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

Optimal. Leaf size=105 \[ \frac{121 (10679-6744 x)}{8464 \sqrt{2 x^2-x+3}}+\frac{125}{16} x \sqrt{2 x^2-x+3}+\frac{3175}{64} \sqrt{2 x^2-x+3}-\frac{1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}-\frac{7495 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}} \]

[Out]

(-1331*(17 - 45*x))/(1104*(3 - x + 2*x^2)^(3/2)) + (121*(10679 - 6744*x))/(8464*
Sqrt[3 - x + 2*x^2]) + (3175*Sqrt[3 - x + 2*x^2])/64 + (125*x*Sqrt[3 - x + 2*x^2
])/16 - (7495*ArcSinh[(1 - 4*x)/Sqrt[23]])/(128*Sqrt[2])

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Rubi [A]  time = 0.176466, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{121 (10679-6744 x)}{8464 \sqrt{2 x^2-x+3}}+\frac{125}{16} x \sqrt{2 x^2-x+3}+\frac{3175}{64} \sqrt{2 x^2-x+3}-\frac{1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}-\frac{7495 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-1331*(17 - 45*x))/(1104*(3 - x + 2*x^2)^(3/2)) + (121*(10679 - 6744*x))/(8464*
Sqrt[3 - x + 2*x^2]) + (3175*Sqrt[3 - x + 2*x^2])/64 + (125*x*Sqrt[3 - x + 2*x^2
])/16 - (7495*ArcSinh[(1 - 4*x)/Sqrt[23]])/(128*Sqrt[2])

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Rubi in Sympy [A]  time = 72.9896, size = 144, normalized size = 1.37 \[ - \frac{2 \left (- 1116 x + 26\right ) \left (5 x^{2} + 3 x + 2\right )^{2}}{1587 \sqrt{2 x^{2} - x + 3}} - \frac{2 \left (- 4 x + 1\right ) \left (5 x^{2} + 3 x + 2\right )^{3}}{69 \left (2 x^{2} - x + 3\right )^{\frac{3}{2}}} - \frac{\left (807600 x + 990060\right ) \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )}{190440} + \frac{\left (29748900 x + 165587715\right ) \sqrt{2 x^{2} - x + 3}}{1523520} + \frac{7495 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x - 1\right )}{4 \sqrt{2 x^{2} - x + 3}} \right )}}{256} \]

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)**(5/2),x)

[Out]

-2*(-1116*x + 26)*(5*x**2 + 3*x + 2)**2/(1587*sqrt(2*x**2 - x + 3)) - 2*(-4*x +
1)*(5*x**2 + 3*x + 2)**3/(69*(2*x**2 - x + 3)**(3/2)) - (807600*x + 990060)*sqrt
(2*x**2 - x + 3)*(5*x**2 + 3*x + 2)/190440 + (29748900*x + 165587715)*sqrt(2*x**
2 - x + 3)/1523520 + 7495*sqrt(2)*atanh(sqrt(2)*(4*x - 1)/(4*sqrt(2*x**2 - x + 3
)))/256

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Mathematica [A]  time = 0.11998, size = 65, normalized size = 0.62 \[ \frac{3174000 x^5+16980900 x^4-29423976 x^3+101546529 x^2-62463282 x+89784565}{101568 \left (2 x^2-x+3\right )^{3/2}}+\frac{7495 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{128 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(89784565 - 62463282*x + 101546529*x^2 - 29423976*x^3 + 16980900*x^4 + 3174000*x
^5)/(101568*(3 - x + 2*x^2)^(3/2)) + (7495*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(128*Sq
rt[2])

