimal. Leaf size=37 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a x^n}}\right )}{\sqrt {b} (2-n)} \]
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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2004, 2033,
212} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x^n+b x^2}}\right )}{\sqrt {b} (2-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2004
Rule 2033
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x^2 \left (b+a x^{-2+n}\right )}} \, dx &=\int \frac {1}{\sqrt {b x^2+a x^n}} \, dx\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+a x^n}}\right )}{2-n}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a x^n}}\right )}{\sqrt {b} (2-n)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(78\) vs. \(2(37)=74\).
time = 0.02, size = 78, normalized size = 2.11 \begin {gather*} -\frac {2 \sqrt {a} x^{n/2} \sqrt {1+\frac {b x^{2-n}}{a}} \sinh ^{-1}\left (\frac {\sqrt {b} x^{1-\frac {n}{2}}}{\sqrt {a}}\right )}{\sqrt {b} (-2+n) \sqrt {b x^2+a x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {x^{2} \left (b +a \,x^{-2+n}\right )}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.39, size = 109, normalized size = 2.95 \begin {gather*} \left [\frac {\sqrt {b} \log \left (\frac {a x x^{n - 2} + 2 \, b x - 2 \, \sqrt {a x^{2} x^{n - 2} + b x^{2}} \sqrt {b}}{x x^{n - 2}}\right )}{b n - 2 \, b}, \frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {a x^{2} x^{n - 2} + b x^{2}} \sqrt {-b}}{b x}\right )}{b n - 2 \, b}\right ] \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} \left (a x^{n - 2} + b\right )}}\, dx \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.32, size = 67, normalized size = 1.81 \begin {gather*} \frac {\sqrt {a}\,x^{n/2}\,\mathrm {asin}\left (\frac {\sqrt {b}\,x^{1-\frac {n}{2}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,\sqrt {\frac {b\,x^{2-n}}{a}+1}\,1{}\mathrm {i}}{\sqrt {b}\,\left (\frac {n}{2}-1\right )\,\sqrt {a\,x^n+b\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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