Optimal. Leaf size=33 \[ \frac {x^{m+1} \sqrt {b-\frac {a}{x^2}}}{m \sqrt {a-b x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {515, 23, 30} \[ \frac {x^{m+1} \sqrt {b-\frac {a}{x^2}}}{m \sqrt {a-b x^2}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rule 515
Rubi steps
\begin {align*} \int \frac {\sqrt {b-\frac {a}{x^2}} x^m}{\sqrt {a-b x^2}} \, dx &=\frac {\left (\sqrt {b-\frac {a}{x^2}} x\right ) \int \frac {x^{-1+m} \sqrt {-a+b x^2}}{\sqrt {a-b x^2}} \, dx}{\sqrt {-a+b x^2}}\\ &=\frac {\left (\sqrt {b-\frac {a}{x^2}} x\right ) \int x^{-1+m} \, dx}{\sqrt {a-b x^2}}\\ &=\frac {\sqrt {b-\frac {a}{x^2}} x^{1+m}}{m \sqrt {a-b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \[ \frac {x^{m+1} \sqrt {b-\frac {a}{x^2}}}{m \sqrt {a-b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 44, normalized size = 1.33 \[ -\frac {\sqrt {-b x^{2} + a} x x^{m} \sqrt {\frac {b x^{2} - a}{x^{2}}}}{b m x^{2} - a m} \]
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 \[ \int \frac {\sqrt {b - \frac {a}{x^{2}}} x^{m}}{\sqrt {-b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 1.06 \[ \frac {\sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}\, x^{m +1}}{\sqrt {-b \,x^{2}+a}\, m} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.07, size = 8, normalized size = 0.24 \[ -\frac {i \, x^{m}}{m} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.36, size = 29, normalized size = 0.88 \[ \frac {x^{m+1}\,\sqrt {b-\frac {a}{x^2}}}{m\,\sqrt {a-b\,x^2}} \]
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 \[ \int \frac {x^{m} \sqrt {- \frac {a}{x^{2}} + b}}{\sqrt {a - b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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