3.58 \(\int \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^4 \, dx\)

Optimal. Leaf size=208 \[ -\frac{83948353 \left (2 x^2-x+3\right )^{3/2} x^2}{2293760}+\frac{804243809 \left (2 x^2-x+3\right )^{3/2} x}{36700160}+\frac{27185733541 \left (2 x^2-x+3\right )^{3/2}}{440401920}-\frac{359471503 (1-4 x) \sqrt{2 x^2-x+3}}{67108864}+\frac{125}{4} \left (2 x^2-x+3\right )^{3/2} x^7+\frac{14125}{144} \left (2 x^2-x+3\right )^{3/2} x^6+\frac{233225 \left (2 x^2-x+3\right )^{3/2} x^5}{1536}+\frac{4796405 \left (2 x^2-x+3\right )^{3/2} x^4}{43008}+\frac{8325631 \left (2 x^2-x+3\right )^{3/2} x^3}{1032192}-\frac{8267844569 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{134217728 \sqrt{2}} \]

[Out]

(-359471503*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/67108864 + (27185733541*(3 - x + 2*x^
2)^(3/2))/440401920 + (804243809*x*(3 - x + 2*x^2)^(3/2))/36700160 - (83948353*x
^2*(3 - x + 2*x^2)^(3/2))/2293760 + (8325631*x^3*(3 - x + 2*x^2)^(3/2))/1032192
+ (4796405*x^4*(3 - x + 2*x^2)^(3/2))/43008 + (233225*x^5*(3 - x + 2*x^2)^(3/2))
/1536 + (14125*x^6*(3 - x + 2*x^2)^(3/2))/144 + (125*x^7*(3 - x + 2*x^2)^(3/2))/
4 - (8267844569*ArcSinh[(1 - 4*x)/Sqrt[23]])/(134217728*Sqrt[2])

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Rubi [A]  time = 0.483532, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{83948353 \left (2 x^2-x+3\right )^{3/2} x^2}{2293760}+\frac{804243809 \left (2 x^2-x+3\right )^{3/2} x}{36700160}+\frac{27185733541 \left (2 x^2-x+3\right )^{3/2}}{440401920}-\frac{359471503 (1-4 x) \sqrt{2 x^2-x+3}}{67108864}+\frac{125}{4} \left (2 x^2-x+3\right )^{3/2} x^7+\frac{14125}{144} \left (2 x^2-x+3\right )^{3/2} x^6+\frac{233225 \left (2 x^2-x+3\right )^{3/2} x^5}{1536}+\frac{4796405 \left (2 x^2-x+3\right )^{3/2} x^4}{43008}+\frac{8325631 \left (2 x^2-x+3\right )^{3/2} x^3}{1032192}-\frac{8267844569 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{134217728 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-359471503*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/67108864 + (27185733541*(3 - x + 2*x^
2)^(3/2))/440401920 + (804243809*x*(3 - x + 2*x^2)^(3/2))/36700160 - (83948353*x
^2*(3 - x + 2*x^2)^(3/2))/2293760 + (8325631*x^3*(3 - x + 2*x^2)^(3/2))/1032192
+ (4796405*x^4*(3 - x + 2*x^2)^(3/2))/43008 + (233225*x^5*(3 - x + 2*x^2)^(3/2))
/1536 + (14125*x^6*(3 - x + 2*x^2)^(3/2))/144 + (125*x^7*(3 - x + 2*x^2)^(3/2))/
4 - (8267844569*ArcSinh[(1 - 4*x)/Sqrt[23]])/(134217728*Sqrt[2])

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Rubi in Sympy [A]  time = 102.482, size = 192, normalized size = 0.92 \[ - \frac{\left (- \frac{3847264125 x}{8} + \frac{6356151165}{32}\right ) \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{2}}{604800000} - \frac{\left (- \frac{170205 x}{2} + \frac{6162015}{8}\right ) \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{3}}{1008000} + \frac{\left (90 x + \frac{241}{2}\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}}{360} + \frac{\left (\frac{31895887634775 x}{32} + \frac{185297635332855}{128}\right ) \left (- \frac{2126392508985 x^{2}}{64} - \frac{440352854355 x}{64} + \frac{122592566805}{32}\right ) \sqrt{2 x^{2} - x + 3}}{964531642075596000000} + \frac{\left (\frac{1259687335261370714967506625 x}{8192} + \frac{43862534956680368824216701675}{32768}\right ) \sqrt{2 x^{2} - x + 3}}{7716253136604768000000} + \frac{8267844569 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x - 1\right )}{4 \sqrt{2 x^{2} - x + 3}} \right )}}{268435456} \]

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)**(1/2),x)

[Out]

-(-3847264125*x/8 + 6356151165/32)*sqrt(2*x**2 - x + 3)*(5*x**2 + 3*x + 2)**2/60
4800000 - (-170205*x/2 + 6162015/8)*sqrt(2*x**2 - x + 3)*(5*x**2 + 3*x + 2)**3/1
008000 + (90*x + 241/2)*(2*x**2 - x + 3)**(3/2)*(5*x**2 + 3*x + 2)**3/360 + (318
95887634775*x/32 + 185297635332855/128)*(-2126392508985*x**2/64 - 440352854355*x
/64 + 122592566805/32)*sqrt(2*x**2 - x + 3)/964531642075596000000 + (12596873352
61370714967506625*x/8192 + 43862534956680368824216701675/32768)*sqrt(2*x**2 - x
+ 3)/7716253136604768000000 + 8267844569*sqrt(2)*atanh(sqrt(2)*(4*x - 1)/(4*sqrt
(2*x**2 - x + 3)))/268435456

