3.89 \(\int \frac{2+3 x+5 x^2}{\left (3-x+2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{11 (3 x+5)}{23 \sqrt{2 x^2-x+3}}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2 \sqrt{2}} \]

[Out]

(-11*(5 + 3*x))/(23*Sqrt[3 - x + 2*x^2]) - (5*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2*Sq
rt[2])

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Rubi [A]  time = 0.0544134, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{11 (3 x+5)}{23 \sqrt{2 x^2-x+3}}-\frac{5 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x + 5*x^2)/(3 - x + 2*x^2)^(3/2),x]

[Out]

(-11*(5 + 3*x))/(23*Sqrt[3 - x + 2*x^2]) - (5*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2*Sq
rt[2])

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Rubi in Sympy [A]  time = 9.59914, size = 51, normalized size = 1.13 \[ - \frac{33 x + 55}{23 \sqrt{2 x^{2} - x + 3}} + \frac{5 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x - 1\right )}{4 \sqrt{2 x^{2} - x + 3}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**2+3*x+2)/(2*x**2-x+3)**(3/2),x)

[Out]

-(33*x + 55)/(23*sqrt(2*x**2 - x + 3)) + 5*sqrt(2)*atanh(sqrt(2)*(4*x - 1)/(4*sq
rt(2*x**2 - x + 3)))/4

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Mathematica [A]  time = 0.0520673, size = 45, normalized size = 1. \[ \frac{5 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{2 \sqrt{2}}-\frac{11 (3 x+5)}{23 \sqrt{2 x^2-x+3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x + 5*x^2)/(3 - x + 2*x^2)^(3/2),x]

[Out]

(-11*(5 + 3*x))/(23*Sqrt[3 - x + 2*x^2]) + (5*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(2*S
qrt[2])

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Maple [A]  time = 0.007, size = 64, normalized size = 1.4 \[{\frac{196\,x-49}{184}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{17}{8}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{5\,x}{2}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{5\,\sqrt{2}}{4}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2+3*x+2)/(2*x^2-x+3)^(3/2),x)

[Out]

49/184*(4*x-1)/(2*x^2-x+3)^(1/2)-17/8/(2*x^2-x+3)^(1/2)-5/2*x/(2*x^2-x+3)^(1/2)+
5/4*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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Maxima [A]  time = 0.781412, size = 62, normalized size = 1.38 \[ \frac{5}{4} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{33 \, x}{23 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{55}{23 \, \sqrt{2 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 3*x + 2)/(2*x^2 - x + 3)^(3/2),x, algorithm="maxima")

[Out]

5/4*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 33/23*x/sqrt(2*x^2 - x + 3) - 55/
23/sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 0.277572, size = 119, normalized size = 2.64 \[ -\frac{\sqrt{2}{\left (44 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (3 \, x + 5\right )} - 115 \,{\left (2 \, x^{2} - x + 3\right )} \log \left (-\sqrt{2}{\left (32 \, x^{2} - 16 \, x + 25\right )} - 8 \, \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )}\right )\right )}}{184 \,{\left (2 \, x^{2} - x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 3*x + 2)/(2*x^2 - x + 3)^(3/2),x, algorithm="fricas")

[Out]

-1/184*sqrt(2)*(44*sqrt(2)*sqrt(2*x^2 - x + 3)*(3*x + 5) - 115*(2*x^2 - x + 3)*l
og(-sqrt(2)*(32*x^2 - 16*x + 25) - 8*sqrt(2*x^2 - x + 3)*(4*x - 1)))/(2*x^2 - x
+ 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{5 x^{2} + 3 x + 2}{\left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+3*x+2)/(2*x**2-x+3)**(3/2),x)

[Out]

Integral((5*x**2 + 3*x + 2)/(2*x**2 - x + 3)**(3/2), x)

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GIAC/XCAS [A]  time = 0.272084, size = 72, normalized size = 1.6 \[ -\frac{5}{4} \, \sqrt{2}{\rm ln}\left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac{11 \,{\left (3 \, x + 5\right )}}{23 \, \sqrt{2 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 3*x + 2)/(2*x^2 - x + 3)^(3/2),x, algorithm="giac")

[Out]

-5/4*sqrt(2)*ln(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) - 11/23*(3*x +
5)/sqrt(2*x^2 - x + 3)