3.2450 \(\int (5-x) \left (2+5 x+3 x^2\right )^{7/2} \, dx\)

Optimal. Leaf size=149 \[ -\frac{1}{27} \left (3 x^2+5 x+2\right )^{9/2}+\frac{35}{288} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac{245 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{20736}+\frac{1225 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{995328}-\frac{1225 (6 x+5) \sqrt{3 x^2+5 x+2}}{7962624}+\frac{1225 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{15925248 \sqrt{3}} \]

[Out]

(-1225*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/7962624 + (1225*(5 + 6*x)*(2 + 5*x + 3*x
^2)^(3/2))/995328 - (245*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/20736 + (35*(5 + 6*x
)*(2 + 5*x + 3*x^2)^(7/2))/288 - (2 + 5*x + 3*x^2)^(9/2)/27 + (1225*ArcTanh[(5 +
6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(15925248*Sqrt[3])

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Rubi [A]  time = 0.117345, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{1}{27} \left (3 x^2+5 x+2\right )^{9/2}+\frac{35}{288} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac{245 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{20736}+\frac{1225 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{995328}-\frac{1225 (6 x+5) \sqrt{3 x^2+5 x+2}}{7962624}+\frac{1225 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{15925248 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(5 - x)*(2 + 5*x + 3*x^2)^(7/2),x]

[Out]

(-1225*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/7962624 + (1225*(5 + 6*x)*(2 + 5*x + 3*x
^2)^(3/2))/995328 - (245*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/20736 + (35*(5 + 6*x
)*(2 + 5*x + 3*x^2)^(7/2))/288 - (2 + 5*x + 3*x^2)^(9/2)/27 + (1225*ArcTanh[(5 +
6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(15925248*Sqrt[3])

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Rubi in Sympy [A]  time = 8.86648, size = 138, normalized size = 0.93 \[ \frac{35 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{288} - \frac{245 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{20736} + \frac{1225 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{995328} - \frac{1225 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{7962624} - \frac{\left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{27} + \frac{1225 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{47775744} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+5*x+2)**(7/2),x)

[Out]

35*(6*x + 5)*(3*x**2 + 5*x + 2)**(7/2)/288 - 245*(6*x + 5)*(3*x**2 + 5*x + 2)**(
5/2)/20736 + 1225*(6*x + 5)*(3*x**2 + 5*x + 2)**(3/2)/995328 - 1225*(6*x + 5)*sq
rt(3*x**2 + 5*x + 2)/7962624 - (3*x**2 + 5*x + 2)**(9/2)/27 + 1225*sqrt(3)*atanh
(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/47775744

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Mathematica [A]  time = 0.0948673, size = 90, normalized size = 0.6 \[ \frac{1225 \sqrt{3} \log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )-6 \sqrt{3 x^2+5 x+2} \left (23887872 x^8+2488320 x^7-452625408 x^6-1507127040 x^5-2320737408 x^4-2013572880 x^3-1014795048 x^2-278256050 x-32198883\right )}{47775744} \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - x)*(2 + 5*x + 3*x^2)^(7/2),x]

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-32198883 - 278256050*x - 1014795048*x^2 - 2013572880
*x^3 - 2320737408*x^4 - 1507127040*x^5 - 452625408*x^6 + 2488320*x^7 + 23887872*
x^8) + 1225*Sqrt[3]*Log[5 + 6*x + 2*Sqrt[6 + 15*x + 9*x^2]])/47775744

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Maple [A]  time = 0.007, size = 121, normalized size = 0.8 \[ -{\frac{1}{27} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{175+210\,x}{288} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{1225+1470\,x}{20736} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{6125+7350\,x}{995328} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{6125+7350\,x}{7962624}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{1225\,\sqrt{3}}{47775744}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+5*x+2)^(7/2),x)

[Out]

-1/27*(3*x^2+5*x+2)^(9/2)+35/288*(5+6*x)*(3*x^2+5*x+2)^(7/2)-245/20736*(5+6*x)*(
3*x^2+5*x+2)^(5/2)+1225/995328*(5+6*x)*(3*x^2+5*x+2)^(3/2)-1225/7962624*(5+6*x)*
(3*x^2+5*x+2)^(1/2)+1225/47775744*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*
3^(1/2)

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Maxima [A]  time = 0.794465, size = 215, normalized size = 1.44 \[ -\frac{1}{27} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} + \frac{35}{48} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{175}{288} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{245}{3456} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{1225}{20736} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{1225}{165888} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{6125}{995328} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{1225}{1327104} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{1225}{47775744} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{6125}{7962624} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(x - 5),x, algorithm="maxima")

[Out]

-1/27*(3*x^2 + 5*x + 2)^(9/2) + 35/48*(3*x^2 + 5*x + 2)^(7/2)*x + 175/288*(3*x^2
+ 5*x + 2)^(7/2) - 245/3456*(3*x^2 + 5*x + 2)^(5/2)*x - 1225/20736*(3*x^2 + 5*x
+ 2)^(5/2) + 1225/165888*(3*x^2 + 5*x + 2)^(3/2)*x + 6125/995328*(3*x^2 + 5*x +
2)^(3/2) - 1225/1327104*sqrt(3*x^2 + 5*x + 2)*x + 1225/47775744*sqrt(3)*log(2*s
qrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 6125/7962624*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.277488, size = 135, normalized size = 0.91 \[ -\frac{1}{95551488} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (23887872 \, x^{8} + 2488320 \, x^{7} - 452625408 \, x^{6} - 1507127040 \, x^{5} - 2320737408 \, x^{4} - 2013572880 \, x^{3} - 1014795048 \, x^{2} - 278256050 \, x - 32198883\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 1225 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(x - 5),x, algorithm="fricas")

[Out]

-1/95551488*sqrt(3)*(4*sqrt(3)*(23887872*x^8 + 2488320*x^7 - 452625408*x^6 - 150
7127040*x^5 - 2320737408*x^4 - 2013572880*x^3 - 1014795048*x^2 - 278256050*x - 3
2198883)*sqrt(3*x^2 + 5*x + 2) - 1225*log(sqrt(3)*(72*x^2 + 120*x + 49) + 12*sqr
t(3*x^2 + 5*x + 2)*(6*x + 5)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- 292 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 870 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1339 x^{3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1090 x^{4} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 396 x^{5} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 27 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 40 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3*x**2+5*x+2)**(7/2),x)

[Out]

-Integral(-292*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-870*x**2*sqrt(3*x**2 + 5
*x + 2), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1090*x*
*4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2), x) -
Integral(27*x**7*sqrt(3*x**2 + 5*x + 2), x) - Integral(-40*sqrt(3*x**2 + 5*x + 2
), x)

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GIAC/XCAS [A]  time = 0.270737, size = 120, normalized size = 0.81 \[ -\frac{1}{7962624} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (2 \,{\left (48 \, x + 5\right )} x - 1819\right )} x - 218045\right )} x - 2014529\right )} x - 13983145\right )} x - 42283127\right )} x - 139128025\right )} x - 32198883\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{1225}{47775744} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(x - 5),x, algorithm="giac")

[Out]

-1/7962624*(2*(12*(6*(8*(6*(36*(2*(48*x + 5)*x - 1819)*x - 218045)*x - 2014529)*
x - 13983145)*x - 42283127)*x - 139128025)*x - 32198883)*sqrt(3*x^2 + 5*x + 2) -
1225/47775744*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5
))