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

Optimal. Leaf size=166 \[ \frac{531681 \left (2 x^2-x+3\right )^{3/2} x^2}{71680}-\frac{9627393 \left (2 x^2-x+3\right )^{3/2} x}{1146880}-\frac{22548119 \left (2 x^2-x+3\right )^{3/2}}{4587520}-\frac{6766097 (1-4 x) \sqrt{2 x^2-x+3}}{2097152}+\frac{125}{16} \left (2 x^2-x+3\right )^{3/2} x^5+\frac{8825}{448} \left (2 x^2-x+3\right )^{3/2} x^4+\frac{247435 \left (2 x^2-x+3\right )^{3/2} x^3}{10752}-\frac{155620231 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4194304 \sqrt{2}} \]

[Out]

(-6766097*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/2097152 - (22548119*(3 - x + 2*x^2)^(3/
2))/4587520 - (9627393*x*(3 - x + 2*x^2)^(3/2))/1146880 + (531681*x^2*(3 - x + 2
*x^2)^(3/2))/71680 + (247435*x^3*(3 - x + 2*x^2)^(3/2))/10752 + (8825*x^4*(3 - x
+ 2*x^2)^(3/2))/448 + (125*x^5*(3 - x + 2*x^2)^(3/2))/16 - (155620231*ArcSinh[(
1 - 4*x)/Sqrt[23]])/(4194304*Sqrt[2])

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Rubi [A]  time = 0.311495, 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 \[ \frac{531681 \left (2 x^2-x+3\right )^{3/2} x^2}{71680}-\frac{9627393 \left (2 x^2-x+3\right )^{3/2} x}{1146880}-\frac{22548119 \left (2 x^2-x+3\right )^{3/2}}{4587520}-\frac{6766097 (1-4 x) \sqrt{2 x^2-x+3}}{2097152}+\frac{125}{16} \left (2 x^2-x+3\right )^{3/2} x^5+\frac{8825}{448} \left (2 x^2-x+3\right )^{3/2} x^4+\frac{247435 \left (2 x^2-x+3\right )^{3/2} x^3}{10752}-\frac{155620231 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4194304 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-6766097*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/2097152 - (22548119*(3 - x + 2*x^2)^(3/
2))/4587520 - (9627393*x*(3 - x + 2*x^2)^(3/2))/1146880 + (531681*x^2*(3 - x + 2
*x^2)^(3/2))/71680 + (247435*x^3*(3 - x + 2*x^2)^(3/2))/10752 + (8825*x^4*(3 - x
+ 2*x^2)^(3/2))/448 + (125*x^5*(3 - x + 2*x^2)^(3/2))/16 - (155620231*ArcSinh[(
1 - 4*x)/Sqrt[23]])/(4194304*Sqrt[2])

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Rubi in Sympy [A]  time = 68.4031, size = 153, normalized size = 0.92 \[ - \frac{\left (- \frac{836463255 x}{8} + \frac{1760991321}{32}\right ) \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )}{80640000} - \frac{\left (- \frac{83975 x}{2} + \frac{2357587}{8}\right ) \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{2}}{336000} + \frac{\left (70 x + \frac{185}{2}\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}}{224} - \frac{\left (\frac{306519103821 x}{32} + \frac{1490800160271}{128}\right ) \sqrt{2 x^{2} - x + 3}}{645120000} + \frac{155620231 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x - 1\right )}{4 \sqrt{2 x^{2} - x + 3}} \right )}}{8388608} \]

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]

-(-836463255*x/8 + 1760991321/32)*sqrt(2*x**2 - x + 3)*(5*x**2 + 3*x + 2)/806400
00 - (-83975*x/2 + 2357587/8)*sqrt(2*x**2 - x + 3)*(5*x**2 + 3*x + 2)**2/336000
+ (70*x + 185/2)*(2*x**2 - x + 3)**(3/2)*(5*x**2 + 3*x + 2)**2/224 - (3065191038
21*x/32 + 1490800160271/128)*sqrt(2*x**2 - x + 3)/645120000 + 155620231*sqrt(2)*
atanh(sqrt(2)*(4*x - 1)/(4*sqrt(2*x**2 - x + 3)))/8388608

