Optimal. Leaf size=31 \[ \frac {\sqrt {b-\frac {a}{x^2}} x^3}{2 \sqrt {a-b x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 31, 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^3 \sqrt {b-\frac {a}{x^2}}}{2 \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^2}{\sqrt {a-b x^2}} \, dx &=\frac {\left (\sqrt {b-\frac {a}{x^2}} x\right ) \int \frac {x \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 \, dx}{\sqrt {a-b x^2}}\\ &=\frac {\sqrt {b-\frac {a}{x^2}} x^3}{2 \sqrt {a-b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 32, normalized size = 1.03 \begin {gather*} -\frac {\sqrt {b-\frac {a}{x^2}} x \sqrt {a-b x^2}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 31, normalized size = 1.00
method | result | size |
gosper | \(\frac {x^{3} \sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}}{2 \sqrt {-b \,x^{2}+a}}\) | \(31\) |
default | \(\frac {x^{3} \sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}}{2 \sqrt {-b \,x^{2}+a}}\) | \(31\) |
risch | \(\frac {i x^{3} \sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}\, \left (b \,x^{2}-a \right ) \sqrt {\frac {-b \,x^{2}+a}{b \,x^{2}-a}}}{2 \left (-b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.30, size = 5, normalized size = 0.16 \begin {gather*} -\frac {1}{2} i \, x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 41, normalized size = 1.32 \begin {gather*} -\frac {\sqrt {-b x^{2} + a} x^{3} \sqrt {\frac {b x^{2} - a}{x^{2}}}}{2 \, {\left (b x^{2} - a\right )}} \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^{2} \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 [C] Result contains complex when optimal does not.
time = 4.73, size = 15, normalized size = 0.48 \begin {gather*} -\frac {i \, b x^{2} - i \, a}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.14, size = 25, normalized size = 0.81 \begin {gather*} \frac {x^3\,\sqrt {b-\frac {a}{x^2}}}{2\,\sqrt {a-b\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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