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

Optimal. Leaf size=143 \[ -\frac{3387 \sqrt{2 x^2-x+3} x^2}{1024}-\frac{372783 \sqrt{2 x^2-x+3} x}{8192}-\frac{203373 \sqrt{2 x^2-x+3}}{32768}+\frac{125}{12} \sqrt{2 x^2-x+3} x^5+\frac{1355}{48} \sqrt{2 x^2-x+3} x^4+\frac{8185}{256} \sqrt{2 x^2-x+3} x^3-\frac{9267707 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{65536 \sqrt{2}} \]

[Out]

(-203373*Sqrt[3 - x + 2*x^2])/32768 - (372783*x*Sqrt[3 - x + 2*x^2])/8192 - (338
7*x^2*Sqrt[3 - x + 2*x^2])/1024 + (8185*x^3*Sqrt[3 - x + 2*x^2])/256 + (1355*x^4
*Sqrt[3 - x + 2*x^2])/48 + (125*x^5*Sqrt[3 - x + 2*x^2])/12 - (9267707*ArcSinh[(
1 - 4*x)/Sqrt[23]])/(65536*Sqrt[2])

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Rubi [A]  time = 0.286926, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{3387 \sqrt{2 x^2-x+3} x^2}{1024}-\frac{372783 \sqrt{2 x^2-x+3} x}{8192}-\frac{203373 \sqrt{2 x^2-x+3}}{32768}+\frac{125}{12} \sqrt{2 x^2-x+3} x^5+\frac{1355}{48} \sqrt{2 x^2-x+3} x^4+\frac{8185}{256} \sqrt{2 x^2-x+3} x^3-\frac{9267707 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{65536 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-203373*Sqrt[3 - x + 2*x^2])/32768 - (372783*x*Sqrt[3 - x + 2*x^2])/8192 - (338
7*x^2*Sqrt[3 - x + 2*x^2])/1024 + (8185*x^3*Sqrt[3 - x + 2*x^2])/256 + (1355*x^4
*Sqrt[3 - x + 2*x^2])/48 + (125*x^5*Sqrt[3 - x + 2*x^2])/12 - (9267707*ArcSinh[(
1 - 4*x)/Sqrt[23]])/(65536*Sqrt[2])

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Rubi in Sympy [A]  time = 45.5465, size = 124, normalized size = 0.87 \[ \frac{\left (- \frac{11685 x}{2} + \frac{257403}{8}\right ) \left (- \frac{779 x^{2}}{4} + \frac{16103 x}{4} + \frac{2207}{2}\right ) \sqrt{2 x^{2} - x + 3}}{1121760} + \frac{\left (50 x + \frac{151}{2}\right ) \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{2}}{120} - \frac{\left (\frac{47223864585 x}{32} + \frac{46377749763}{128}\right ) \sqrt{2 x^{2} - x + 3}}{8974080} + \frac{9267707 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x - 1\right )}{4 \sqrt{2 x^{2} - x + 3}} \right )}}{131072} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-11685*x/2 + 257403/8)*(-779*x**2/4 + 16103*x/4 + 2207/2)*sqrt(2*x**2 - x + 3)/
1121760 + (50*x + 151/2)*sqrt(2*x**2 - x + 3)*(5*x**2 + 3*x + 2)**2/120 - (47223
864585*x/32 + 46377749763/128)*sqrt(2*x**2 - x + 3)/8974080 + 9267707*sqrt(2)*at
anh(sqrt(2)*(4*x - 1)/(4*sqrt(2*x**2 - x + 3)))/131072

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Mathematica [A]  time = 0.0860002, size = 65, normalized size = 0.45 \[ \frac{4 \sqrt{2 x^2-x+3} \left (1024000 x^5+2775040 x^4+3143040 x^3-325152 x^2-4473396 x-610119\right )[phone] \sqrt{2} \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{393216} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-610119 - 4473396*x - 325152*x^2 + 3143040*x^3 + 2775040
*x^4 + 1024000*x^5) + 27803121*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]])/393216

