Optimal. Leaf size=192 \[ -\frac {\sqrt {x^2+2} (a-2 b) \Pi \left (1-\frac {b}{a};\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} a b \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}}+\frac {\sqrt {x^2+2} x}{b \sqrt {x^2+1}}+\frac {\sqrt {x^2+2} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} b \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}}-\frac {\sqrt {2} \sqrt {x^2+2} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{b \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}} \]
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Rubi [A] time = 0.09, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {534, 422, 418, 492, 411, 539} \[ -\frac {\sqrt {x^2+2} (a-2 b) \Pi \left (1-\frac {b}{a};\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} a b \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}}+\frac {\sqrt {x^2+2} x}{b \sqrt {x^2+1}}+\frac {\sqrt {x^2+2} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} b \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}}-\frac {\sqrt {2} \sqrt {x^2+2} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{b \sqrt {x^2+1} \sqrt {\frac {x^2+2}{x^2+1}}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 422
Rule 492
Rule 534
Rule 539
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^2} \sqrt {2+x^2}}{a+b x^2} \, dx &=\frac {\int \frac {\sqrt {1+x^2}}{\sqrt {2+x^2}} \, dx}{b}+\frac {(-a+2 b) \int \frac {\sqrt {1+x^2}}{\sqrt {2+x^2} \left (a+b x^2\right )} \, dx}{b}\\ &=-\frac {(a-2 b) \sqrt {2+x^2} \Pi \left (1-\frac {b}{a};\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} a b \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}}+\frac {\int \frac {1}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx}{b}+\frac {\int \frac {x^2}{\sqrt {1+x^2} \sqrt {2+x^2}} \, dx}{b}\\ &=\frac {x \sqrt {2+x^2}}{b \sqrt {1+x^2}}+\frac {\sqrt {2+x^2} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} b \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}}-\frac {(a-2 b) \sqrt {2+x^2} \Pi \left (1-\frac {b}{a};\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} a b \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}}-\frac {\int \frac {\sqrt {2+x^2}}{\left (1+x^2\right )^{3/2}} \, dx}{b}\\ &=\frac {x \sqrt {2+x^2}}{b \sqrt {1+x^2}}-\frac {\sqrt {2} \sqrt {2+x^2} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{b \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}}+\frac {\sqrt {2+x^2} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} b \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}}-\frac {(a-2 b) \sqrt {2+x^2} \Pi \left (1-\frac {b}{a};\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} a b \sqrt {1+x^2} \sqrt {\frac {2+x^2}{1+x^2}}}\\ \end {align*}
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Mathematica [C] time = 0.25, size = 71, normalized size = 0.37 \[ \frac {i \left ((a-b) \left (a F\left (i \sinh ^{-1}(x)|\frac {1}{2}\right )-(a-2 b) \Pi \left (\frac {b}{a};i \sinh ^{-1}(x)|\frac {1}{2}\right )\right )-2 a b E\left (i \sinh ^{-1}(x)|\frac {1}{2}\right )\right )}{\sqrt {2} a b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{2} + 2} \sqrt {x^{2} + 1}}{b x^{2} + a}, x\right ) \]
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 \[ \int \frac {\sqrt {x^{2} + 2} \sqrt {x^{2} + 1}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 120, normalized size = 0.62 \[ -\frac {i \left (-a^{2} \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+a^{2} \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 b}{a}, \sqrt {2}\right )+a b \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )+2 a b \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-3 a b \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 b}{a}, \sqrt {2}\right )+2 b^{2} \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 b}{a}, \sqrt {2}\right )\right )}{a \,b^{2}} \]
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 \[ \int \frac {\sqrt {x^{2} + 2} \sqrt {x^{2} + 1}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {x^2+1}\,\sqrt {x^2+2}}{b\,x^2+a} \,d x \]
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 \[ \int \frac {\sqrt {x^{2} + 1} \sqrt {x^{2} + 2}}{a + b x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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