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

Optimal. Leaf size=254 \[ \frac{122595067 \left (2 x^2-x+3\right )^{7/2} x^2}{19169280}+\frac{112244125 \left (2 x^2-x+3\right )^{7/2} x}{122683392}+\frac{25250178739 \left (2 x^2-x+3\right )^{7/2}}{5725224960}-\frac{401135647 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{335544320}-\frac{9226119881 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{2147483648}-\frac{636602271789 (1-4 x) \sqrt{2 x^2-x+3}}{34359738368}+\frac{625}{28} \left (2 x^2-x+3\right )^{7/2} x^7+\frac{13875}{208} \left (2 x^2-x+3\right )^{7/2} x^6+\frac{1046225 \left (2 x^2-x+3\right )^{7/2} x^5}{9984}+\frac{3684995 \left (2 x^2-x+3\right )^{7/2} x^4}{39936}+\frac{23460839 \left (2 x^2-x+3\right )^{7/2} x^3}{532480}-\frac{14641852251147 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{68719476736 \sqrt{2}} \]

[Out]

(-636602271789*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/34359738368 - (9226119881*(1 - 4*x
)*(3 - x + 2*x^2)^(3/2))/2147483648 - (401135647*(1 - 4*x)*(3 - x + 2*x^2)^(5/2)
)/335544320 + (25250178739*(3 - x + 2*x^2)^(7/2))/5725224960 + (112244125*x*(3 -
x + 2*x^2)^(7/2))/122683392 + (122595067*x^2*(3 - x + 2*x^2)^(7/2))/19169280 +
(23460839*x^3*(3 - x + 2*x^2)^(7/2))/532480 + (3684995*x^4*(3 - x + 2*x^2)^(7/2)
)/39936 + (1046225*x^5*(3 - x + 2*x^2)^(7/2))/9984 + (13875*x^6*(3 - x + 2*x^2)^
(7/2))/208 + (625*x^7*(3 - x + 2*x^2)^(7/2))/28 - (14641852251147*ArcSinh[(1 - 4
*x)/Sqrt[23]])/(68719476736*Sqrt[2])

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Rubi [A]  time = 0.604329, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{122595067 \left (2 x^2-x+3\right )^{7/2} x^2}{19169280}+\frac{112244125 \left (2 x^2-x+3\right )^{7/2} x}{122683392}+\frac{25250178739 \left (2 x^2-x+3\right )^{7/2}}{5725224960}-\frac{401135647 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{335544320}-\frac{9226119881 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{2147483648}-\frac{636602271789 (1-4 x) \sqrt{2 x^2-x+3}}{34359738368}+\frac{625}{28} \left (2 x^2-x+3\right )^{7/2} x^7+\frac{13875}{208} \left (2 x^2-x+3\right )^{7/2} x^6+\frac{1046225 \left (2 x^2-x+3\right )^{7/2} x^5}{9984}+\frac{3684995 \left (2 x^2-x+3\right )^{7/2} x^4}{39936}+\frac{23460839 \left (2 x^2-x+3\right )^{7/2} x^3}{532480}-\frac{14641852251147 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{68719476736 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-636602271789*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/34359738368 - (9226119881*(1 - 4*x
)*(3 - x + 2*x^2)^(3/2))/2147483648 - (401135647*(1 - 4*x)*(3 - x + 2*x^2)^(5/2)
)/335544320 + (25250178739*(3 - x + 2*x^2)^(7/2))/5725224960 + (112244125*x*(3 -
x + 2*x^2)^(7/2))/122683392 + (122595067*x^2*(3 - x + 2*x^2)^(7/2))/19169280 +
(23460839*x^3*(3 - x + 2*x^2)^(7/2))/532480 + (3684995*x^4*(3 - x + 2*x^2)^(7/2)
)/39936 + (1046225*x^5*(3 - x + 2*x^2)^(7/2))/9984 + (13875*x^6*(3 - x + 2*x^2)^
(7/2))/208 + (625*x^7*(3 - x + 2*x^2)^(7/2))/28 - (14641852251147*ArcSinh[(1 - 4
*x)/Sqrt[23]])/(68719476736*Sqrt[2])

