Optimal. Leaf size=78 \[ \frac {\sqrt {b x^2+2} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right ),1-\frac {3 b}{2 d}\right )}{\sqrt {2} \sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}} \]
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Rubi [A] time = 0.01, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {418} \[ \frac {\sqrt {b x^2+2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {2} \sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}} \]
Antiderivative was successfully verified.
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Rule 418
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx &=\frac {\sqrt {2+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {2} \sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 0.47 \[ \frac {\operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {-b} x}{\sqrt {2}}\right ),\frac {2 d}{3 b}\right )}{\sqrt {3} \sqrt {-b}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}}{b d x^{4} + {\left (3 \, b + 2 \, d\right )} x^{2} + 6}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 38, normalized size = 0.49 \[ \frac {\sqrt {2}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-d}\, x}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right )}{2 \sqrt {-d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {b\,x^2+2}\,\sqrt {d\,x^2+3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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