Optimal. Leaf size=28 \[ \frac {\sqrt {b-\frac {a}{x^2}} x^2}{\sqrt {a-b x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {529, 23, 8}
\begin {gather*} \frac {x^2 \sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 23
Rule 529
Rubi steps
\begin {align*} \int \frac {\sqrt {b-\frac {a}{x^2}} x}{\sqrt {a-b x^2}} \, dx &=\frac {\left (\sqrt {b-\frac {a}{x^2}} x\right ) \int \frac {\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 1 \, dx}{\sqrt {a-b x^2}}\\ &=\frac {\sqrt {b-\frac {a}{x^2}} x^2}{\sqrt {a-b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 28, normalized size = 1.00 \begin {gather*} \frac {\sqrt {b-\frac {a}{x^2}} x^2}{\sqrt {a-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 30, normalized size = 1.07
method | result | size |
default | \(\frac {x^{2} \sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}}{\sqrt {-b \,x^{2}+a}}\) | \(30\) |
risch | \(\frac {i x^{2} \sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}\, \left (b \,x^{2}-a \right ) \sqrt {\frac {-b \,x^{2}+a}{b \,x^{2}-a}}}{\left (-b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.28, size = 7, normalized size = 0.25 \begin {gather*} -i \, \sqrt {x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 41, normalized size = 1.46 \begin {gather*} -\frac {\sqrt {-b x^{2} + a} x^{2} \sqrt {\frac {b x^{2} - a}{x^{2}}}}{b x^{2} - a} \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 \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 = 3.10, size = 27, normalized size = 0.96 \begin {gather*} \frac {\sqrt {b\,x^2-a}\,\sqrt {x^2}}{\sqrt {a-b\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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