Optimal. Leaf size=24 \[ x \sqrt {a+b x^2} \sqrt {c+d x^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {1604}
\begin {gather*} x \sqrt {a+b x^2} \sqrt {c+d x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 1604
Rubi steps
\begin {align*} \int \frac {a c+2 (b c+a d) x^2+3 b d x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx &=x \sqrt {a+b x^2} \sqrt {c+d x^2}\\ \end {align*}
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Mathematica [A]
time = 5.75, size = 24, normalized size = 1.00 \begin {gather*} x \sqrt {a+b x^2} \sqrt {c+d x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 21, normalized size = 0.88
method | result | size |
gosper | \(x \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\) | \(21\) |
default | \(x \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\) | \(21\) |
risch | \(x \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\) | \(21\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, x \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 20, normalized size = 0.83 \begin {gather*} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 20, normalized size = 0.83 \begin {gather*} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x \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 {a c + 2 a d x^{2} + 2 b c x^{2} + 3 b d x^{4}}{\sqrt {a + b x^{2}} \sqrt {c + d 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 = 5.59, size = 20, normalized size = 0.83 \begin {gather*} x\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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