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

Optimal. Leaf size=124 \[ \frac{1825}{64} \sqrt{2 x^2-x+3} x^2+\frac{15565}{512} \sqrt{2 x^2-x+3} x-\frac{181561 \sqrt{2 x^2-x+3}}{2048}-\frac{1331 (17-45 x)}{368 \sqrt{2 x^2-x+3}}+\frac{125}{16} \sqrt{2 x^2-x+3} x^3+\frac{1168881 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4096 \sqrt{2}} \]

[Out]

(-1331*(17 - 45*x))/(368*Sqrt[3 - x + 2*x^2]) - (181561*Sqrt[3 - x + 2*x^2])/204
8 + (15565*x*Sqrt[3 - x + 2*x^2])/512 + (1825*x^2*Sqrt[3 - x + 2*x^2])/64 + (125
*x^3*Sqrt[3 - x + 2*x^2])/16 + (1168881*ArcSinh[(1 - 4*x)/Sqrt[23]])/(4096*Sqrt[
2])

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Rubi [A]  time = 0.217738, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{1825}{64} \sqrt{2 x^2-x+3} x^2+\frac{15565}{512} \sqrt{2 x^2-x+3} x-\frac{181561 \sqrt{2 x^2-x+3}}{2048}-\frac{1331 (17-45 x)}{368 \sqrt{2 x^2-x+3}}+\frac{125}{16} \sqrt{2 x^2-x+3} x^3+\frac{1168881 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4096 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-1331*(17 - 45*x))/(368*Sqrt[3 - x + 2*x^2]) - (181561*Sqrt[3 - x + 2*x^2])/204
8 + (15565*x*Sqrt[3 - x + 2*x^2])/512 + (1825*x^2*Sqrt[3 - x + 2*x^2])/64 + (125
*x^3*Sqrt[3 - x + 2*x^2])/16 + (1168881*ArcSinh[(1 - 4*x)/Sqrt[23]])/(4096*Sqrt[
2])

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Rubi in Sympy [A]  time = 72.1264, size = 144, normalized size = 1.16 \[ - \frac{\left (- 476859375 x + \frac{7171882875}{4}\right ) \sqrt{2 x^{2} - x + 3}}{13248000} - \frac{2 \left (- 4 x + 1\right ) \left (5 x^{2} + 3 x + 2\right )^{3}}{23 \sqrt{2 x^{2} - x + 3}} - \frac{\left (6000 x + 5100\right ) \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{2}}{6900} + \frac{\left (8527500 x + 24325875\right ) \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )}{1656000} - \frac{1168881 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x - 1\right )}{4 \sqrt{2 x^{2} - x + 3}} \right )}}{8192} \]

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)**(3/2),x)

[Out]

-(-476859375*x + 7171882875/4)*sqrt(2*x**2 - x + 3)/13248000 - 2*(-4*x + 1)*(5*x
**2 + 3*x + 2)**3/(23*sqrt(2*x**2 - x + 3)) - (6000*x + 5100)*sqrt(2*x**2 - x +
3)*(5*x**2 + 3*x + 2)**2/6900 + (8527500*x + 24325875)*sqrt(2*x**2 - x + 3)*(5*x
**2 + 3*x + 2)/1656000 - 1168881*sqrt(2)*atanh(sqrt(2)*(4*x - 1)/(4*sqrt(2*x**2
- x + 3)))/8192

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Mathematica [A]  time = 0.0972838, size = 65, normalized size = 0.52 \[ \frac{\frac{4 \left (736000 x^5+2318400 x^4+2624760 x^3-5754186 x^2+16138403 x-15423965\right )}{\sqrt{2 x^2-x+3}}-26884263 \sqrt{2} \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{188416} \]

Antiderivative was successfully verified.

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

[Out]

((4*(-15423965 + 16138403*x - 5754186*x^2 + 2624760*x^3 + 2318400*x^4 + 736000*x
^5))/Sqrt[3 - x + 2*x^2] - 26884263*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]])/188416

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Maple [A]  time = 0.01, size = 132, normalized size = 1.1 \[{\frac{21570172\,x-5392543}{376832}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{5130399}{16384}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{1168881\,x}{4096}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{1168881\,\sqrt{2}}{8192}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{125091\,{x}^{2}}{1024}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{14265\,{x}^{3}}{256}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{1575\,{x}^{4}}{32}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{125\,{x}^{5}}{8}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5392543/376832*(4*x-1)/(2*x^2-x+3)^(1/2)-5130399/16384/(2*x^2-x+3)^(1/2)+1168881
/4096*x/(2*x^2-x+3)^(1/2)-1168881/8192*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-12
5091/1024*x^2/(2*x^2-x+3)^(1/2)+14265/256*x^3/(2*x^2-x+3)^(1/2)+1575/32*x^4/(2*x
^2-x+3)^(1/2)+125/8*x^5/(2*x^2-x+3)^(1/2)

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Maxima [A]  time = 0.780091, size = 154, normalized size = 1.24 \[ \frac{125 \, x^{5}}{8 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{1575 \, x^{4}}{32 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{14265 \, x^{3}}{256 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{125091 \, x^{2}}{1024 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{1168881}{8192} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{16138403 \, x}{47104 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{15423965}{47104 \, \sqrt{2 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/8*x^5/sqrt(2*x^2 - x + 3) + 1575/32*x^4/sqrt(2*x^2 - x + 3) + 14265/256*x^3/
sqrt(2*x^2 - x + 3) - 125091/1024*x^2/sqrt(2*x^2 - x + 3) - 1168881/8192*sqrt(2)
*arcsinh(1/23*sqrt(23)*(4*x - 1)) + 16138403/47104*x/sqrt(2*x^2 - x + 3) - 15423
965/47104/sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 0.285335, size = 146, normalized size = 1.18 \[ \frac{\sqrt{2}{\left (4 \, \sqrt{2}{\left (736000 \, x^{5} + 2318400 \, x^{4} + 2624760 \, x^{3} - 5754186 \, x^{2} + 16138403 \, x - 15423965\right )} \sqrt{2 \, x^{2} - x + 3} + 26884263 \,{\left (2 \, x^{2} - x + 3\right )} \log \left (-\sqrt{2}{\left (32 \, x^{2} - 16 \, x + 25\right )} + 8 \, \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )}\right )\right )}}{376832 \,{\left (2 \, x^{2} - x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/376832*sqrt(2)*(4*sqrt(2)*(736000*x^5 + 2318400*x^4 + 2624760*x^3 - 5754186*x^
2 + 16138403*x - 15423965)*sqrt(2*x^2 - x + 3) + 26884263*(2*x^2 - x + 3)*log(-s
qrt(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. \[ \int \frac{\left (5 x^{2} + 3 x + 2\right )^{3}}{\left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272944, size = 97, normalized size = 0.78 \[ \frac{1168881}{8192} \, \sqrt{2}{\rm ln}\left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{{\left (46 \,{\left (20 \,{\left (40 \,{\left (20 \, x + 63\right )} x + 2853\right )} x - 125091\right )} x + 16138403\right )} x - 15423965}{47104 \, \sqrt{2 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1168881/8192*sqrt(2)*ln(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/47
104*((46*(20*(40*(20*x + 63)*x + 2853)*x - 125091)*x + 16138403)*x - 15423965)/s
qrt(2*x^2 - x + 3)