Optimal. Leaf size=19 \[ -\frac {2}{(2+n) x \sqrt {b x^n}} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30}
\begin {gather*} -\frac {2}{(n+2) x \sqrt {b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {b x^n}} \, dx &=\frac {x^{n/2} \int x^{-2-\frac {n}{2}} \, dx}{\sqrt {b x^n}}\\ &=-\frac {2}{(2+n) x \sqrt {b x^n}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} -\frac {2}{(2+n) x \sqrt {b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 18, normalized size = 0.95
method | result | size |
gosper | \(-\frac {2}{\left (2+n \right ) x \sqrt {b \,x^{n}}}\) | \(18\) |
risch | \(-\frac {2}{\left (2+n \right ) x \sqrt {b \,x^{n}}}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 17, normalized size = 0.89 \begin {gather*} -\frac {2}{\sqrt {b x^{n}} {\left (n + 2\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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*} \begin {cases} - \frac {2}{n x \sqrt {b x^{n}} + 2 x \sqrt {b x^{n}}} & \text {for}\: n \neq -2 \\\int \frac {1}{x^{2} \sqrt {\frac {b}{x^{2}}}}\, dx & \text {otherwise} \end {cases} \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 = 0.97, size = 24, normalized size = 1.26 \begin {gather*} -\frac {2\,\sqrt {b\,x^n}}{b\,x^{n+1}\,\left (n+2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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