Optimal. Leaf size=55 \[ -\frac{13 \sqrt{3 x^2+2}}{35 (2 x+3)}-\frac{41 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{35 \sqrt{35}} \]
[Out]
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Rubi [A] time = 0.0764289, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{13 \sqrt{3 x^2+2}}{35 (2 x+3)}-\frac{41 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{35 \sqrt{35}} \]
Antiderivative was successfully verified.
[In] Int[(5 - x)/((3 + 2*x)^2*Sqrt[2 + 3*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 9.62118, size = 49, normalized size = 0.89 \[ - \frac{41 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{1225} - \frac{13 \sqrt{3 x^{2} + 2}}{35 \left (2 x + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5-x)/(3+2*x)**2/(3*x**2+2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.093619, size = 70, normalized size = 1.27 \[ \frac{-\frac{455 \sqrt{3 x^2+2}}{2 x+3}-41 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )+41 \sqrt{35} \log (2 x+3)}{1225} \]
Antiderivative was successfully verified.
[In] Integrate[(5 - x)/((3 + 2*x)^2*Sqrt[2 + 3*x^2]),x]
[Out]
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Maple [A] time = 0.016, size = 53, normalized size = 1. \[ -{\frac{13}{70}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{41\,\sqrt{35}}{1225}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5-x)/(2*x+3)^2/(3*x^2+2)^(1/2),x)
[Out]
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Maxima [A] time = 0.762794, size = 72, normalized size = 1.31 \[ \frac{41}{1225} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) - \frac{13 \, \sqrt{3 \, x^{2} + 2}}{35 \,{\left (2 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - 5)/(sqrt(3*x^2 + 2)*(2*x + 3)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.278601, size = 108, normalized size = 1.96 \[ \frac{\sqrt{35}{\left (41 \,{\left (2 \, x + 3\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right ) - 26 \, \sqrt{35} \sqrt{3 \, x^{2} + 2}\right )}}{2450 \,{\left (2 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - 5)/(sqrt(3*x^2 + 2)*(2*x + 3)^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{4 x^{2} \sqrt{3 x^{2} + 2} + 12 x \sqrt{3 x^{2} + 2} + 9 \sqrt{3 x^{2} + 2}}\, dx - \int \left (- \frac{5}{4 x^{2} \sqrt{3 x^{2} + 2} + 12 x \sqrt{3 x^{2} + 2} + 9 \sqrt{3 x^{2} + 2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5-x)/(3+2*x)**2/(3*x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{x - 5}{\sqrt{3 \, x^{2} + 2}{\left (2 \, x + 3\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - 5)/(sqrt(3*x^2 + 2)*(2*x + 3)^2),x, algorithm="giac")
[Out]