Optimal. Leaf size=31 \[ \frac {x^3 \sqrt {b-\frac {a}{x^2}}}{2 \sqrt {a-b x^2}} \]
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Rubi [A] time = 0.04, 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 = {515, 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 515
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 = 0.01, size = 31, normalized size = 1.00 \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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IntegrateAlgebraic [A] time = 0.12, size = 43, normalized size = 1.39 \begin {gather*} \frac {x \sqrt {b-\frac {a}{x^2}} \left (a-b x^2\right )^{3/2}}{2 b \left (b x^2-a\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, 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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giac [C] time = 0.33, 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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maple [A] time = 0.00, size = 31, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}\, x^{3}}{2 \sqrt {-b \,x^{2}+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.02, 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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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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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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