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

Optimal. Leaf size=212 \[ \frac{80483 \left (2 x^2-x+3\right )^{7/2} x^2}{9216}+\frac{509257 \left (2 x^2-x+3\right )^{7/2} x}{294912}-\frac{1696165 \left (2 x^2-x+3\right )^{7/2}}{2752512}-\frac{57915 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{2097152}-\frac{6660225 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{67108864}-\frac{459555525 (1-4 x) \sqrt{2 x^2-x+3}}{1073741824}+\frac{125}{24} \left (2 x^2-x+3\right )^{7/2} x^5+\frac{1175}{96} \left (2 x^2-x+3\right )^{7/2} x^4+\frac{3823}{256} \left (2 x^2-x+3\right )^{7/2} x^3-\frac{10569777075 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2147483648 \sqrt{2}} \]

[Out]

(-459555525*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/1073741824 - (6660225*(1 - 4*x)*(3 -
x + 2*x^2)^(3/2))/67108864 - (57915*(1 - 4*x)*(3 - x + 2*x^2)^(5/2))/2097152 - (
1696165*(3 - x + 2*x^2)^(7/2))/2752512 + (509257*x*(3 - x + 2*x^2)^(7/2))/294912
+ (80483*x^2*(3 - x + 2*x^2)^(7/2))/9216 + (3823*x^3*(3 - x + 2*x^2)^(7/2))/256
+ (1175*x^4*(3 - x + 2*x^2)^(7/2))/96 + (125*x^5*(3 - x + 2*x^2)^(7/2))/24 - (1
0569777075*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2147483648*Sqrt[2])

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Rubi [A]  time = 0.369172, antiderivative size = 212, 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 \[ \frac{80483 \left (2 x^2-x+3\right )^{7/2} x^2}{9216}+\frac{509257 \left (2 x^2-x+3\right )^{7/2} x}{294912}-\frac{1696165 \left (2 x^2-x+3\right )^{7/2}}{2752512}-\frac{57915 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{2097152}-\frac{6660225 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{67108864}-\frac{459555525 (1-4 x) \sqrt{2 x^2-x+3}}{1073741824}+\frac{125}{24} \left (2 x^2-x+3\right )^{7/2} x^5+\frac{1175}{96} \left (2 x^2-x+3\right )^{7/2} x^4+\frac{3823}{256} \left (2 x^2-x+3\right )^{7/2} x^3-\frac{10569777075 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2147483648 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-459555525*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/1073741824 - (6660225*(1 - 4*x)*(3 -
x + 2*x^2)^(3/2))/67108864 - (57915*(1 - 4*x)*(3 - x + 2*x^2)^(5/2))/2097152 - (
1696165*(3 - x + 2*x^2)^(7/2))/2752512 + (509257*x*(3 - x + 2*x^2)^(7/2))/294912
+ (80483*x^2*(3 - x + 2*x^2)^(7/2))/9216 + (3823*x^3*(3 - x + 2*x^2)^(7/2))/256
+ (1175*x^4*(3 - x + 2*x^2)^(7/2))/96 + (125*x^5*(3 - x + 2*x^2)^(7/2))/24 - (1
0569777075*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2147483648*Sqrt[2])

