Optimal. Leaf size=298 \[ \frac {3 \sqrt {d x^2+2} (2 b-a d) \Pi \left (1-\frac {3 b}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} a b \sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}+\frac {f x \sqrt {d x^2+2}}{b \sqrt {f x^2+3}}+\frac {3 d \sqrt {d x^2+2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} b \sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}-\frac {\sqrt {2} \sqrt {f} \sqrt {d x^2+2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{b \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}} \]
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Rubi [A] time = 0.18, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {535, 422, 418, 492, 411, 539} \[ \frac {3 \sqrt {d x^2+2} (2 b-a d) \Pi \left (1-\frac {3 b}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} a b \sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}+\frac {f x \sqrt {d x^2+2}}{b \sqrt {f x^2+3}}+\frac {3 d \sqrt {d x^2+2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} b \sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}-\frac {\sqrt {2} \sqrt {f} \sqrt {d x^2+2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{b \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 422
Rule 492
Rule 535
Rule 539
Rubi steps
\begin {align*} \int \frac {\sqrt {2+d x^2} \sqrt {3+f x^2}}{a+b x^2} \, dx &=\frac {d \int \frac {\sqrt {3+f x^2}}{\sqrt {2+d x^2}} \, dx}{b}+\frac {(2 b-a d) \int \frac {\sqrt {3+f x^2}}{\left (a+b x^2\right ) \sqrt {2+d x^2}} \, dx}{b}\\ &=\frac {3 (2 b-a d) \sqrt {2+d x^2} \Pi \left (1-\frac {3 b}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} a b \sqrt {f} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}+\frac {(3 d) \int \frac {1}{\sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx}{b}+\frac {(d f) \int \frac {x^2}{\sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx}{b}\\ &=\frac {f x \sqrt {2+d x^2}}{b \sqrt {3+f x^2}}+\frac {3 d \sqrt {2+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} b \sqrt {f} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}+\frac {3 (2 b-a d) \sqrt {2+d x^2} \Pi \left (1-\frac {3 b}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} a b \sqrt {f} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}-\frac {(3 f) \int \frac {\sqrt {2+d x^2}}{\left (3+f x^2\right )^{3/2}} \, dx}{b}\\ &=\frac {f x \sqrt {2+d x^2}}{b \sqrt {3+f x^2}}-\frac {\sqrt {2} \sqrt {f} \sqrt {2+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{b \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}+\frac {3 d \sqrt {2+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} b \sqrt {f} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}+\frac {3 (2 b-a d) \sqrt {2+d x^2} \Pi \left (1-\frac {3 b}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} a b \sqrt {f} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}\\ \end {align*}
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Mathematica [C] time = 0.40, size = 134, normalized size = 0.45 \[ \frac {i \left ((a d-2 b) \left ((3 b-a f) \Pi \left (\frac {2 b}{a d};i \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )+a f F\left (i \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )\right )-3 a b d E\left (i \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )\right )}{\sqrt {3} a b^2 \sqrt {d}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {\sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 293, normalized size = 0.98 \[ -\frac {\left (a^{2} d f \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )-a^{2} d f \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {3 b}{a f}, \frac {\sqrt {2}\, \sqrt {-d}\, \sqrt {3}}{2 \sqrt {-f}}\right )-3 a b d \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )+3 a b d \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {3 b}{a f}, \frac {\sqrt {2}\, \sqrt {-d}\, \sqrt {3}}{2 \sqrt {-f}}\right )-2 a b f \EllipticE \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )+2 a b f \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {3 b}{a f}, \frac {\sqrt {2}\, \sqrt {-d}\, \sqrt {3}}{2 \sqrt {-f}}\right )-6 b^{2} \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {3 b}{a f}, \frac {\sqrt {2}\, \sqrt {-d}\, \sqrt {3}}{2 \sqrt {-f}}\right )\right ) \sqrt {2}}{2 \sqrt {-f}\, a \,b^{2}} \]
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 \[ \int \frac {\sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {d\,x^2+2}\,\sqrt {f\,x^2+3}}{b\,x^2+a} \,d x \]
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 \[ \int \frac {\sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}}{a + b x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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