Optimal. Leaf size=24 \[ x \sqrt{a+b x^2} \sqrt{c+d x^2} \]
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Rubi [A] time = 0.0525376, antiderivative size = 24, normalized size of antiderivative = 1., 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 = {1590} \[ x \sqrt{a+b x^2} \sqrt{c+d x^2} \]
Antiderivative was successfully verified.
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Rule 1590
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*}
Mathematica [A] time = 0.148485, size = 24, normalized size = 1. \[ x \sqrt{a+b x^2} \sqrt{c+d x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 21, normalized size = 0.9 \begin{align*} x\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10455, size = 27, normalized size = 1.12 \begin{align*} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61799, size = 47, normalized size = 1.96 \begin{align*} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3 \, b d x^{4} + 2 \,{\left (b c + a d\right )} x^{2} + a c}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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