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

Optimal. Leaf size=189 \[ \frac{384739 \left (2 x^2-x+3\right )^{5/2} x^2}{43008}-\frac{81685 \left (2 x^2-x+3\right )^{5/2} x}{114688}-\frac{4625907 \left (2 x^2-x+3\right )^{5/2}}{2293760}-\frac{667795 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{2097152}-\frac{46077855 (1-4 x) \sqrt{2 x^2-x+3}}{33554432}+\frac{25}{4} \left (2 x^2-x+3\right )^{5/2} x^5+\frac{725}{48} \left (2 x^2-x+3\right )^{5/2} x^4+\frac{27785 \left (2 x^2-x+3\right )^{5/2} x^3}{1536}-\frac{1059790665 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{67108864 \sqrt{2}} \]

[Out]

(-46077855*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/33554432 - (667795*(1 - 4*x)*(3 - x +
2*x^2)^(3/2))/2097152 - (4625907*(3 - x + 2*x^2)^(5/2))/2293760 - (81685*x*(3 -
x + 2*x^2)^(5/2))/114688 + (384739*x^2*(3 - x + 2*x^2)^(5/2))/43008 + (27785*x^3
*(3 - x + 2*x^2)^(5/2))/1536 + (725*x^4*(3 - x + 2*x^2)^(5/2))/48 + (25*x^5*(3 -
x + 2*x^2)^(5/2))/4 - (1059790665*ArcSinh[(1 - 4*x)/Sqrt[23]])/(67108864*Sqrt[2
])

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Rubi [A]  time = 0.31363, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{384739 \left (2 x^2-x+3\right )^{5/2} x^2}{43008}-\frac{81685 \left (2 x^2-x+3\right )^{5/2} x}{114688}-\frac{4625907 \left (2 x^2-x+3\right )^{5/2}}{2293760}-\frac{667795 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{2097152}-\frac{46077855 (1-4 x) \sqrt{2 x^2-x+3}}{33554432}+\frac{25}{4} \left (2 x^2-x+3\right )^{5/2} x^5+\frac{725}{48} \left (2 x^2-x+3\right )^{5/2} x^4+\frac{27785 \left (2 x^2-x+3\right )^{5/2} x^3}{1536}-\frac{1059790665 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{67108864 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-46077855*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/33554432 - (667795*(1 - 4*x)*(3 - x +
2*x^2)^(3/2))/2097152 - (4625907*(3 - x + 2*x^2)^(5/2))/2293760 - (81685*x*(3 -
x + 2*x^2)^(5/2))/114688 + (384739*x^2*(3 - x + 2*x^2)^(5/2))/43008 + (27785*x^3
*(3 - x + 2*x^2)^(5/2))/1536 + (725*x^4*(3 - x + 2*x^2)^(5/2))/48 + (25*x^5*(3 -
x + 2*x^2)^(5/2))/4 - (1059790665*ArcSinh[(1 - 4*x)/Sqrt[23]])/(67108864*Sqrt[2
])

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Rubi in Sympy [A]  time = 96.9671, size = 201, normalized size = 1.06 \[ - \frac{32 \left (- \frac{162474157526316622978033338525 x}{8192} + \frac{303537532424394352210249465395}{32768}\right ) \sqrt{2 x^{2} - x + 3} \left (\frac{194050502594980298595 x^{2}}{1024} - \frac{39088192706870460075 x}{1024} + \frac{97626563836837610535}{512}\right )}{5356417031858863054634751704213813807578125} - \frac{\left (- \frac{276465 x}{2} + \frac{3734733}{8}\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}}{1008000} + \frac{\left (90 x + \frac{219}{2}\right ) \left (2 x^{2} - x + 3\right )^{\frac{5}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}}{360} + \frac{\left (\frac{6977314125 x}{8} + \frac{35164168335}{32}\right ) \sqrt{2 x^{2} - x + 3} \left (\frac{279092565 x^{2}}{16} - \frac{205812081 x}{16} + \frac{14206635}{8}\right )^{2}}{18402140886793466906250} - \frac{4 \left (\frac{1266161437051083776619266308303527744115784816975625 x}{33554432} + \frac{2787236768504426756518680483616037764540168023447075}{134217728}\right ) \sqrt{2 x^{2} - x + 3}}{5356417031858863054634751704213813807578125} + \frac{1059790665 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x - 1\right )}{4 \sqrt{2 x^{2} - x + 3}} \right )}}{134217728} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32*(-162474157526316622978033338525*x/8192 + 303537532424394352210249465395/327
68)*sqrt(2*x**2 - x + 3)*(194050502594980298595*x**2/1024 - 39088192706870460075
*x/1024 + 97626563836837610535/512)/5356417031858863054634751704213813807578125
- (-276465*x/2 + 3734733/8)*(2*x**2 - x + 3)**(3/2)*(5*x**2 + 3*x + 2)**2/100800
0 + (90*x + 219/2)*(2*x**2 - x + 3)**(5/2)*(5*x**2 + 3*x + 2)**2/360 + (69773141
25*x/8 + 35164168335/32)*sqrt(2*x**2 - x + 3)*(279092565*x**2/16 - 205812081*x/1
6 + 14206635/8)**2/18402140886793466906250 - 4*(12661614370510837766192663083035
27744115784816975625*x/33554432 + 2787236768504426756518680483616037764540168023
447075/134217728)*sqrt(2*x**2 - x + 3)/53564170318588630546347517042138138075781
25 + 1059790665*sqrt(2)*atanh(sqrt(2)*(4*x - 1)/(4*sqrt(2*x**2 - x + 3)))/134217
728

