3.2494 \(\int \frac{(5-x) (3+2 x)^4}{\sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=137 \[ -\frac{1}{15} \sqrt{3 x^2+5 x+2} (2 x+3)^4+\frac{53}{60} \sqrt{3 x^2+5 x+2} (2 x+3)^3+\frac{391}{135} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{1}{648} (9650 x+27519) \sqrt{3 x^2+5 x+2}+\frac{28051 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1296 \sqrt{3}} \]

[Out]

(391*(3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2])/135 + (53*(3 + 2*x)^3*Sqrt[2 + 5*x + 3*x
^2])/60 - ((3 + 2*x)^4*Sqrt[2 + 5*x + 3*x^2])/15 + ((27519 + 9650*x)*Sqrt[2 + 5*
x + 3*x^2])/648 + (28051*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(
1296*Sqrt[3])

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Rubi [A]  time = 0.296324, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{1}{15} \sqrt{3 x^2+5 x+2} (2 x+3)^4+\frac{53}{60} \sqrt{3 x^2+5 x+2} (2 x+3)^3+\frac{391}{135} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{1}{648} (9650 x+27519) \sqrt{3 x^2+5 x+2}+\frac{28051 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1296 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(3 + 2*x)^4)/Sqrt[2 + 5*x + 3*x^2],x]

[Out]

(391*(3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2])/135 + (53*(3 + 2*x)^3*Sqrt[2 + 5*x + 3*x
^2])/60 - ((3 + 2*x)^4*Sqrt[2 + 5*x + 3*x^2])/15 + ((27519 + 9650*x)*Sqrt[2 + 5*
x + 3*x^2])/648 + (28051*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 = 35.3375, size = 124, normalized size = 0.91 \[ - \frac{\left (2 x + 3\right )^{4} \sqrt{3 x^{2} + 5 x + 2}}{15} + \frac{53 \left (2 x + 3\right )^{3} \sqrt{3 x^{2} + 5 x + 2}}{60} + \frac{391 \left (2 x + 3\right )^{2} \sqrt{3 x^{2} + 5 x + 2}}{135} + \frac{\left (434250 x + 1238355\right ) \sqrt{3 x^{2} + 5 x + 2}}{29160} + \frac{28051 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{3888} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**4/(3*x**2+5*x+2)**(1/2),x)

[Out]

-(2*x + 3)**4*sqrt(3*x**2 + 5*x + 2)/15 + 53*(2*x + 3)**3*sqrt(3*x**2 + 5*x + 2)
/60 + 391*(2*x + 3)**2*sqrt(3*x**2 + 5*x + 2)/135 + (434250*x + 1238355)*sqrt(3*
x**2 + 5*x + 2)/29160 + 28051*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5
*x + 2)))/3888

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Mathematica [A]  time = 0.103195, size = 70, normalized size = 0.51 \[ \frac{140255 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (3456 x^4-2160 x^3-93912 x^2-268750 x-281829\right )}{19440} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(3 + 2*x)^4)/Sqrt[2 + 5*x + 3*x^2],x]

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-281829 - 268750*x - 93912*x^2 - 2160*x^3 + 3456*x^4)
+ 140255*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/19440

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Maple [A]  time = 0.017, size = 111, normalized size = 0.8 \[{\frac{28051\,\sqrt{3}}{3888}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{93943}{1080}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{26875\,x}{324}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{3913\,{x}^{2}}{135}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{2\,{x}^{3}}{3}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{16\,{x}^{4}}{15}\sqrt{3\,{x}^{2}+5\,x+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3+2*x)^4/(3*x^2+5*x+2)^(1/2),x)

[Out]

28051/3888*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+93943/1080*(3*x
^2+5*x+2)^(1/2)+26875/324*x*(3*x^2+5*x+2)^(1/2)+3913/135*x^2*(3*x^2+5*x+2)^(1/2)
+2/3*x^3*(3*x^2+5*x+2)^(1/2)-16/15*x^4*(3*x^2+5*x+2)^(1/2)

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Maxima [A]  time = 0.796875, size = 147, normalized size = 1.07 \[ -\frac{16}{15} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{4} + \frac{2}{3} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{3} + \frac{3913}{135} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{2} + \frac{26875}{324} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{28051}{3888} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{93943}{1080} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/sqrt(3*x^2 + 5*x + 2),x, algorithm="maxima")

[Out]

-16/15*sqrt(3*x^2 + 5*x + 2)*x^4 + 2/3*sqrt(3*x^2 + 5*x + 2)*x^3 + 3913/135*sqrt
(3*x^2 + 5*x + 2)*x^2 + 26875/324*sqrt(3*x^2 + 5*x + 2)*x + 28051/3888*sqrt(3)*l
og(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 93943/1080*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.27239, size = 108, normalized size = 0.79 \[ -\frac{1}{38880} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (3456 \, x^{4} - 2160 \, x^{3} - 93912 \, x^{2} - 268750 \, x - 281829\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 140255 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/sqrt(3*x^2 + 5*x + 2),x, algorithm="fricas")

[Out]

-1/38880*sqrt(3)*(4*sqrt(3)*(3456*x^4 - 2160*x^3 - 93912*x^2 - 268750*x - 281829
)*sqrt(3*x^2 + 5*x + 2) - 140255*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. \[ - \int \left (- \frac{999 x}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{864 x^{2}}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{264 x^{3}}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{16 x^{4}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \frac{16 x^{5}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{405}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**4/(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(-999*x/sqrt(3*x**2 + 5*x + 2), x) - Integral(-864*x**2/sqrt(3*x**2 + 5
*x + 2), x) - Integral(-264*x**3/sqrt(3*x**2 + 5*x + 2), x) - Integral(16*x**4/s
qrt(3*x**2 + 5*x + 2), x) - Integral(16*x**5/sqrt(3*x**2 + 5*x + 2), x) - Integr
al(-405/sqrt(3*x**2 + 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.290838, size = 93, normalized size = 0.68 \[ -\frac{1}{3240} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \, x - 5\right )} x - 3913\right )} x - 134375\right )} x - 281829\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{28051}{3888} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/sqrt(3*x^2 + 5*x + 2),x, algorithm="giac")

[Out]

-1/3240*(2*(12*(18*(8*x - 5)*x - 3913)*x - 134375)*x - 281829)*sqrt(3*x^2 + 5*x
+ 2) - 28051/3888*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))
- 5))