3.86 \(\int \frac{\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=166 \[ -\frac{111315 \sqrt{2 x^2-x+3} x^2}{2048}-\frac{8992487 \sqrt{2 x^2-x+3} x}{16384}-\frac{31009685 \sqrt{2 x^2-x+3}}{65536}-\frac{14641 (79 x+101)}{1472 \sqrt{2 x^2-x+3}}+\frac{625}{24} \sqrt{2 x^2-x+3} x^5+\frac{10075}{96} \sqrt{2 x^2-x+3} x^4+\frac{79425}{512} \sqrt{2 x^2-x+3} x^3-\frac{310445587 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{131072 \sqrt{2}} \]

[Out]

(-14641*(101 + 79*x))/(1472*Sqrt[3 - x + 2*x^2]) - (31009685*Sqrt[3 - x + 2*x^2]
)/65536 - (8992487*x*Sqrt[3 - x + 2*x^2])/16384 - (111315*x^2*Sqrt[3 - x + 2*x^2
])/2048 + (79425*x^3*Sqrt[3 - x + 2*x^2])/512 + (10075*x^4*Sqrt[3 - x + 2*x^2])/
96 + (625*x^5*Sqrt[3 - x + 2*x^2])/24 - (310445587*ArcSinh[(1 - 4*x)/Sqrt[23]])/
(131072*Sqrt[2])

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Rubi [A]  time = 0.335245, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{111315 \sqrt{2 x^2-x+3} x^2}{2048}-\frac{8992487 \sqrt{2 x^2-x+3} x}{16384}-\frac{31009685 \sqrt{2 x^2-x+3}}{65536}-\frac{14641 (79 x+101)}{1472 \sqrt{2 x^2-x+3}}+\frac{625}{24} \sqrt{2 x^2-x+3} x^5+\frac{10075}{96} \sqrt{2 x^2-x+3} x^4+\frac{79425}{512} \sqrt{2 x^2-x+3} x^3-\frac{310445587 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{131072 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x + 5*x^2)^4/(3 - x + 2*x^2)^(3/2),x]

[Out]

(-14641*(101 + 79*x))/(1472*Sqrt[3 - x + 2*x^2]) - (31009685*Sqrt[3 - x + 2*x^2]
)/65536 - (8992487*x*Sqrt[3 - x + 2*x^2])/16384 - (111315*x^2*Sqrt[3 - x + 2*x^2
])/2048 + (79425*x^3*Sqrt[3 - x + 2*x^2])/512 + (10075*x^4*Sqrt[3 - x + 2*x^2])/
96 + (625*x^5*Sqrt[3 - x + 2*x^2])/24 - (310445587*ArcSinh[(1 - 4*x)/Sqrt[23]])/
(131072*Sqrt[2])

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Rubi in Sympy [A]  time = 97.0207, size = 175, normalized size = 1.05 \[ - \frac{2 \left (- 4 x + 1\right ) \left (5 x^{2} + 3 x + 2\right )^{4}}{23 \sqrt{2 x^{2} - x + 3}} - \frac{\left (11200 x + 9520\right ) \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{3}}{12880} + \frac{\left (35770000 x + 99473500\right ) \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{2}}{7728000} - \frac{\left (56221252500 x + 33377245125\right ) \left (- 1874041750 x^{2} + 4439548750 x + 1495749500\right ) \sqrt{2 x^{2} - x + 3}}{695164542912000000} - \frac{\left (4040855648915524031250 x + \frac{8690385309739832671875}{2}\right ) \sqrt{2 x^{2} - x + 3}}{5561316343296000000} + \frac{310445587 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x - 1\right )}{4 \sqrt{2 x^{2} - x + 3}} \right )}}{262144} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**2+3*x+2)**4/(2*x**2-x+3)**(3/2),x)

[Out]

-2*(-4*x + 1)*(5*x**2 + 3*x + 2)**4/(23*sqrt(2*x**2 - x + 3)) - (11200*x + 9520)
*sqrt(2*x**2 - x + 3)*(5*x**2 + 3*x + 2)**3/12880 + (35770000*x + 99473500)*sqrt
(2*x**2 - x + 3)*(5*x**2 + 3*x + 2)**2/7728000 - (56221252500*x + 33377245125)*(
-1874041750*x**2 + 4439548750*x + 1495749500)*sqrt(2*x**2 - x + 3)/6951645429120
00000 - (4040855648915524031250*x + 8690385309739832671875/2)*sqrt(2*x**2 - x +
3)/5561316343296000000 + 310445587*sqrt(2)*atanh(sqrt(2)*(4*x - 1)/(4*sqrt(2*x**
2 - x + 3)))/262144

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Mathematica [A]  time = 0.0946359, size = 95, normalized size = 0.57 \[ \sqrt{2 x^2-x+3} \left (\frac{625 x^5}{24}+\frac{10075 x^4}{96}+\frac{79425 x^3}{512}-\frac{111315 x^2}{2048}-\frac{14641 (79 x+101)}{1472 \left (2 x^2-x+3\right )}-\frac{8992487 x}{16384}-\frac{31009685}{65536}\right )+\frac{310445587 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{131072 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x + 5*x^2)^4/(3 - x + 2*x^2)^(3/2),x]

[Out]

