Optimal. Leaf size=33 \[ \frac {\sqrt {b-\frac {a}{x^2}} x^{1+m}}{m \sqrt {a-b x^2}} \]
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Rubi [A]
time = 0.02, 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 = {529, 23, 30}
\begin {gather*} \frac {x^{m+1} \sqrt {b-\frac {a}{x^2}}}{m \sqrt {a-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rule 529
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 = 0.06, size = 33, normalized size = 1.00 \begin {gather*} \frac {\sqrt {b-\frac {a}{x^2}} x^{1+m}}{m \sqrt {a-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 35, normalized size = 1.06
method | result | size |
gosper | \(\frac {x^{1+m} \sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}}{m \sqrt {-b \,x^{2}+a}}\) | \(35\) |
risch | \(\frac {i \sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}\, \left (b \,x^{2}-a \right ) x \sqrt {\frac {-b \,x^{2}+a}{b \,x^{2}-a}}\, x^{m}}{\left (-b \,x^{2}+a \right )^{\frac {3}{2}} m}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.29, size = 8, normalized size = 0.24 \begin {gather*} -\frac {i \, x^{m}}{m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 44, normalized size = 1.33 \begin {gather*} -\frac {\sqrt {-b x^{2} + a} x x^{m} \sqrt {\frac {b x^{2} - a}{x^{2}}}}{b m x^{2} - a m} \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 {x^{m} \sqrt {- \frac {a}{x^{2}} + b}}{\sqrt {a - b x^{2}}}\, 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 = 3.36, size = 29, normalized size = 0.88 \begin {gather*} \frac {x^{m+1}\,\sqrt {b-\frac {a}{x^2}}}{m\,\sqrt {a-b\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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