Optimal. Leaf size=192 \[ \frac{\sqrt{x^2+2} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{\sqrt{2} b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}-\frac{\sqrt{x^2+2} (a-2 b) \Pi \left (1-\frac{b}{a};\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} a b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}+\frac{\sqrt{x^2+2} x}{b \sqrt{x^2+1}}-\frac{\sqrt{2} \sqrt{x^2+2} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}} \]
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Rubi [A] time = 0.0918102, antiderivative size = 192, normalized size of antiderivative = 1., 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 534
Rule 422
Rule 418
Rule 492
Rule 411
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*}
Mathematica [C] time = 0.191546, size = 71, normalized size = 0.37 \[ \frac{i \left ((a-b) \left (a \text{EllipticF}\left (i \sinh ^{-1}(x),\frac{1}{2}\right )-(a-2 b) \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )\right )-2 a b E\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )\right )}{\sqrt{2} a b^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.008, size = 120, normalized size = 0.6 \begin{align*}{\frac{-i}{a{b}^{2}} \left ({\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) ba-{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){a}^{2}+2\,{\it EllipticF} \left ( i/2x\sqrt{2},\sqrt{2} \right ) ba+{a}^{2}{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ) -3\,{\it EllipticPi} \left ( i/2x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ) ba+2\,{\it EllipticPi} \left ( i/2x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ){b}^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} + 2} \sqrt{x^{2} + 1}}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{2} + 2} \sqrt{x^{2} + 1}}{b x^{2} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} + 1} \sqrt{x^{2} + 2}}{a + b x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} + 2} \sqrt{x^{2} + 1}}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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