3.61 \(\int \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right ) \, dx\)

Optimal. Leaf size=82 \[ \frac{5}{8} x \left (2 x^2-x+3\right )^{3/2}+\frac{73}{96} \left (2 x^2-x+3\right )^{3/2}-\frac{81}{512} (1-4 x) \sqrt{2 x^2-x+3}-\frac{1863 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1024 \sqrt{2}} \]

[Out]

(-81*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/512 + (73*(3 - x + 2*x^2)^(3/2))/96 + (5*x*(
3 - x + 2*x^2)^(3/2))/8 - (1863*ArcSinh[(1 - 4*x)/Sqrt[23]])/(1024*Sqrt[2])

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Rubi [A]  time = 0.0854822, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5}{8} x \left (2 x^2-x+3\right )^{3/2}+\frac{73}{96} \left (2 x^2-x+3\right )^{3/2}-\frac{81}{512} (1-4 x) \sqrt{2 x^2-x+3}-\frac{1863 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1024 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-81*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/512 + (73*(3 - x + 2*x^2)^(3/2))/96 + (5*x*(
3 - x + 2*x^2)^(3/2))/8 - (1863*ArcSinh[(1 - 4*x)/Sqrt[23]])/(1024*Sqrt[2])

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Rubi in Sympy [A]  time = 10.173, size = 73, normalized size = 0.89 \[ - \frac{81 \left (- 4 x + 1\right ) \sqrt{2 x^{2} - x + 3}}{512} + \frac{\left (30 x + \frac{73}{2}\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}{48} + \frac{1863 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x - 1\right )}{4 \sqrt{2 x^{2} - x + 3}} \right )}}{2048} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**2+3*x+2)*(2*x**2-x+3)**(1/2),x)

[Out]

-81*(-4*x + 1)*sqrt(2*x**2 - x + 3)/512 + (30*x + 73/2)*(2*x**2 - x + 3)**(3/2)/
48 + 1863*sqrt(2)*atanh(sqrt(2)*(4*x - 1)/(4*sqrt(2*x**2 - x + 3)))/2048

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Mathematica [A]  time = 0.0511416, size = 55, normalized size = 0.67 \[ \frac{4 \sqrt{2 x^2-x+3} \left (1920 x^3+1376 x^2+2684 x+3261\right )+5589 \sqrt{2} \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{6144} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(3261 + 2684*x + 1376*x^2 + 1920*x^3) + 5589*Sqrt[2]*ArcS
inh[(-1 + 4*x)/Sqrt[23]])/6144

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Maple [A]  time = 0.008, size = 64, normalized size = 0.8 \[{\frac{324\,x-81}{512}\sqrt{2\,{x}^{2}-x+3}}+{\frac{1863\,\sqrt{2}}{2048}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{73}{96} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{5\,x}{8} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2+3*x+2)*(2*x^2-x+3)^(1/2),x)

[Out]

81/512*(4*x-1)*(2*x^2-x+3)^(1/2)+1863/2048*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4)
)+73/96*(2*x^2-x+3)^(3/2)+5/8*x*(2*x^2-x+3)^(3/2)

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Maxima [A]  time = 0.771803, size = 101, normalized size = 1.23 \[ \frac{5}{8} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{73}{96} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{81}{128} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{1863}{2048} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{81}{512} \, \sqrt{2 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/8*(2*x^2 - x + 3)^(3/2)*x + 73/96*(2*x^2 - x + 3)^(3/2) + 81/128*sqrt(2*x^2 -
x + 3)*x + 1863/2048*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 81/512*sqrt(2*x^
2 - x + 3)

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Fricas [A]  time = 0.277699, size = 103, normalized size = 1.26 \[ \frac{1}{12288} \, \sqrt{2}{\left (4 \, \sqrt{2}{\left (1920 \, x^{3} + 1376 \, x^{2} + 2684 \, x + 3261\right )} \sqrt{2 \, x^{2} - x + 3} + 5589 \, \log \left (-\sqrt{2}{\left (32 \, x^{2} - 16 \, x + 25\right )} - 8 \, \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12288*sqrt(2)*(4*sqrt(2)*(1920*x^3 + 1376*x^2 + 2684*x + 3261)*sqrt(2*x^2 - x
+ 3) + 5589*log(-sqrt(2)*(32*x^2 - 16*x + 25) - 8*sqrt(2*x^2 - x + 3)*(4*x - 1))
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+3*x+2)*(2*x**2-x+3)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.270001, size = 85, normalized size = 1.04 \[ \frac{1}{1536} \,{\left (4 \,{\left (8 \,{\left (60 \, x + 43\right )} x + 671\right )} x + 3261\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{1863}{2048} \, \sqrt{2}{\rm ln}\left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1536*(4*(8*(60*x + 43)*x + 671)*x + 3261)*sqrt(2*x^2 - x + 3) - 1863/2048*sqrt
(2)*ln(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)