Sqrt[3 - x + 2*x^2]*(-31009685/65536 - (8992487*x)/16384 - (111315*x^2)/2048 + (
79425*x^3)/512 + (10075*x^4)/96 + (625*x^5)/24 - (14641*(101 + 79*x))/(1472*(3 -
x + 2*x^2))) + (310445587*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(131072*Sqrt[2])

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Maple [A]  time = 0.036, size = 166, normalized size = 1. \[{\frac{4936178060\,x-1234044515}{12058624}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{1217267299}{524288}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{310445587\,x}{131072}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{310445587\,\sqrt{2}}{262144}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{18367831\,{x}^{2}}{32768}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{4734827\,{x}^{3}}{8192}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{52235\,{x}^{4}}{1024}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{217675\,{x}^{5}}{768}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{8825\,{x}^{6}}{48}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{625\,{x}^{7}}{12}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2+3*x+2)^4/(2*x^2-x+3)^(3/2),x)

[Out]

1234044515/12058624*(4*x-1)/(2*x^2-x+3)^(1/2)-1217267299/524288/(2*x^2-x+3)^(1/2
)-310445587/131072*x/(2*x^2-x+3)^(1/2)[phone]/262144*2^(1/2)*arcsinh(4/23*23^
(1/2)*(x-1/4))-18367831/32768*x^2/(2*x^2-x+3)^(1/2)-4734827/8192*x^3/(2*x^2-x+3)
^(1/2)+52235/1024*x^4/(2*x^2-x+3)^(1/2)+217675/768*x^5/(2*x^2-x+3)^(1/2)+8825/48
*x^6/(2*x^2-x+3)^(1/2)+625/12*x^7/(2*x^2-x+3)^(1/2)

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Maxima [A]  time = 0.781994, size = 200, normalized size = 1.2 \[ \frac{625 \, x^{7}}{12 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{8825 \, x^{6}}{48 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{217675 \, x^{5}}{768 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{52235 \, x^{4}}{1024 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{4734827 \, x^{3}}{8192 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{18367831 \, x^{2}}{32768 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{310445587}{262144} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{2953101993 \, x}{1507328 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{3653899049}{1507328 \, \sqrt{2 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 3*x + 2)^4/(2*x^2 - x + 3)^(3/2),x, algorithm="maxima")

[Out]

625/12*x^7/sqrt(2*x^2 - x + 3) + 8825/48*x^6/sqrt(2*x^2 - x + 3) + 217675/768*x^
5/sqrt(2*x^2 - x + 3) + 52235/1024*x^4/sqrt(2*x^2 - x + 3) - 4734827/8192*x^3/sq
rt(2*x^2 - x + 3) - 18367831/32768*x^2/sqrt(2*x^2 - x + 3) + 310445587/262144*sq
rt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 2953101993/1507328*x/sqrt(2*x^2 - x + 3
) - 3653899049/1507328/sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 0.290256, size = 159, normalized size = 0.96 \[ \frac{\sqrt{2}{\left (4 \, \sqrt{2}{\left (235520000 \, x^{7} + 831385600 \, x^{6} + 1281670400 \, x^{5} + 230669760 \, x^{4} - 2613624504 \, x^{3} - 2534760678 \, x^{2} - 8859305979 \, x - 10961697147\right )} \sqrt{2 \, x^{2} - x + 3} + 21420745503 \,{\left (2 \, x^{2} - x + 3\right )} \log \left (-\sqrt{2}{\left (32 \, x^{2} - 16 \, x + 25\right )} - 8 \, \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )}\right )\right )}}{36175872 \,{\left (2 \, x^{2} - x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 3*x + 2)^4/(2*x^2 - x + 3)^(3/2),x, algorithm="fricas")

[Out]

1/36175872*sqrt(2)*(4*sqrt(2)*(235520000*x^7 + 831385600*x^6 + 1281670400*x^5 +
230669760*x^4 - 2613624504*x^3 - 2534760678*x^2 - 8859305979*x - 10961697147)*sq
rt(2*x^2 - x + 3) + 21420745503*(2*x^2 - x + 3)*log(-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. \[ \int \frac{\left (5 x^{2} + 3 x + 2\right )^{4}}{\left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+3*x+2)**4/(2*x**2-x+3)**(3/2),x)

[Out]

Integral((5*x**2 + 3*x + 2)**4/(2*x**2 - x + 3)**(3/2), x)

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GIAC/XCAS [A]  time = 0.273171, size = 111, normalized size = 0.67 \[ -\frac{310445587}{262144} \, \sqrt{2}{\rm ln}\left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{{\left (46 \,{\left (4 \,{\left (40 \,{\left (20 \,{\left (16 \,{\left (100 \, x + 353\right )} x + 8707\right )} x + 31341\right )} x - 14204481\right )} x - 55103493\right )} x - 8859305979\right )} x - 10961697147}{4521984 \, \sqrt{2 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 3*x + 2)^4/(2*x^2 - x + 3)^(3/2),x, algorithm="giac")

[Out]

-310445587/262144*sqrt(2)*ln(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) +
1/4521984*((46*(4*(40*(20*(16*(100*x + 353)*x + 8707)*x + 31341)*x - 14204481)*
x - 55103493)*x - 8859305979)*x - 10961697147)/sqrt(2*x^2 - x + 3)