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Rubi in Sympy [A]  time = 96.8315, size = 221, normalized size = 1.04 \[ - \frac{32 \left (- \frac{12567702364502818458541932241695 x}{8192} + \frac{5982091748009989536759139882665}{32768}\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (\frac{3139688559102733218339 x^{2}}{1024} + \frac{249212689792938092133 x}{1024} + \frac{1101764683633170644703}{512}\right )}{7844060888663380369127335209622566358625214375} - \frac{\left (- \frac{647955 x}{2} + \frac{5493279}{8}\right ) \left (2 x^{2} - x + 3\right )^{\frac{5}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}}{2376000} - \frac{459555525 \left (- 4 x + 1\right ) \sqrt{2 x^{2} - x + 3}}{1073741824} + \frac{\left (110 x + \frac{253}{2}\right ) \left (2 x^{2} - x + 3\right )^{\frac{7}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}}{528} + \frac{\left (\frac{28019950035 x}{8} + \frac{111669293241}{32}\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (\frac{800570001 x^{2}}{16} - \frac{390192693 x}{16} + \frac{62395707}{8}\right )^{2}}{666228363397935031039500} - \frac{2 \left (\frac{4592512541975180182314768057999433328279346523437598275 x}{33554432} + \frac{15162300112699042851601654176430304348808713213789752725}{134217728}\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}{23532182665990141107382005628867699075875643125} + \frac{10569777075 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x - 1\right )}{4 \sqrt{2 x^{2} - x + 3}} \right )}}{4294967296} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32*(-12567702364502818458541932241695*x/8192 + 5982091748009989536759139882665/
32768)*(2*x**2 - x + 3)**(3/2)*(3139688559102733218339*x**2/1024 + 2492126897929
38092133*x/1024 + 1101764683633170644703/512)/7844060888663380369127335209622566
358625214375 - (-647955*x/2 + 5493279/8)*(2*x**2 - x + 3)**(5/2)*(5*x**2 + 3*x +
2)**2/2376000 - 459555525*(-4*x + 1)*sqrt(2*x**2 - x + 3)/1073741824 + (110*x +
253/2)*(2*x**2 - x + 3)**(7/2)*(5*x**2 + 3*x + 2)**2/528 + (28019950035*x/8 + 1
11669293241/32)*(2*x**2 - x + 3)**(3/2)*(800570001*x**2/16 - 390192693*x/16 + 62
395707/8)**2/666228363397935031039500 - 2*(4592512541975180182314768057999433328
279346523437598275*x/33554432 + 151623001126990428516016541764303043488087132137
89752725/134217728)*(2*x**2 - x + 3)**(3/2)/235321826659901411073820056288676990
75875643125 + 10569777075*sqrt(2)*atanh(sqrt(2)*(4*x - 1)/(4*sqrt(2*x**2 - x + 3
)))/4294967296

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Mathematica [A]  time = 0.125707, size = 95, normalized size = 0.45 \[ \frac{665895955725 \sqrt{2} \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )+4 \sqrt{2 x^2-x+3} \left (2818572288000 x^{11}[phone] x^{10}[phone] x^9+14341894045696 x^8+27835561148416 x^7+28347538538496 x^6+34378613923840 x^5+26186527209472 x^4+20384824684416 x^3+10060731582048 x^2+4560943728924 x-1191399152715\right )}{270582939648} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-1191399152715 + 4560943728924*x + 10060731582048*x^2 +
20384824684416*x^3 + 26186527209472*x^4 + 34378613923840*x^5 + 28347538538496*x^
6 + 27835561148416*x^7 + 14341894045696*x^8 + 12943588589568*x^9 + 2395786444800
*x^10 + 2818572288000*x^11) + 665895955725*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]])
/270582939648

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Maple [A]  time = 0.01, size = 170, normalized size = 0.8 \[{\frac{231660\,x-57915}{2097152} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{26640900\,x-6660225}{67108864} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{1838222100\,x-459555525}{1073741824}\sqrt{2\,{x}^{2}-x+3}}+{\frac{10569777075\,\sqrt{2}}{4294967296}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{1696165}{2752512} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{509257\,x}{294912} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{80483\,{x}^{2}}{9216} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{3823\,{x}^{3}}{256} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{1175\,{x}^{4}}{96} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{125\,{x}^{5}}{24} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

