Optimal. Leaf size=20 \[ b x^{-n/2} \log (x) \sqrt {b x^n} \]
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Rubi [A] time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {15, 29} \[ b x^{-n/2} \log (x) \sqrt {b x^n} \]
Antiderivative was successfully verified.
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Rule 15
Rule 29
Rubi steps
\begin {align*} \int x^{-1-\frac {3 n}{2}} \left (b x^n\right )^{3/2} \, dx &=\left (b x^{-n/2} \sqrt {b x^n}\right ) \int \frac {1}{x} \, dx\\ &=b x^{-n/2} \sqrt {b x^n} \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 0.95 \[ x^{-3 n/2} \log (x) \left (b x^n\right )^{3/2} \]
Antiderivative was successfully verified.
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fricas [B] time = 9.71, size = 110, normalized size = 5.50 \[ \left [\frac {1}{2} \, b^{\frac {3}{2}} \log \left (\frac {b x^{4} + {\left (x^{4} - 1\right )} \sqrt {b} x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}} \sqrt {\frac {b}{x^{\frac {2}{3}} x^{-n - \frac {2}{3}}}} + b}{x^{2}}\right ), -\sqrt {-b} b \arctan \left (\frac {{\left (x^{2} + 1\right )} \sqrt {-b} x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}} \sqrt {\frac {b}{x^{\frac {2}{3}} x^{-n - \frac {2}{3}}}}}{b x^{2} - b}\right )\right ] \]
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 \[ \int \left (b x^{n}\right )^{\frac {3}{2}} x^{-\frac {3}{2} \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 23, normalized size = 1.15 \[ \sqrt {b \,x^{n}}\, b \,x^{-\frac {n}{2}} \ln \relax (x ) \]
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 \[ \int \left (b x^{n}\right )^{\frac {3}{2}} x^{-\frac {3}{2} \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (b\,x^n\right )}^{3/2}}{x^{\frac {3\,n}{2}+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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