3.134 \(\int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt{6+17 x+12 x^2} \, dx\)

Optimal. Leaf size=103 \[ -\frac{1}{20} \left (12 x^2+17 x+6\right )^{5/2}+\frac{97}{768} (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}-\frac{97 (24 x+17) \sqrt{12 x^2+17 x+6}}{24576}+\frac{97 \tanh ^{-1}\left (\frac{24 x+17}{4 \sqrt{3} \sqrt{12 x^2+17 x+6}}\right )}{98304 \sqrt{3}} \]

[Out]

(-97*(17 + 24*x)*Sqrt[6 + 17*x + 12*x^2])/24576 + (97*(17 + 24*x)*(6 + 17*x + 12
*x^2)^(3/2))/768 - (6 + 17*x + 12*x^2)^(5/2)/20 + (97*ArcTanh[(17 + 24*x)/(4*Sqr
t[3]*Sqrt[6 + 17*x + 12*x^2])])/(98304*Sqrt[3])

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Rubi [A]  time = 0.113357, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{1}{20} \left (12 x^2+17 x+6\right )^{5/2}+\frac{97}{768} (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}-\frac{97 (24 x+17) \sqrt{12 x^2+17 x+6}}{24576}+\frac{97 \tanh ^{-1}\left (\frac{24 x+17}{4 \sqrt{3} \sqrt{12 x^2+17 x+6}}\right )}{98304 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)*(30 + 31*x - 12*x^2)*Sqrt[6 + 17*x + 12*x^2],x]

[Out]

(-97*(17 + 24*x)*Sqrt[6 + 17*x + 12*x^2])/24576 + (97*(17 + 24*x)*(6 + 17*x + 12
*x^2)^(3/2))/768 - (6 + 17*x + 12*x^2)^(5/2)/20 + (97*ArcTanh[(17 + 24*x)/(4*Sqr
t[3]*Sqrt[6 + 17*x + 12*x^2])])/(98304*Sqrt[3])

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Rubi in Sympy [A]  time = 17.7073, size = 94, normalized size = 0.91 \[ \frac{97 \left (24 x + 17\right ) \left (12 x^{2} + 17 x + 6\right )^{\frac{3}{2}}}{768} - \frac{97 \left (24 x + 17\right ) \sqrt{12 x^{2} + 17 x + 6}}{24576} - \frac{\left (12 x^{2} + 17 x + 6\right )^{\frac{5}{2}}}{20} + \frac{97 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (24 x + 17\right )}{12 \sqrt{12 x^{2} + 17 x + 6}} \right )}}{294912} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)*(-12*x**2+31*x+30)*(12*x**2+17*x+6)**(1/2),x)

[Out]

97*(24*x + 17)*(12*x**2 + 17*x + 6)**(3/2)/768 - 97*(24*x + 17)*sqrt(12*x**2 + 1
7*x + 6)/24576 - (12*x**2 + 17*x + 6)**(5/2)/20 + 97*sqrt(3)*atanh(sqrt(3)*(24*x
+ 17)/(12*sqrt(12*x**2 + 17*x + 6)))/294912

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Mathematica [A]  time = 0.0840474, size = 70, normalized size = 0.68 \[ \frac{485 \sqrt{3} \log \left (4 \sqrt{36 x^2+51 x+18}+24 x+17\right )+12 \sqrt{12 x^2+17 x+6} \left (-884736 x^4+1963008 x^3+6837888 x^2+5455144 x+1353611\right )}{1474560} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)*(30 + 31*x - 12*x^2)*Sqrt[6 + 17*x + 12*x^2],x]

[Out]

(12*Sqrt[6 + 17*x + 12*x^2]*(1353611 + 5455144*x + 6837888*x^2 + 1963008*x^3 - 8
84736*x^4) + 485*Sqrt[3]*Log[17 + 24*x + 4*Sqrt[18 + 51*x + 36*x^2]])/1474560

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Maple [A]  time = 0.009, size = 96, normalized size = 0.9 \[ -{\frac{1649+2328\,x}{24576}\sqrt{12\,{x}^{2}+17\,x+6}}+{\frac{97\,\sqrt{12}}{589824}\ln \left ({\frac{\sqrt{12}}{12} \left ({\frac{17}{2}}+12\,x \right ) }+\sqrt{12\,{x}^{2}+17\,x+6} \right ) }+{\frac{7093}{3840} \left ( 12\,{x}^{2}+17\,x+6 \right ) ^{{\frac{3}{2}}}}+{\frac{349\,x}{160} \left ( 12\,{x}^{2}+17\,x+6 \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{x}^{2}}{5} \left ( 12\,{x}^{2}+17\,x+6 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)*(-12*x^2+31*x+30)*(12*x^2+17*x+6)^(1/2),x)

[Out]

