Optimal. Leaf size=93 \[ \frac {2 \sqrt {f x^2+3} \Pi \left (1-\frac {2 b}{a d};\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|1-\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {d} \sqrt {d x^2+2} \sqrt {\frac {f x^2+3}{d x^2+2}}} \]
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Rubi [A] time = 0.04, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {539} \[ \frac {2 \sqrt {f x^2+3} \Pi \left (1-\frac {2 b}{a d};\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|1-\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {d} \sqrt {d x^2+2} \sqrt {\frac {f x^2+3}{d x^2+2}}} \]
Antiderivative was successfully verified.
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Rule 539
Rubi steps
\begin {align*} \int \frac {\sqrt {2+d x^2}}{\left (a+b x^2\right ) \sqrt {3+f x^2}} \, dx &=\frac {2 \sqrt {3+f x^2} \Pi \left (1-\frac {2 b}{a d};\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|1-\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {d} \sqrt {2+d x^2} \sqrt {\frac {3+f x^2}{2+d x^2}}}\\ \end {align*}
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Mathematica [C] time = 0.27, size = 94, normalized size = 1.01 \[ -\frac {i \left ((2 b-a d) \Pi \left (\frac {2 b}{a d};i \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )+a d F\left (i \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )\right )}{\sqrt {3} a b \sqrt {d}} \]
Antiderivative was successfully verified.
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fricas [F] time = 155.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}}{b f x^{4} + {\left (a f + 3 \, b\right )} x^{2} + 3 \, a}, x\right ) \]
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 \[ \int \frac {\sqrt {d x^{2} + 2}}{{\left (b x^{2} + a\right )} \sqrt {f x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 133, normalized size = 1.43 \[ \frac {\sqrt {2}\, \left (a d \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )-a d \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {3 b}{a f}, \frac {\sqrt {2}\, \sqrt {-d}\, \sqrt {3}}{2 \sqrt {-f}}\right )+2 b \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {3 b}{a f}, \frac {\sqrt {2}\, \sqrt {-d}\, \sqrt {3}}{2 \sqrt {-f}}\right )\right )}{2 \sqrt {-f}\, a b} \]
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 \[ \int \frac {\sqrt {d x^{2} + 2}}{{\left (b x^{2} + a\right )} \sqrt {f x^{2} + 3}}\,{d x} \]
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 \[ \int \frac {\sqrt {d\,x^2+2}}{\left (b\,x^2+a\right )\,\sqrt {f\,x^2+3}} \,d x \]
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 \[ \int \frac {\sqrt {d x^{2} + 2}}{\left (a + b x^{2}\right ) \sqrt {f x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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