3.2495 \(\int \frac{(5-x) (3+2 x)^3}{\sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=112 \[ -\frac{1}{12} \sqrt{3 x^2+5 x+2} (2 x+3)^3+\frac{32}{27} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{5}{648} (1078 x+3261) \sqrt{3 x^2+5 x+2}+\frac{19405 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1296 \sqrt{3}} \]

[Out]

(32*(3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2])/27 - ((3 + 2*x)^3*Sqrt[2 + 5*x + 3*x^2])/
12 + (5*(3261 + 1078*x)*Sqrt[2 + 5*x + 3*x^2])/648 + (19405*ArcTanh[(5 + 6*x)/(2
*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1296*Sqrt[3])

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Rubi [A]  time = 0.211943, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{1}{12} \sqrt{3 x^2+5 x+2} (2 x+3)^3+\frac{32}{27} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{5}{648} (1078 x+3261) \sqrt{3 x^2+5 x+2}+\frac{19405 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1296 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(32*(3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2])/27 - ((3 + 2*x)^3*Sqrt[2 + 5*x + 3*x^2])/
12 + (5*(3261 + 1078*x)*Sqrt[2 + 5*x + 3*x^2])/648 + (19405*ArcTanh[(5 + 6*x)/(2
*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1296*Sqrt[3])

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Rubi in Sympy [A]  time = 26.0339, size = 100, normalized size = 0.89 \[ - \frac{\left (2 x + 3\right )^{3} \sqrt{3 x^{2} + 5 x + 2}}{12} + \frac{32 \left (2 x + 3\right )^{2} \sqrt{3 x^{2} + 5 x + 2}}{27} + \frac{\left (16170 x + 48915\right ) \sqrt{3 x^{2} + 5 x + 2}}{1944} + \frac{19405 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{3888} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**3/(3*x**2+5*x+2)**(1/2),x)

[Out]

-(2*x + 3)**3*sqrt(3*x**2 + 5*x + 2)/12 + 32*(2*x + 3)**2*sqrt(3*x**2 + 5*x + 2)
/27 + (16170*x + 48915)*sqrt(3*x**2 + 5*x + 2)/1944 + 19405*sqrt(3)*atanh(sqrt(3
)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/3888

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Mathematica [A]  time = 0.0742255, size = 65, normalized size = 0.58 \[ \frac{19405 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (432 x^3-1128 x^2-11690 x-21759\right )}{3888} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-21759 - 11690*x - 1128*x^2 + 432*x^3) + 19405*Sqrt[3
]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/3888

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Maple [A]  time = 0.008, size = 94, normalized size = 0.8 \[{\frac{19405\,\sqrt{3}}{3888}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{7253}{216}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{5845\,x}{324}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{47\,{x}^{2}}{27}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{2\,{x}^{3}}{3}\sqrt{3\,{x}^{2}+5\,x+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3+2*x)^3/(3*x^2+5*x+2)^(1/2),x)

[Out]

19405/3888*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+7253/216*(3*x^2
+5*x+2)^(1/2)+5845/324*x*(3*x^2+5*x+2)^(1/2)+47/27*x^2*(3*x^2+5*x+2)^(1/2)-2/3*x
^3*(3*x^2+5*x+2)^(1/2)

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Maxima [A]  time = 0.803017, size = 124, normalized size = 1.11 \[ -\frac{2}{3} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{3} + \frac{47}{27} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{2} + \frac{5845}{324} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{19405}{3888} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{7253}{216} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*sqrt(3*x^2 + 5*x + 2)*x^3 + 47/27*sqrt(3*x^2 + 5*x + 2)*x^2 + 5845/324*sqrt
(3*x^2 + 5*x + 2)*x + 19405/3888*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6
*x + 5) + 7253/216*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.270737, size = 101, normalized size = 0.9 \[ -\frac{1}{7776} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (432 \, x^{3} - 1128 \, x^{2} - 11690 \, x - 21759\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 19405 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7776*sqrt(3)*(4*sqrt(3)*(432*x^3 - 1128*x^2 - 11690*x - 21759)*sqrt(3*x^2 + 5
*x + 2) - 19405*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. \[ - \int \left (- \frac{243 x}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{126 x^{2}}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{4 x^{3}}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{8 x^{4}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{135}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**3/(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(-243*x/sqrt(3*x**2 + 5*x + 2), x) - Integral(-126*x**2/sqrt(3*x**2 + 5
*x + 2), x) - Integral(-4*x**3/sqrt(3*x**2 + 5*x + 2), x) - Integral(8*x**4/sqrt
(3*x**2 + 5*x + 2), x) - Integral(-135/sqrt(3*x**2 + 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.272488, size = 86, normalized size = 0.77 \[ -\frac{1}{648} \,{\left (2 \,{\left (12 \,{\left (18 \, x - 47\right )} x - 5845\right )} x - 21759\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{19405}{3888} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/648*(2*(12*(18*x - 47)*x - 5845)*x - 21759)*sqrt(3*x^2 + 5*x + 2) - 19405/388
8*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))