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Mathematica [A]  time = 0.112816, size = 85, normalized size = 0.41 \[ \frac{4 \sqrt{2 x^2-x+3} \left (1321205760000 x^9+3486515200000 x^8+6327795712000 x^7+7725962035200 x^6+7612808028160 x^5+5354741991424 x^4+2211683657856 x^3-174418077792 x^2+537752185764 x+3801512106459\right )[phone] \sqrt{2} \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{84557168640} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(3801512106459 + 537752185764*x - 174418077792*x^2 + 2211
683657856*x^3 + 5354741991424*x^4 + 7612808028160*x^5 + 7725962035200*x^6 + 6327
795712000*x^7 + 3486515200000*x^8 + 1321205760000*x^9) + 2604371039235*Sqrt[2]*A
rcSinh[(-1 + 4*x)/Sqrt[23]])/84557168640

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Maple [A]  time = 0.039, size = 166, normalized size = 0.8 \[{\frac{1437886012\,x-359471503}{67108864}\sqrt{2\,{x}^{2}-x+3}}+{\frac{8267844569\,\sqrt{2}}{268435456}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{27185733541}{440401920} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{804243809\,x}{36700160} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{83948353\,{x}^{2}}{2293760} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{8325631\,{x}^{3}}{1032192} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{4796405\,{x}^{4}}{43008} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{233225\,{x}^{5}}{1536} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{14125\,{x}^{6}}{144} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{125\,{x}^{7}}{4} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2+3*x+2)^4*(2*x^2-x+3)^(1/2),x)

[Out]

359471503/67108864*(4*x-1)*(2*x^2-x+3)^(1/2)[phone]/268435456*2^(1/2)*arcsin
h(4/23*23^(1/2)*(x-1/4))[phone]/440401920*(2*x^2-x+3)^(3/2)[phone]/36700
160*x*(2*x^2-x+3)^(3/2)-83948353/2293760*x^2*(2*x^2-x+3)^(3/2)+8325631/1032192*x
^3*(2*x^2-x+3)^(3/2)+4796405/43008*x^4*(2*x^2-x+3)^(3/2)+233225/1536*x^5*(2*x^2-
x+3)^(3/2)+14125/144*x^6*(2*x^2-x+3)^(3/2)+125/4*x^7*(2*x^2-x+3)^(3/2)

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Maxima [A]  time = 0.795224, size = 239, normalized size = 1.15 \[ \frac{125}{4} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{7} + \frac{14125}{144} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{6} + \frac{233225}{1536} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{5} + \frac{4796405}{43008} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} + \frac{8325631}{1032192} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{83948353}{2293760} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{804243809}{36700160} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{27185733541}{440401920} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{359471503}{16777216} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{8267844569}{268435456} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{359471503}{67108864} \, \sqrt{2 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/4*(2*x^2 - x + 3)^(3/2)*x^7 + 14125/144*(2*x^2 - x + 3)^(3/2)*x^6 + 233225/1
536*(2*x^2 - x + 3)^(3/2)*x^5 + 4796405/43008*(2*x^2 - x + 3)^(3/2)*x^4 + 832563
1/1032192*(2*x^2 - x + 3)^(3/2)*x^3 - 83948353/2293760*(2*x^2 - x + 3)^(3/2)*x^2
+ 804243809/36700160*(2*x^2 - x + 3)^(3/2)*x + 27185733541/440401920*(2*x^2 - x
+ 3)^(3/2) + 359471503/16777216*sqrt(2*x^2 - x + 3)*x + 8267844569/268435456*sq
rt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 359471503/67108864*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 0.283938, size = 143, normalized size = 0.69 \[ \frac{1}{169114337280} \, \sqrt{2}{\left (4 \, \sqrt{2}{\left (1321205760000 \, x^{9} + 3486515200000 \, x^{8} + 6327795712000 \, x^{7} + 7725962035200 \, x^{6} + 7612808028160 \, x^{5} + 5354741991424 \, x^{4} + 2211683657856 \, x^{3} - 174418077792 \, x^{2} + 537752185764 \, x + 3801512106459\right )} \sqrt{2 \, x^{2} - x + 3} + 2604371039235 \, \log \left (-\sqrt{2}{\left (32 \, x^{2} - 16 \, x + 25\right )} - 8 \, \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/169114337280*sqrt(2)*(4*sqrt(2)*(1321205760000*x^9 + 3486515200000*x^8 + 63277
95712000*x^7 + 7725962035200*x^6 + 7612808028160*x^5 + 5354741991424*x^4 + 22116
83657856*x^3 - 174418077792*x^2 + 537752185764*x + 3801512106459)*sqrt(2*x^2 - x
+ 3) + 2604371039235*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. \[ \int \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+3*x+2)**4*(2*x**2-x+3)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.270402, size = 126, normalized size = 0.61 \[ \frac{1}{21139292160} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (20 \,{\left (40 \,{\left (140 \,{\left (160 \,{\left (36 \, x + 95\right )} x + 27587\right )} x + 4715553\right )} x + 185859571\right )} x + 2614620113\right )} x + 17278778577\right )} x - 5450564931\right )} x + 134438046441\right )} x + 3801512106459\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{8267844569}{268435456} \, \sqrt{2}{\rm ln}\left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21139292160*(4*(8*(4*(16*(20*(40*(140*(160*(36*x + 95)*x + 27587)*x + 4715553)
*x + 185859571)*x + 2614620113)*x + 17278778577)*x - 5450564931)*x + 13443804644
1)*x + 3801512106459)*sqrt(2*x^2 - x + 3) - 8267844569/268435456*sqrt(2)*ln(-2*s
qrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)