57915/2097152*(4*x-1)*(2*x^2-x+3)^(5/2)+6660225/67108864*(4*x-1)*(2*x^2-x+3)^(3/
2)[phone]/1073741824*(4*x-1)*(2*x^2-x+3)^(1/2)[phone]/4294967296*2^(1/2)
*arcsinh(4/23*23^(1/2)*(x-1/4))-1696165/2752512*(2*x^2-x+3)^(7/2)+509257/294912*
x*(2*x^2-x+3)^(7/2)+80483/9216*x^2*(2*x^2-x+3)^(7/2)+3823/256*x^3*(2*x^2-x+3)^(7
/2)+1175/96*x^4*(2*x^2-x+3)^(7/2)+125/24*x^5*(2*x^2-x+3)^(7/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.790145, size = 271, normalized size = 1.28 \[ \frac{125}{24} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{5} + \frac{1175}{96} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{4} + \frac{3823}{256} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{3} + \frac{80483}{9216} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{2} + \frac{509257}{294912} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x - \frac{1696165}{2752512} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} + \frac{57915}{524288} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{57915}{2097152} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{6660225}{16777216} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{6660225}{67108864} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{459555525}{268435456} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{10569777075}{4294967296} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{459555525}{1073741824} \, \sqrt{2 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/24*(2*x^2 - x + 3)^(7/2)*x^5 + 1175/96*(2*x^2 - x + 3)^(7/2)*x^4 + 3823/256*
(2*x^2 - x + 3)^(7/2)*x^3 + 80483/9216*(2*x^2 - x + 3)^(7/2)*x^2 + 509257/294912
*(2*x^2 - x + 3)^(7/2)*x - 1696165/2752512*(2*x^2 - x + 3)^(7/2) + 57915/524288*
(2*x^2 - x + 3)^(5/2)*x - 57915/2097152*(2*x^2 - x + 3)^(5/2) + 6660225/16777216
*(2*x^2 - x + 3)^(3/2)*x - 6660225/67108864*(2*x^2 - x + 3)^(3/2) + 459555525/26
8435456*sqrt(2*x^2 - x + 3)*x + 10569777075/4294967296*sqrt(2)*arcsinh(1/23*sqrt
(23)*(4*x - 1)) - 459555525/1073741824*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 0.286218, size = 157, normalized size = 0.74 \[ \frac{1}{541165879296} \, \sqrt{2}{\left (4 \, \sqrt{2}{\left (2818572288000 \, x^{11} + 2395786444800 \, x^{10} + 12943588589568 \, x^{9} + 14341894045696 \, x^{8} + 27835561148416 \, x^{7} + 28347538538496 \, x^{6} + 34378613923840 \, x^{5} + 26186527209472 \, x^{4} + 20384824684416 \, x^{3} + 10060731582048 \, x^{2} + 4560943728924 \, x - 1191399152715\right )} \sqrt{2 \, x^{2} - x + 3} + 665895955725 \, \log \left (-\sqrt{2}{\left (32 \, x^{2} - 16 \, x + 25\right )} - 8 \, \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/541165879296*sqrt(2)*(4*sqrt(2)*(2818572288000*x^11 + 2395786444800*x^10 + 129
43588589568*x^9 + 14341894045696*x^8 + 27835561148416*x^7 + 28347538538496*x^6 +
34378613923840*x^5 + 26186527209472*x^4 + 20384824684416*x^3 + 10060731582048*x
^2 + 4560943728924*x - 1191399152715)*sqrt(2*x^2 - x + 3) + 665895955725*log(-sq
rt(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. \[ \int \left (2 x^{2} - x + 3\right )^{\frac{5}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273106, size = 139, normalized size = 0.66 \[ \frac{1}{67645734912} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (4 \,{\left (8 \,{\left (28 \,{\left (32 \,{\left (12 \,{\left (200 \,{\left (20 \, x + 17\right )} x + 18369\right )} x + 244241\right )} x + 15169177\right )} x + 432549111\right )} x + 4196608145\right )} x + 12786390239\right )} x + 159256442847\right )} x + 314397861939\right )} x + 1140235932231\right )} x - 1191399152715\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{10569777075}{4294967296} \, \sqrt{2}{\rm ln}\left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/67645734912*(4*(8*(4*(16*(4*(8*(28*(32*(12*(200*(20*x + 17)*x + 18369)*x + 244
241)*x + 15169177)*x + 432549111)*x + 4196608145)*x + 12786390239)*x + 159256442
847)*x + 314397861939)*x + 1140235932231)*x - 1191399152715)*sqrt(2*x^2 - x + 3)
- 10569777075/4294967296*sqrt(2)*ln(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)
) + 1)