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Mathematica [A]  time = 0.10956, size = 85, normalized size = 0.45 \[ \frac{4 \sqrt{2 x^2-x+3} \left (88080384000 x^9+124780544000 x^8+328328806400 x^7+430820229120 x^6+571298324480 x^5+487891884032 x^4+389257196928 x^3+199615064544 x^2+53985432012 x-72152399943\right )[phone] \sqrt{2} \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{14092861440} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-72152399943 + 53985432012*x + 199615064544*x^2 + 389257
196928*x^3 + 487891884032*x^4 + 571298324480*x^5 + 430820229120*x^6 + 3283288064
00*x^7 + 124780544000*x^8 + 88080384000*x^9) + 111278019825*Sqrt[2]*ArcSinh[(-1
+ 4*x)/Sqrt[23]])/14092861440

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Maple [A]  time = 0.01, size = 151, normalized size = 0.8 \[{\frac{2671180\,x-667795}{2097152} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{184311420\,x-46077855}{33554432}\sqrt{2\,{x}^{2}-x+3}}+{\frac{1059790665\,\sqrt{2}}{134217728}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{4625907}{2293760} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{81685\,x}{114688} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{384739\,{x}^{2}}{43008} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{27785\,{x}^{3}}{1536} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{725\,{x}^{4}}{48} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{25\,{x}^{5}}{4} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

667795/2097152*(4*x-1)*(2*x^2-x+3)^(3/2)[phone]/33554432*(4*x-1)*(2*x^2-x+3)^(
1/2)[phone]/134217728*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-4625907/2293760
*(2*x^2-x+3)^(5/2)-81685/114688*x*(2*x^2-x+3)^(5/2)+384739/43008*x^2*(2*x^2-x+3)
^(5/2)+27785/1536*x^3*(2*x^2-x+3)^(5/2)+725/48*x^4*(2*x^2-x+3)^(5/2)+25/4*x^5*(2
*x^2-x+3)^(5/2)

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Maxima [A]  time = 0.770314, size = 232, normalized size = 1.23 \[ \frac{25}{4} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{5} + \frac{725}{48} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{4} + \frac{27785}{1536} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{3} + \frac{384739}{43008} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} - \frac{81685}{114688} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{4625907}{2293760} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{667795}{524288} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{667795}{2097152} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{46077855}{8388608} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{1059790665}{134217728} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{46077855}{33554432} \, \sqrt{2 \, x^{2} - x + 3} \]

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]

25/4*(2*x^2 - x + 3)^(5/2)*x^5 + 725/48*(2*x^2 - x + 3)^(5/2)*x^4 + 27785/1536*(
2*x^2 - x + 3)^(5/2)*x^3 + 384739/43008*(2*x^2 - x + 3)^(5/2)*x^2 - 81685/114688
*(2*x^2 - x + 3)^(5/2)*x - 4625907/2293760*(2*x^2 - x + 3)^(5/2) + 667795/524288
*(2*x^2 - x + 3)^(3/2)*x - 667795/2097152*(2*x^2 - x + 3)^(3/2) + 46077855/83886
08*sqrt(2*x^2 - x + 3)*x + 1059790665/134217728*sqrt(2)*arcsinh(1/23*sqrt(23)*(4
*x - 1)) - 46077855/33554432*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 0.287477, size = 143, normalized size = 0.76 \[ \frac{1}{28185722880} \, \sqrt{2}{\left (4 \, \sqrt{2}{\left (88080384000 \, x^{9} + 124780544000 \, x^{8} + 328328806400 \, x^{7} + 430820229120 \, x^{6} + 571298324480 \, x^{5} + 487891884032 \, x^{4} + 389257196928 \, x^{3} + 199615064544 \, x^{2} + 53985432012 \, x - 72152399943\right )} \sqrt{2 \, x^{2} - x + 3} + 111278019825 \, \log \left (-\sqrt{2}{\left (32 \, x^{2} - 16 \, x + 25\right )} - 8 \, \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/28185722880*sqrt(2)*(4*sqrt(2)*(88080384000*x^9 + 124780544000*x^8 + 328328806
400*x^7 + 430820229120*x^6 + 571298324480*x^5 + 487891884032*x^4 + 389257196928*
x^3 + 199615064544*x^2 + 53985432012*x - 72152399943)*sqrt(2*x^2 - x + 3) + 1112
78019825*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. \[ \int \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269215, size = 126, normalized size = 0.67 \[ \frac{1}{3523215360} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (20 \,{\left (8 \,{\left (140 \,{\left (160 \,{\left (12 \, x + 17\right )} x + 7157\right )} x + 1314759\right )} x + 13947713\right )} x + 238228459\right )} x + 3041071851\right )} x + 6237970767\right )} x + 13496358003\right )} x - 72152399943\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{1059790665}{134217728} \, \sqrt{2}{\rm ln}\left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3523215360*(4*(8*(4*(16*(20*(8*(140*(160*(12*x + 17)*x + 7157)*x + 1314759)*x
+ 13947713)*x + 238228459)*x + 3041071851)*x + 6237970767)*x + 13496358003)*x -
72152399943)*sqrt(2*x^2 - x + 3) - 1059790665/134217728*sqrt(2)*ln(-2*sqrt(2)*(s
qrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)