3.2422 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx\)

Optimal. Leaf size=123 \[ \frac{1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac{1}{128} (175-414 x) \sqrt{3 x^2+5 x+2}-\frac{2011 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{256 \sqrt{3}}+\frac{65}{32} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

((175 - 414*x)*Sqrt[2 + 5*x + 3*x^2])/128 + ((47 - 6*x)*(2 + 5*x + 3*x^2)^(3/2))
/48 - (2011*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(256*Sqrt[3])
+ (65*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/32

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Rubi [A]  time = 0.231574, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac{1}{128} (175-414 x) \sqrt{3 x^2+5 x+2}-\frac{2011 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{256 \sqrt{3}}+\frac{65}{32} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((175 - 414*x)*Sqrt[2 + 5*x + 3*x^2])/128 + ((47 - 6*x)*(2 + 5*x + 3*x^2)^(3/2))
/48 - (2011*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(256*Sqrt[3])
+ (65*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/32

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Rubi in Sympy [A]  time = 32.1941, size = 112, normalized size = 0.91 \[ \frac{\left (- 7452 x + 3150\right ) \sqrt{3 x^{2} + 5 x + 2}}{2304} + \frac{\left (- 18 x + 141\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{144} - \frac{2011 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{768} - \frac{65 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-7452*x + 3150)*sqrt(3*x**2 + 5*x + 2)/2304 + (-18*x + 141)*(3*x**2 + 5*x + 2)*
*(3/2)/144 - 2011*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/76
8 - 65*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/32

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Mathematica [A]  time = 0.0974627, size = 152, normalized size = 1.24 \[ \frac{1}{768} \left (1776 \sqrt{3 x^2+5 x+2} x^2+1084 \sqrt{3 x^2+5 x+2} x+2554 \sqrt{3 x^2+5 x+2}-1560 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-2011 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-288 \sqrt{3 x^2+5 x+2} x^3+1560 \sqrt{5} \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2554*Sqrt[2 + 5*x + 3*x^2] + 1084*x*Sqrt[2 + 5*x + 3*x^2] + 1776*x^2*Sqrt[2 + 5
*x + 3*x^2] - 288*x^3*Sqrt[2 + 5*x + 3*x^2] + 1560*Sqrt[5]*Log[3 + 2*x] - 1560*S
qrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]] - 2011*Sqrt[3]*Log[-5 - 6
*x - 2*Sqrt[6 + 15*x + 9*x^2]])/768

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Maple [A]  time = 0.011, size = 183, normalized size = 1.5 \[ -{\frac{5+6\,x}{48} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{5+6\,x}{384}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{\sqrt{3}}{2304}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{13}{12} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{65+78\,x}{24}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{377\,\sqrt{3}}{144}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }+{\frac{65}{32}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{65\,\sqrt{5}}{32}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(5+6*x)*(3*x^2+5*x+2)^(3/2)+1/384*(5+6*x)*(3*x^2+5*x+2)^(1/2)-1/2304*ln(1/
3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+13/12*(3*(x+3/2)^2-4*x-19/4)^(3
/2)-13/24*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-377/144*ln(1/3*(5/2+3*x)*3^(1/2)+
(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)+65/32*(12*(x+3/2)^2-16*x-19)^(1/2)-65/32*5
^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))

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Maxima [A]  time = 0.770367, size = 173, normalized size = 1.41 \[ -\frac{1}{8} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{47}{48} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{207}{64} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{2011}{768} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{65}{32} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{175}{128} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(3*x^2 + 5*x + 2)^(3/2)*x + 47/48*(3*x^2 + 5*x + 2)^(3/2) - 207/64*sqrt(3*x
^2 + 5*x + 2)*x - 2011/768*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2
) - 65/32*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x +
3) - 2) + 175/128*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.287875, size = 174, normalized size = 1.41 \[ -\frac{1}{4608} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (144 \, x^{3} - 888 \, x^{2} - 542 \, x - 1277\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 1560 \, \sqrt{5} \sqrt{3} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 6033 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4608*sqrt(3)*(4*sqrt(3)*(144*x^3 - 888*x^2 - 542*x - 1277)*sqrt(3*x^2 + 5*x +
2) - 1560*sqrt(5)*sqrt(3)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*
x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) - 6033*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. \[ - \int \left (- \frac{10 \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac{23 x \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac{10 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \frac{3 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-23*x*sqrt(3*x**2
+ 5*x + 2)/(2*x + 3), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x
) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x)

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GIAC/XCAS [A]  time = 0.311026, size = 184, normalized size = 1.5 \[ -\frac{1}{384} \,{\left (2 \,{\left (12 \,{\left (6 \, x - 37\right )} x - 271\right )} x - 1277\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{65}{32} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{2011}{768} \, \sqrt{3}{\rm ln}\left ({\left | -6 \, \sqrt{3} x - 5 \, \sqrt{3} + 6 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/384*(2*(12*(6*x - 37)*x - 271)*x - 1277)*sqrt(3*x^2 + 5*x + 2) + 65/32*sqrt(5
)*ln(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*
sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) + 2011/768*sqrt(3)
*ln(abs(-6*sqrt(3)*x - 5*sqrt(3) + 6*sqrt(3*x^2 + 5*x + 2)))