Optimal. Leaf size=215 \[ \frac{2 b^2 \sqrt{x^2+1} \Pi \left (1-\frac{2 b}{a};\left .\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-1\right )}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2} (a-b)^2}+\frac{x \sqrt{x^2+2}}{3 \left (x^2+1\right )^{3/2} (a-b)}-\frac{\sqrt{2} \sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)}+\frac{\sqrt{2} \sqrt{x^2+2} (a-2 b) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)^2} \]
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Rubi [A] time = 0.425159, 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 \[ \frac{2 b^2 \sqrt{x^2+1} \Pi \left (1-\frac{2 b}{a};\left .\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-1\right )}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2} (a-b)^2}+\frac{x \sqrt{x^2+2}}{3 \left (x^2+1\right )^{3/2} (a-b)}-\frac{\sqrt{2} \sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)}+\frac{\sqrt{2} \sqrt{x^2+2} (a-2 b) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)^2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2 + x^2]/((1 + x^2)^(5/2)*(a + b*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 55.113, size = 190, normalized size = 0.88 \[ \frac{x \sqrt{x^{2} + 2}}{3 \left (a - b\right ) \left (x^{2} + 1\right )^{\frac{3}{2}}} + \frac{\sqrt{2} \left (a - 2 b\right ) \sqrt{x^{2} + 2} E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{\sqrt{\frac{x^{2} + 2}{x^{2} + 1}} \left (a - b\right )^{2} \sqrt{x^{2} + 1}} - \frac{\sqrt{2} \sqrt{x^{2} + 2} F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{3 \sqrt{\frac{x^{2} + 2}{x^{2} + 1}} \left (a - b\right ) \sqrt{x^{2} + 1}} + \frac{2 \sqrt{2} b^{2} \sqrt{x^{2} + 1} \Pi \left (1 - \frac{2 b}{a}; \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -1\right )}{a \sqrt{\frac{2 x^{2} + 2}{x^{2} + 2}} \left (a - b\right )^{2} \sqrt{x^{2} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2+2)**(1/2)/(x**2+1)**(5/2)/(b*x**2+a),x)
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Mathematica [C] time = 0.47772, size = 357, normalized size = 1.66 \[ \frac{8 a^2 \sqrt{x^2+1} \sqrt{x^2+2} x+6 a^2 \sqrt{x^2+1} \sqrt{x^2+2} x^3-6 i \sqrt{2} b^2 x^4 \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )-12 i \sqrt{2} b^2 x^2 \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )-6 i \sqrt{2} b^2 \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )+3 i \sqrt{2} a b x^4 \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )-14 a b \sqrt{x^2+1} \sqrt{x^2+2} x-i \sqrt{2} a \left (x^2+1\right )^2 (4 a-7 b) F\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )+6 i \sqrt{2} a \left (x^2+1\right )^2 (a-2 b) E\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )+6 i \sqrt{2} a b x^2 \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )-12 a b \sqrt{x^2+1} \sqrt{x^2+2} x^3+3 i \sqrt{2} a b \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )}{6 a \left (x^2+1\right )^2 (a-b)^2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[2 + x^2]/((1 + x^2)^(5/2)*(a + b*x^2)),x]
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Maple [B] time = 0.051, size = 477, normalized size = 2.2 \[ -{\frac{1}{3\, \left ( a-b \right ) ^{2}a} \left ( -3\,i{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){a}^{2}\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}+6\,i{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ){x}^{2}{b}^{2}\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}+6\,i{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){x}^{2}ab\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}+6\,i{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ){b}^{2}\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-3\,i{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){x}^{2}{a}^{2}\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-i{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) ab\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-3\,{x}^{5}{a}^{2}+6\,{x}^{5}ab+i{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){a}^{2}\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}+6\,i{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) ab\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-3\,i{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ){x}^{2}ab\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-3\,i{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ) ab\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-i{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){x}^{2}ab\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}+i{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){x}^{2}{a}^{2}\sqrt{{x}^{2}+2}\sqrt{{x}^{2}+1}-10\,{x}^{3}{a}^{2}+19\,ab{x}^{3}-8\,x{a}^{2}+14\,abx \right ){\frac{1}{\sqrt{{x}^{2}+2}}} \left ({x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2+2)^(1/2)/(x^2+1)^(5/2)/(b*x^2+a),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} + 2}}{{\left (b x^{2} + a\right )}{\left (x^{2} + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 2)/((b*x^2 + a)*(x^2 + 1)^(5/2)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{x^{2} + 2}}{{\left (b x^{6} +{\left (a + 2 \, b\right )} x^{4} +{\left (2 \, a + b\right )} x^{2} + a\right )} \sqrt{x^{2} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 2)/((b*x^2 + a)*(x^2 + 1)^(5/2)),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2+2)**(1/2)/(x**2+1)**(5/2)/(b*x**2+a),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} + 2}}{{\left (b x^{2} + a\right )}{\left (x^{2} + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 2)/((b*x^2 + a)*(x^2 + 1)^(5/2)),x, algorithm="giac")
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