Optimal. Leaf size=215 \[ -\frac{\sqrt{2} \sqrt{x^2+2} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{3 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)}+\frac{2 b^2 \sqrt{x^2+1} \Pi \left (1-\frac{2 b}{a};\left .\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-1\right )}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2} (a-b)^2}+\frac{x \sqrt{x^2+2}}{3 \left (x^2+1\right )^{3/2} (a-b)}+\frac{\sqrt{2} \sqrt{x^2+2} (a-2 b) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)^2} \]
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Rubi [A] time = 0.144995, antiderivative size = 215, 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 = {546, 539, 526, 525, 418, 411} \[ \frac{2 b^2 \sqrt{x^2+1} \Pi \left (1-\frac{2 b}{a};\left .\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-1\right )}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2} (a-b)^2}+\frac{x \sqrt{x^2+2}}{3 \left (x^2+1\right )^{3/2} (a-b)}-\frac{\sqrt{2} \sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)}+\frac{\sqrt{2} \sqrt{x^2+2} (a-2 b) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)^2} \]
Antiderivative was successfully verified.
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Rule 546
Rule 539
Rule 526
Rule 525
Rule 418
Rule 411
Rubi steps
\begin{align*} \int \frac{\sqrt{2+x^2}}{\left (1+x^2\right )^{5/2} \left (a+b x^2\right )} \, dx &=-\frac{\int \frac{\sqrt{2+x^2} \left (-a+2 b+b x^2\right )}{\left (1+x^2\right )^{5/2}} \, dx}{(a-b)^2}+\frac{b^2 \int \frac{\sqrt{2+x^2}}{\sqrt{1+x^2} \left (a+b x^2\right )} \, dx}{(a-b)^2}\\ &=\frac{x \sqrt{2+x^2}}{3 (a-b) \left (1+x^2\right )^{3/2}}+\frac{2 b^2 \sqrt{1+x^2} \Pi \left (1-\frac{2 b}{a};\left .\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-1\right )}{a (a-b)^2 \sqrt{\frac{1+x^2}{2+x^2}} \sqrt{2+x^2}}+\frac{\int \frac{2 (2 a-5 b)+(a-4 b) x^2}{\left (1+x^2\right )^{3/2} \sqrt{2+x^2}} \, dx}{3 (a-b)^2}\\ &=\frac{x \sqrt{2+x^2}}{3 (a-b) \left (1+x^2\right )^{3/2}}+\frac{2 b^2 \sqrt{1+x^2} \Pi \left (1-\frac{2 b}{a};\left .\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-1\right )}{a (a-b)^2 \sqrt{\frac{1+x^2}{2+x^2}} \sqrt{2+x^2}}+\frac{(a-2 b) \int \frac{\sqrt{2+x^2}}{\left (1+x^2\right )^{3/2}} \, dx}{(a-b)^2}-\frac{2 \int \frac{1}{\sqrt{1+x^2} \sqrt{2+x^2}} \, dx}{3 (a-b)}\\ &=\frac{x \sqrt{2+x^2}}{3 (a-b) \left (1+x^2\right )^{3/2}}+\frac{\sqrt{2} (a-2 b) \sqrt{2+x^2} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{(a-b)^2 \sqrt{1+x^2} \sqrt{\frac{2+x^2}{1+x^2}}}-\frac{\sqrt{2} \sqrt{2+x^2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 (a-b) \sqrt{1+x^2} \sqrt{\frac{2+x^2}{1+x^2}}}+\frac{2 b^2 \sqrt{1+x^2} \Pi \left (1-\frac{2 b}{a};\left .\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-1\right )}{a (a-b)^2 \sqrt{\frac{1+x^2}{2+x^2}} \sqrt{2+x^2}}\\ \end{align*}
Mathematica [C] time = 0.377982, size = 357, normalized size = 1.66 \[ \frac{-i \sqrt{2} a \left (x^2+1\right )^2 (4 a-7 b) \text{EllipticF}\left (i \sinh ^{-1}(x),\frac{1}{2}\right )+6 a^2 \sqrt{x^2+1} \sqrt{x^2+2} x^3+8 a^2 \sqrt{x^2+1} \sqrt{x^2+2} x-6 i \sqrt{2} b^2 x^4 \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )-12 i \sqrt{2} b^2 x^2 \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )-6 i \sqrt{2} b^2 \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )-12 a b \sqrt{x^2+1} \sqrt{x^2+2} x^3-14 a b \sqrt{x^2+1} \sqrt{x^2+2} x+6 i \sqrt{2} a \left (x^2+1\right )^2 (a-2 b) E\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )+3 i \sqrt{2} a b x^4 \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )+6 i \sqrt{2} a b x^2 \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )+3 i \sqrt{2} a b \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )}{6 a \left (x^2+1\right )^2 (a-b)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 477, normalized size = 2.2 \begin{align*} -{\frac{1}{3\, \left ( a-b \right ) ^{2}a} \left ( 6\,i{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){x}^{2}ab\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-i{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) ab\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-3\,i{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){a}^{2}\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}+i{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){x}^{2}{a}^{2}\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-3\,i{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){x}^{2}{a}^{2}\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-3\,i{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ) ab\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-3\,{x}^{5}{a}^{2}+6\,{x}^{5}ab+6\,i{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) ab\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-i{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){x}^{2}ab\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-3\,i{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ){x}^{2}ab\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}+6\,i{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ){x}^{2}{b}^{2}\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}+i{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){a}^{2}\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}+6\,i{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ){b}^{2}\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-10\,{x}^{3}{a}^{2}+19\,{x}^{3}ab-8\,{a}^{2}x+14\,xab \right ){\frac{1}{\sqrt{{x}^{2}+2}}} \left ({x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \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}}{{\left (b x^{2} + a\right )}{\left (x^{2} + 1\right )}^{\frac{5}{2}}}\,{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^{8} +{\left (a + 3 \, b\right )} x^{6} + 3 \,{\left (a + b\right )} x^{4} +{\left (3 \, a + b\right )} x^{2} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}}{{\left (b x^{2} + a\right )}{\left (x^{2} + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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