Optimal. Leaf size=49 \[ \frac {\Pi \left (\frac {2 b}{a d};\sin ^{-1}\left (\frac {\sqrt {-d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {-d}} \]
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Rubi [A]
time = 0.03, antiderivative size = 49, 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 = {551}
\begin {gather*} \frac {\Pi \left (\frac {2 b}{a d};\text {ArcSin}\left (\frac {\sqrt {-d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {-d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 551
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx &=\frac {\Pi \left (\frac {2 b}{a d};\sin ^{-1}\left (\frac {\sqrt {-d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {-d}}\\ \end {align*}
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Mathematica [A]
time = 1.86, size = 49, normalized size = 1.00 \begin {gather*} \frac {\Pi \left (\frac {2 b}{a d};\sin ^{-1}\left (\frac {\sqrt {-d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {-d}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 53, normalized size = 1.08
method | result | size |
default | \(\frac {\sqrt {2}\, \EllipticPi \left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {3 b}{a f}, \frac {\sqrt {2}\, \sqrt {-d}\, \sqrt {3}}{2 \sqrt {-f}}\right )}{2 \sqrt {-f}\, a}\) | \(53\) |
elliptic | \(\frac {\sqrt {\left (f \,x^{2}+3\right ) \left (d \,x^{2}+2\right )}\, \sqrt {1+\frac {f \,x^{2}}{3}}\, \sqrt {1+\frac {d \,x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {f}{3}}\, x , \frac {3 b}{a f}, \frac {\sqrt {-\frac {d}{2}}}{\sqrt {-\frac {f}{3}}}\right )}{\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}\, a \sqrt {-\frac {f}{3}}\, \sqrt {d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}}\) | \(115\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {1}{\left (a + b x^{2}\right ) \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\left (b\,x^2+a\right )\,\sqrt {d\,x^2+2}\,\sqrt {f\,x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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