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Rubi in Sympy [A]  time = 128.005, size = 246, normalized size = 0.97 \[ - \frac{\left (- \frac{68243649705 x}{8} + \frac{362629622751}{32}\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}}{21621600000} - \frac{\left (- \frac{945065 x}{2} + \frac{11877899}{8}\right ) \left (2 x^{2} - x + 3\right )^{\frac{5}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}}{4804800} - \frac{636602271789 \left (- 4 x + 1\right ) \sqrt{2 x^{2} - x + 3}}{34359738368} + \frac{\left (130 x + \frac{309}{2}\right ) \left (2 x^{2} - x + 3\right )^{\frac{7}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}}{728} + \frac{\left (\frac{1719653051422845 x}{32} + \frac{2254414002500583}{128}\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}}{24216192000000} + \frac{\left (\frac{7211421563490750375 x}{128} + \frac{41400525906772259535}{512}\right ) \left (- \frac{288456862539630015 x^{2}}{256} - \frac{382553649481777389 x}{256} + \frac{30255323481274815}{128}\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}{3274371922020603774470100000000} + \frac{\left (\frac{52558098024608188084793573247595880295 x}{131072} + \frac{1598024049869061092244937230038073366225}{524288}\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}{157169852256988981174564800000000} + \frac{14641852251147 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x - 1\right )}{4 \sqrt{2 x^{2} - x + 3}} \right )}}{137438953472} \]

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)**4,x)

[Out]

-(-68243649705*x/8 + 362629622751/32)*(2*x**2 - x + 3)**(3/2)*(5*x**2 + 3*x + 2)
**3/21621600000 - (-945065*x/2 + 11877899/8)*(2*x**2 - x + 3)**(5/2)*(5*x**2 + 3
*x + 2)**3/4804800 - 636602271789*(-4*x + 1)*sqrt(2*x**2 - x + 3)/34359738368 +
(130*x + 309/2)*(2*x**2 - x + 3)**(7/2)*(5*x**2 + 3*x + 2)**3/728 + (17196530514
22845*x/32 + 2254414002500583/128)*(2*x**2 - x + 3)**(3/2)*(5*x**2 + 3*x + 2)**2
/24216192000000 + (7211421563490750375*x/128 + 41400525906772259535/512)*(-28845
6862539630015*x**2/256 - 382553649481777389*x/256 + 30255323481274815/128)*(2*x*
*2 - x + 3)**(3/2)/3274371922020603774470100000000 + (52558098024608188084793573
247595880295*x/131072 + 1598024049869061092244937230038073366225/524288)*(2*x**2
- x + 3)**(3/2)/157169852256988981174564800000000 + 14641852251147*sqrt(2)*atan
h(sqrt(2)*(4*x - 1)/(4*sqrt(2*x**2 - x + 3)))/137438953472

_______________________________________________________________________________________

Mathematica [A]  time = 0.15309, size = 105, normalized size = 0.41 \[ \frac{59958384968446965 \sqrt{2} \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )+4 \sqrt{2 x^2-x+3} \left (25125558681600000 x^{13}+37398427729920000 x^{12}+137233466130432000 x^{11}+204932411660697600 x^{10}+363646430503501824 x^9+439064558846345216 x^8+530502956133122048 x^7+485091164642279424 x^6+405468382284161024 x^5+257786732552566784 x^4+142490931553577856 x^3+50064174038215008 x^2+12071614275862524 x+10820567498568669\right )}{562812514467840} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(10820567498568669 + 12071614275862524*x + 50064174038215
008*x^2 + 142490931553577856*x^3 + 257786732552566784*x^4 + 405468382284161024*x
^5 + 485091164642279424*x^6 + 530502956133122048*x^7 + 439064558846345216*x^8 +
363646430503501824*x^9 + 204932411660697600*x^10 + 137233466130432000*x^11 + 373
98427729920000*x^12 + 25125558681600000*x^13) + 59958384968446965*Sqrt[2]*ArcSin
h[(-1 + 4*x)/Sqrt[23]])/562812514467840

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Maple [A]  time = 0.054, size = 204, normalized size = 0.8 \[{\frac{1604542588\,x-401135647}{335544320} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{36904479524\,x-9226119881}{2147483648} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{2546409087156\,x-636602271789}{34359738368}\sqrt{2\,{x}^{2}-x+3}}+{\frac{14641852251147\,\sqrt{2}}{137438953472}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{25250178739}{5725224960} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{112244125\,x}{122683392} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{122595067\,{x}^{2}}{19169280} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{23460839\,{x}^{3}}{532480} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{3684995\,{x}^{4}}{39936} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{1046225\,{x}^{5}}{9984} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{13875\,{x}^{6}}{208} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{625\,{x}^{7}}{28} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}} \]

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)^4,x)

[Out]

401135647/335544320*(4*x-1)*(2*x^2-x+3)^(5/2)[phone]/2147483648*(4*x-1)*(2*x
^2-x+3)^(3/2)[phone]/34359738368*(4*x-1)*(2*x^2-x+3)^(1/2)[phone]/
137438953472*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))[phone]/5725224960*(2*x^
2-x+3)^(7/2)[phone]/122683392*x*(2*x^2-x+3)^(7/2)[phone]/19169280*x^2*(2*x
^2-x+3)^(7/2)[phone]/532480*x^3*(2*x^2-x+3)^(7/2)+3684995/39936*x^4*(2*x^2-x+3
)^(7/2)+1046225/9984*x^5*(2*x^2-x+3)^(7/2)+13875/208*x^6*(2*x^2-x+3)^(7/2)+625/2
8*x^7*(2*x^2-x+3)^(7/2)

