Optimal. Leaf size=19 \[ -\frac {2}{(n+2) x \sqrt {b x^n}} \]
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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} \[ -\frac {2}{(n+2) x \sqrt {b x^n}} \]
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 = 0.02, size = 19, normalized size = 1.00 \[ -\frac {2}{(n+2) x \sqrt {b x^n}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {1}{\sqrt {b x^{n}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.95 \[ -\frac {2}{\left (n +2\right ) \sqrt {b \,x^{n}}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 17, normalized size = 0.89 \[ -\frac {2}{\sqrt {b x^{n}} {\left (n + 2\right )} x} \]
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 \[ -\frac {2\,\sqrt {b\,x^n}}{b\,x^{n+1}\,\left (n+2\right )} \]
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 \[ \begin {cases} - \frac {2}{\sqrt {b} n x \sqrt {x^{n}} + 2 \sqrt {b} x \sqrt {x^{n}}} & \text {for}\: n \neq -2 \\\int \frac {1}{x^{2} \sqrt {\frac {b}{x^{2}}}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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