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Maple [B]  time = 0.01, size = 180, normalized size = 1.7 \[ -{\frac{56326844\,x-14081711}{565248} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{13564556\,x-3391139}{203136}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{20961031}{24576} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{281177\,x}{2048} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{222809\,{x}^{2}}{256} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{7495\,{x}^{3}}{192} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{7495\,x}{128}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{7495}{512}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{7495\,\sqrt{2}}{256}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{2675\,{x}^{4}}{16} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{125\,{x}^{5}}{4} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-14081711/565248*(4*x-1)/(2*x^2-x+3)^(3/2)-3391139/203136*(4*x-1)/(2*x^2-x+3)^(1
/2)[phone]/24576/(2*x^2-x+3)^(3/2)-281177/2048*x/(2*x^2-x+3)^(3/2)+222809/256*
x^2/(2*x^2-x+3)^(3/2)-7495/192*x^3/(2*x^2-x+3)^(3/2)-7495/128*x/(2*x^2-x+3)^(1/2
)-7495/512/(2*x^2-x+3)^(1/2)+7495/256*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+267
5/16*x^4/(2*x^2-x+3)^(3/2)+125/4*x^5/(2*x^2-x+3)^(3/2)

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Maxima [A]  time = 0.77999, size = 296, normalized size = 2.82 \[ \frac{125 \, x^{5}}{4 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2675 \, x^{4}}{16 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{7495}{203136} \, x{\left (\frac{284 \, x}{\sqrt{2 \, x^{2} - x + 3}} - \frac{3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{71}{\sqrt{2 \, x^{2} - x + 3}} + \frac{805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}\right )} + \frac{7495}{256} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{532145}{101568} \, \sqrt{2 \, x^{2} - x + 3} - \frac{4515389 \, x}{50784 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{7197 \, x^{2}}{8 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{396211}{50784 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{269783 \, x}{1104 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{1002137}{1104 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/4*x^5/(2*x^2 - x + 3)^(3/2) + 2675/16*x^4/(2*x^2 - x + 3)^(3/2) + 7495/20313
6*x*(284*x/sqrt(2*x^2 - x + 3) - 3174*x^2/(2*x^2 - x + 3)^(3/2) - 71/sqrt(2*x^2
- x + 3) + 805*x/(2*x^2 - x + 3)^(3/2) - 3243/(2*x^2 - x + 3)^(3/2)) + 7495/256*
sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 532145/101568*sqrt(2*x^2 - x + 3) - 4
515389/50784*x/sqrt(2*x^2 - x + 3) + 7197/8*x^2/(2*x^2 - x + 3)^(3/2) + 396211/5
0784/sqrt(2*x^2 - x + 3) - 269783/1104*x/(2*x^2 - x + 3)^(3/2) + 1002137/1104/(2
*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.288325, size = 173, normalized size = 1.65 \[ \frac{\sqrt{2}{\left (4 \, \sqrt{2}{\left (3174000 \, x^{5} + 16980900 \, x^{4} - 29423976 \, x^{3} + 101546529 \, x^{2} - 62463282 \, x + 89784565\right )} \sqrt{2 \, x^{2} - x + 3} + 11894565 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (-\sqrt{2}{\left (32 \, x^{2} - 16 \, x + 25\right )} - 8 \, \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )}\right )\right )}}{812544 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/812544*sqrt(2)*(4*sqrt(2)*(3174000*x^5 + 16980900*x^4 - 29423976*x^3 + 1015465
29*x^2 - 62463282*x + 89784565)*sqrt(2*x^2 - x + 3) + 11894565*(4*x^4 - 4*x^3 +
13*x^2 - 6*x + 9)*log(-sqrt(2)*(32*x^2 - 16*x + 25) - 8*sqrt(2*x^2 - x + 3)*(4*x
- 1)))/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.274439, size = 97, normalized size = 0.92 \[ -\frac{7495}{256} \, \sqrt{2}{\rm ln}\left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{3 \,{\left ({\left (4 \,{\left (13225 \,{\left (20 \, x + 107\right )} x - 2451998\right )} x + 33848843\right )} x - 20821094\right )} x + 89784565}{101568 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7495/256*sqrt(2)*ln(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/10156
8*(3*((4*(13225*(20*x + 107)*x - 2451998)*x + 33848843)*x - 20821094)*x + 897845
65)/(2*x^2 - x + 3)^(3/2)