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Maxima [A]  time = 0.799668, size = 317, normalized size = 1.25 \[ \frac{625}{28} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{7} + \frac{13875}{208} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{6} + \frac{1046225}{9984} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{5} + \frac{3684995}{39936} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{4} + \frac{23460839}{532480} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{3} + \frac{122595067}{19169280} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{2} + \frac{112244125}{122683392} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x + \frac{25250178739}{5725224960} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} + \frac{401135647}{83886080} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{401135647}{335544320} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{9226119881}{536870912} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{9226119881}{2147483648} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{636602271789}{8589934592} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{14641852251147}{137438953472} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{636602271789}{34359738368} \, \sqrt{2 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

625/28*(2*x^2 - x + 3)^(7/2)*x^7 + 13875/208*(2*x^2 - x + 3)^(7/2)*x^6 + 1046225
/9984*(2*x^2 - x + 3)^(7/2)*x^5 + 3684995/39936*(2*x^2 - x + 3)^(7/2)*x^4 + 2346
0839/532480*(2*x^2 - x + 3)^(7/2)*x^3 + 122595067/19169280*(2*x^2 - x + 3)^(7/2)
*x^2 + 112244125/122683392*(2*x^2 - x + 3)^(7/2)*x + 25250178739/5725224960*(2*x
^2 - x + 3)^(7/2) + 401135647/83886080*(2*x^2 - x + 3)^(5/2)*x - 401135647/33554
4320*(2*x^2 - x + 3)^(5/2) + 9226119881/536870912*(2*x^2 - x + 3)^(3/2)*x - 9226
119881/2147483648*(2*x^2 - x + 3)^(3/2) + 636602271789/8589934592*sqrt(2*x^2 - x
+ 3)*x + 14641852251147/137438953472*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) -
636602271789/34359738368*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 0.292602, size = 170, normalized size = 0.67 \[ \frac{1}{1125625028935680} \, \sqrt{2}{\left (4 \, \sqrt{2}{\left (25125558681600000 \, x^{13} + 37398427729920000 \, x^{12} + 137233466130432000 \, x^{11} + 204932411660697600 \, x^{10} + 363646430503501824 \, x^{9} + 439064558846345216 \, x^{8} + 530502956133122048 \, x^{7} + 485091164642279424 \, x^{6} + 405468382284161024 \, x^{5} + 257786732552566784 \, x^{4} + 142490931553577856 \, x^{3} + 50064174038215008 \, x^{2} + 12071614275862524 \, x + 10820567498568669\right )} \sqrt{2 \, x^{2} - x + 3} + 59958384968446965 \, \log \left (-\sqrt{2}{\left (32 \, x^{2} - 16 \, x + 25\right )} - 8 \, \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1125625028935680*sqrt(2)*(4*sqrt(2)*(25125558681600000*x^13 + 3739842772992000
0*x^12 + 137233466130432000*x^11 + 204932411660697600*x^10 + 363646430503501824*
x^9 + 439064558846345216*x^8 + 530502956133122048*x^7 + 485091164642279424*x^6 +
405468382284161024*x^5 + 257786732552566784*x^4 + 142490931553577856*x^3 + 5006
4174038215008*x^2 + 12071614275862524*x + 10820567498568669)*sqrt(2*x^2 - x + 3)
+ 59958384968446965*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. \[ \int \left (2 x^{2} - x + 3\right )^{\frac{5}{2}} \left (5 x^{2} + 3 x + 2\right )^{4}\, dx \]

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)**4,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273533, size = 153, normalized size = 0.6 \[ \frac{1}{140703128616960} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (32 \,{\left (12 \,{\left (200 \,{\left (20 \,{\left (240 \,{\left (260 \, x + 387\right )} x + 340823\right )} x + 10179103\right )} x + 3612502719\right )} x + 52340574127\right )} x + 2023708176167\right )} x + 7401903757359\right )} x + 49495652134297\right )} x + 125872428004183\right )} x + 1113210402762327\right )} x + 1564505438694219\right )} x + 3017903568965631\right )} x + 10820567498568669\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{14641852251147}{137438953472} \, \sqrt{2}{\rm ln}\left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/140703128616960*(4*(8*(4*(16*(4*(8*(4*(32*(12*(200*(20*(240*(260*x + 387)*x +
340823)*x + 10179103)*x + 3612502719)*x + 52340574127)*x + 2023708176167)*x + 74
01903757359)*x + 49495652134297)*x + 125872428004183)*x + 1113210402762327)*x +
1564505438694219)*x + 3017903568965631)*x + 10820567498568669)*sqrt(2*x^2 - x +
3) - 14641852251147/137438953472*sqrt(2)*ln(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 -
x + 3)) + 1)