3.2420 \(\int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=108 \[ \frac{1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac{839 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac{839 (6 x+5) \sqrt{3 x^2+5 x+2}}{20736}+\frac{839 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{41472 \sqrt{3}} \]

[Out]

(-839*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/20736 + (839*(5 + 6*x)*(2 + 5*x + 3*x^2)^
(3/2))/2592 + ((161 - 30*x)*(2 + 5*x + 3*x^2)^(5/2))/270 + (839*ArcTanh[(5 + 6*x
)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(41472*Sqrt[3])

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Rubi [A]  time = 0.0986917, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac{839 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac{839 (6 x+5) \sqrt{3 x^2+5 x+2}}{20736}+\frac{839 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{41472 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-839*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/20736 + (839*(5 + 6*x)*(2 + 5*x + 3*x^2)^
(3/2))/2592 + ((161 - 30*x)*(2 + 5*x + 3*x^2)^(5/2))/270 + (839*ArcTanh[(5 + 6*x
)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(41472*Sqrt[3])

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Rubi in Sympy [A]  time = 9.91414, size = 99, normalized size = 0.92 \[ \frac{\left (- 30 x + 161\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{270} + \frac{839 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{2592} - \frac{839 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{20736} + \frac{839 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{124416} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-30*x + 161)*(3*x**2 + 5*x + 2)**(5/2)/270 + 839*(6*x + 5)*(3*x**2 + 5*x + 2)**
(3/2)/2592 - 839*(6*x + 5)*sqrt(3*x**2 + 5*x + 2)/20736 + 839*sqrt(3)*atanh(sqrt
(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/124416

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Mathematica [A]  time = 0.0686789, size = 81, normalized size = 0.75 \[ \frac{839 \log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )}{41472 \sqrt{3}}-\sqrt{3 x^2+5 x+2} \left (x^5-\frac{61 x^4}{30}-\frac{941 x^3}{48}-\frac{148637 x^2}{4320}-\frac{240695 x}{10368}-\frac{187307}{34560}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[2 + 5*x + 3*x^2]*(-187307/34560 - (240695*x)/10368 - (148637*x^2)/4320 -
(941*x^3)/48 - (61*x^4)/30 + x^5)) + (839*Log[5 + 6*x + 2*Sqrt[6 + 15*x + 9*x^2]
])/(41472*Sqrt[3])

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Maple [A]  time = 0.007, size = 98, normalized size = 0.9 \[{\frac{4195+5034\,x}{2592} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{4195+5034\,x}{20736}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{839\,\sqrt{3}}{124416}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{161}{270} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{x}{9} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

839/2592*(5+6*x)*(3*x^2+5*x+2)^(3/2)-839/20736*(5+6*x)*(3*x^2+5*x+2)^(1/2)+839/1
24416*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+161/270*(3*x^2+5*x+2
)^(5/2)-1/9*x*(3*x^2+5*x+2)^(5/2)

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Maxima [A]  time = 0.776393, size = 157, normalized size = 1.45 \[ -\frac{1}{9} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{161}{270} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{839}{432} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{4195}{2592} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{839}{3456} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{839}{124416} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{4195}{20736} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(3*x^2 + 5*x + 2)^(5/2)*x + 161/270*(3*x^2 + 5*x + 2)^(5/2) + 839/432*(3*x^
2 + 5*x + 2)^(3/2)*x + 4195/2592*(3*x^2 + 5*x + 2)^(3/2) - 839/3456*sqrt(3*x^2 +
5*x + 2)*x + 839/124416*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5)
- 4195/20736*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.276202, size = 115, normalized size = 1.06 \[ -\frac{1}{1244160} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (103680 \, x^{5} - 210816 \, x^{4} - 2032560 \, x^{3} - 3567288 \, x^{2} - 2406950 \, x - 561921\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 4195 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1244160*sqrt(3)*(4*sqrt(3)*(103680*x^5 - 210816*x^4 - 2032560*x^3 - 3567288*x
^2 - 2406950*x - 561921)*sqrt(3*x^2 + 5*x + 2) - 4195*log(sqrt(3)*(72*x^2 + 120*
x + 49) + 12*sqrt(3*x^2 + 5*x + 2)*(6*x + 5)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- 89 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 76 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 11 x^{3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 6 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 30 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-89*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-76*x**2*sqrt(3*x**2 + 5*x
+ 2), x) - Integral(-11*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(6*x**4*sqrt(
3*x**2 + 5*x + 2), x) - Integral(-30*sqrt(3*x**2 + 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.276079, size = 100, normalized size = 0.93 \[ -\frac{1}{103680} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \,{\left (30 \, x - 61\right )} x - 4705\right )} x - 148637\right )} x - 1203475\right )} x - 561921\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{839}{124416} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/103680*(2*(12*(18*(8*(30*x - 61)*x - 4705)*x - 148637)*x - 1203475)*x - 56192
1)*sqrt(3*x^2 + 5*x + 2) - 839/124416*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqr
t(3*x^2 + 5*x + 2)) - 5))