-97/24576*(17+24*x)*(12*x^2+17*x+6)^(1/2)+97/589824*ln(1/12*(17/2+12*x)*12^(1/2)
+(12*x^2+17*x+6)^(1/2))*12^(1/2)+7093/3840*(12*x^2+17*x+6)^(3/2)+349/160*x*(12*x
^2+17*x+6)^(3/2)-3/5*x^2*(12*x^2+17*x+6)^(3/2)

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Maxima [A]  time = 0.779977, size = 140, normalized size = 1.36 \[ -\frac{3}{5} \,{\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac{3}{2}} x^{2} + \frac{349}{160} \,{\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac{3}{2}} x + \frac{7093}{3840} \,{\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac{3}{2}} - \frac{97}{1024} \, \sqrt{12 \, x^{2} + 17 \, x + 6} x + \frac{97}{294912} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{12 \, x^{2} + 17 \, x + 6} + 24 \, x + 17\right ) - \frac{1649}{24576} \, \sqrt{12 \, x^{2} + 17 \, x + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(12*x^2 + 17*x + 6)*(12*x^2 - 31*x - 30)*(3*x + 2),x, algorithm="maxima")

[Out]

-3/5*(12*x^2 + 17*x + 6)^(3/2)*x^2 + 349/160*(12*x^2 + 17*x + 6)^(3/2)*x + 7093/
3840*(12*x^2 + 17*x + 6)^(3/2) - 97/1024*sqrt(12*x^2 + 17*x + 6)*x + 97/294912*s
qrt(3)*log(4*sqrt(3)*sqrt(12*x^2 + 17*x + 6) + 24*x + 17) - 1649/24576*sqrt(12*x
^2 + 17*x + 6)

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Fricas [A]  time = 0.282939, size = 108, normalized size = 1.05 \[ -\frac{1}{2949120} \, \sqrt{3}{\left (8 \, \sqrt{3}{\left (884736 \, x^{4} - 1963008 \, x^{3} - 6837888 \, x^{2} - 5455144 \, x - 1353611\right )} \sqrt{12 \, x^{2} + 17 \, x + 6} - 485 \, \log \left (\sqrt{3}{\left (1152 \, x^{2} + 1632 \, x + 577\right )} + 24 \, \sqrt{12 \, x^{2} + 17 \, x + 6}{\left (24 \, x + 17\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(12*x^2 + 17*x + 6)*(12*x^2 - 31*x - 30)*(3*x + 2),x, algorithm="fricas")

[Out]

-1/2949120*sqrt(3)*(8*sqrt(3)*(884736*x^4 - 1963008*x^3 - 6837888*x^2 - 5455144*
x - 1353611)*sqrt(12*x^2 + 17*x + 6) - 485*log(sqrt(3)*(1152*x^2 + 1632*x + 577)
+ 24*sqrt(12*x^2 + 17*x + 6)*(24*x + 17)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- 152 x \sqrt{12 x^{2} + 17 x + 6}\right )\, dx - \int \left (- 69 x^{2} \sqrt{12 x^{2} + 17 x + 6}\right )\, dx - \int 36 x^{3} \sqrt{12 x^{2} + 17 x + 6}\, dx - \int \left (- 60 \sqrt{12 x^{2} + 17 x + 6}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)*(-12*x**2+31*x+30)*(12*x**2+17*x+6)**(1/2),x)

[Out]

-Integral(-152*x*sqrt(12*x**2 + 17*x + 6), x) - Integral(-69*x**2*sqrt(12*x**2 +
17*x + 6), x) - Integral(36*x**3*sqrt(12*x**2 + 17*x + 6), x) - Integral(-60*sq
rt(12*x**2 + 17*x + 6), x)

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GIAC/XCAS [A]  time = 0.267539, size = 95, normalized size = 0.92 \[ -\frac{1}{122880} \,{\left (8 \,{\left (48 \,{\left (72 \,{\left (32 \, x - 71\right )} x - 17807\right )} x - 681893\right )} x - 1353611\right )} \sqrt{12 \, x^{2} + 17 \, x + 6} - \frac{97}{294912} \, \sqrt{3}{\rm ln}\left ({\left | -4 \, \sqrt{3}{\left (2 \, \sqrt{3} x - \sqrt{12 \, x^{2} + 17 \, x + 6}\right )} - 17 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(12*x^2 + 17*x + 6)*(12*x^2 - 31*x - 30)*(3*x + 2),x, algorithm="giac")

[Out]

-1/122880*(8*(48*(72*(32*x - 71)*x - 17807)*x - 681893)*x - 1353611)*sqrt(12*x^2
+ 17*x + 6) - 97/294912*sqrt(3)*ln(abs(-4*sqrt(3)*(2*sqrt(3)*x - sqrt(12*x^2 +
17*x + 6)) - 17))