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Maple [A]  time = 0.009, size = 113, normalized size = 0.8 \[{\frac{9267707\,\sqrt{2}}{131072}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{203373}{32768}\sqrt{2\,{x}^{2}-x+3}}-{\frac{372783\,x}{8192}\sqrt{2\,{x}^{2}-x+3}}-{\frac{3387\,{x}^{2}}{1024}\sqrt{2\,{x}^{2}-x+3}}+{\frac{8185\,{x}^{3}}{256}\sqrt{2\,{x}^{2}-x+3}}+{\frac{1355\,{x}^{4}}{48}\sqrt{2\,{x}^{2}-x+3}}+{\frac{125\,{x}^{5}}{12}\sqrt{2\,{x}^{2}-x+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9267707/131072*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-203373/32768*(2*x^2-x+3)^(
1/2)-372783/8192*x*(2*x^2-x+3)^(1/2)-3387/1024*x^2*(2*x^2-x+3)^(1/2)+8185/256*x^
3*(2*x^2-x+3)^(1/2)+1355/48*x^4*(2*x^2-x+3)^(1/2)+125/12*x^5*(2*x^2-x+3)^(1/2)

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Maxima [A]  time = 0.790606, size = 154, normalized size = 1.08 \[ \frac{125}{12} \, \sqrt{2 \, x^{2} - x + 3} x^{5} + \frac{1355}{48} \, \sqrt{2 \, x^{2} - x + 3} x^{4} + \frac{8185}{256} \, \sqrt{2 \, x^{2} - x + 3} x^{3} - \frac{3387}{1024} \, \sqrt{2 \, x^{2} - x + 3} x^{2} - \frac{372783}{8192} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{9267707}{131072} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{203373}{32768} \, \sqrt{2 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/12*sqrt(2*x^2 - x + 3)*x^5 + 1355/48*sqrt(2*x^2 - x + 3)*x^4 + 8185/256*sqrt
(2*x^2 - x + 3)*x^3 - 3387/1024*sqrt(2*x^2 - x + 3)*x^2 - 372783/8192*sqrt(2*x^2
- x + 3)*x + 9267707/131072*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 203373/3
2768*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 0.280654, size = 116, normalized size = 0.81 \[ \frac{1}{786432} \, \sqrt{2}{\left (4 \, \sqrt{2}{\left (1024000 \, x^{5} + 2775040 \, x^{4} + 3143040 \, x^{3} - 325152 \, x^{2} - 4473396 \, x - 610119\right )} \sqrt{2 \, x^{2} - x + 3} + 27803121 \, \log \left (-\sqrt{2}{\left (32 \, x^{2} - 16 \, x + 25\right )} - 8 \, \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/786432*sqrt(2)*(4*sqrt(2)*(1024000*x^5 + 2775040*x^4 + 3143040*x^3 - 325152*x^
2 - 4473396*x - 610119)*sqrt(2*x^2 - x + 3) + 27803121*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. \[ \int \frac{\left (5 x^{2} + 3 x + 2\right )^{3}}{\sqrt{2 x^{2} - x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.27489, size = 99, normalized size = 0.69 \[ \frac{1}{98304} \,{\left (4 \,{\left (8 \,{\left (20 \,{\left (16 \,{\left (100 \, x + 271\right )} x + 4911\right )} x - 10161\right )} x - 1118349\right )} x - 610119\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{9267707}{131072} \, \sqrt{2}{\rm ln}\left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/98304*(4*(8*(20*(16*(100*x + 271)*x + 4911)*x - 10161)*x - 1118349)*x - 610119
)*sqrt(2*x^2 - x + 3) - 9267707/131072*sqrt(2)*ln(-2*sqrt(2)*(sqrt(2)*x - sqrt(2
*x^2 - x + 3)) + 1)