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Mathematica [A]  time = 0.0900378, size = 75, normalized size = 0.45 \[ \frac{4 \sqrt{2 x^2-x+3} \left (3440640000 x^7+6955008000 x^6+10958233600 x^5+11212171264 x^4+9872163456 x^3+4583812128 x^2-1621307916 x-3957369321\right )[phone] \sqrt{2} \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{880803840} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-3957369321 - 1621307916*x + 4583812128*x^2 + 9872163456
*x^3 + 11212171264*x^4 + 10958233600*x^5 + 6955008000*x^6 + 3440640000*x^7) + 16
340124255*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]])/880803840

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Maple [A]  time = 0.01, size = 132, normalized size = 0.8 \[{\frac{27064388\,x-6766097}{2097152}\sqrt{2\,{x}^{2}-x+3}}+{\frac{155620231\,\sqrt{2}}{8388608}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{22548119}{4587520} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{9627393\,x}{1146880} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{531681\,{x}^{2}}{71680} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{247435\,{x}^{3}}{10752} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{8825\,{x}^{4}}{448} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{125\,{x}^{5}}{16} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6766097/2097152*(4*x-1)*(2*x^2-x+3)^(1/2)[phone]/8388608*2^(1/2)*arcsinh(4/23
*23^(1/2)*(x-1/4))-22548119/4587520*(2*x^2-x+3)^(3/2)-9627393/1146880*x*(2*x^2-x
+3)^(3/2)+531681/71680*x^2*(2*x^2-x+3)^(3/2)+247435/10752*x^3*(2*x^2-x+3)^(3/2)+
8825/448*x^4*(2*x^2-x+3)^(3/2)+125/16*x^5*(2*x^2-x+3)^(3/2)

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Maxima [A]  time = 0.77788, size = 193, normalized size = 1.16 \[ \frac{125}{16} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{5} + \frac{8825}{448} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} + \frac{247435}{10752} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + \frac{531681}{71680} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{9627393}{1146880} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{22548119}{4587520} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{6766097}{524288} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{155620231}{8388608} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{6766097}{2097152} \, \sqrt{2 \, x^{2} - x + 3} \]

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]

125/16*(2*x^2 - x + 3)^(3/2)*x^5 + 8825/448*(2*x^2 - x + 3)^(3/2)*x^4 + 247435/1
0752*(2*x^2 - x + 3)^(3/2)*x^3 + 531681/71680*(2*x^2 - x + 3)^(3/2)*x^2 - 962739
3/1146880*(2*x^2 - x + 3)^(3/2)*x - 22548119/4587520*(2*x^2 - x + 3)^(3/2) + 676
6097/524288*sqrt(2*x^2 - x + 3)*x + 155620231/8388608*sqrt(2)*arcsinh(1/23*sqrt(
23)*(4*x - 1)) - 6766097/2097152*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 0.281996, size = 130, normalized size = 0.78 \[ \frac{1}{1761607680} \, \sqrt{2}{\left (4 \, \sqrt{2}{\left (3440640000 \, x^{7} + 6955008000 \, x^{6} + 10958233600 \, x^{5} + 11212171264 \, x^{4} + 9872163456 \, x^{3} + 4583812128 \, x^{2} - 1621307916 \, x - 3957369321\right )} \sqrt{2 \, x^{2} - x + 3} + 16340124255 \, \log \left (-\sqrt{2}{\left (32 \, x^{2} - 16 \, x + 25\right )} - 8 \, \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )}\right )\right )} \]

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]

1/1761607680*sqrt(2)*(4*sqrt(2)*(3440640000*x^7 + 6955008000*x^6 + 10958233600*x
^5 + 11212171264*x^4 + 9872163456*x^3 + 4583812128*x^2 - 1621307916*x - 39573693
21)*sqrt(2*x^2 - x + 3) + 16340124255*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. \[ \int \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{3}\, dx \]

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

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GIAC/XCAS [A]  time = 0.270052, size = 112, normalized size = 0.67 \[ \frac{1}{220200960} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \,{\left (120 \,{\left (140 \, x + 283\right )} x + 53507\right )} x + 5474693\right )} x + 77126277\right )} x + 143244129\right )} x - 405326979\right )} x - 3957369321\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{155620231}{8388608} \, \sqrt{2}{\rm ln}\left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \]

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]

1/220200960*(4*(8*(4*(16*(100*(120*(140*x + 283)*x + 53507)*x + 5474693)*x + 771
26277)*x + 143244129)*x - 405326979)*x - 3957369321)*sqrt(2*x^2 - x + 3) - 15562
0231/8388608*sqrt(2)*ln(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)