3.2418 \(\int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=158 \[ -\frac{1}{24} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3+\frac{67}{126} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^2+\frac{(33210 x+75451) \left (3 x^2+5 x+2\right )^{5/2}}{15120}+\frac{12277 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{20736}-\frac{12277 (6 x+5) \sqrt{3 x^2+5 x+2}}{165888}+\frac{12277 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{331776 \sqrt{3}} \]

[Out]

(-12277*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/165888 + (12277*(5 + 6*x)*(2 + 5*x + 3*
x^2)^(3/2))/20736 + (67*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(5/2))/126 - ((3 + 2*x)^3*
(2 + 5*x + 3*x^2)^(5/2))/24 + ((75451 + 33210*x)*(2 + 5*x + 3*x^2)^(5/2))/15120
+ (12277*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(331776*Sqrt[3])

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Rubi [A]  time = 0.248991, antiderivative size = 158, 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 \[ -\frac{1}{24} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3+\frac{67}{126} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^2+\frac{(33210 x+75451) \left (3 x^2+5 x+2\right )^{5/2}}{15120}+\frac{12277 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{20736}-\frac{12277 (6 x+5) \sqrt{3 x^2+5 x+2}}{165888}+\frac{12277 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{331776 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-12277*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/165888 + (12277*(5 + 6*x)*(2 + 5*x + 3*
x^2)^(3/2))/20736 + (67*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(5/2))/126 - ((3 + 2*x)^3*
(2 + 5*x + 3*x^2)^(5/2))/24 + ((75451 + 33210*x)*(2 + 5*x + 3*x^2)^(5/2))/15120
+ (12277*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(331776*Sqrt[3])

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Rubi in Sympy [A]  time = 26.728, size = 144, normalized size = 0.91 \[ - \frac{\left (2 x + 3\right )^{3} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{24} + \frac{67 \left (2 x + 3\right )^{2} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{126} + \frac{12277 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{20736} - \frac{12277 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{165888} + \frac{\left (298890 x + 679059\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{136080} + \frac{12277 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{995328} \]

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)**(3/2),x)

[Out]

-(2*x + 3)**3*(3*x**2 + 5*x + 2)**(5/2)/24 + 67*(2*x + 3)**2*(3*x**2 + 5*x + 2)*
*(5/2)/126 + 12277*(6*x + 5)*(3*x**2 + 5*x + 2)**(3/2)/20736 - 12277*(6*x + 5)*s
qrt(3*x**2 + 5*x + 2)/165888 + (298890*x + 679059)*(3*x**2 + 5*x + 2)**(5/2)/136
080 + 12277*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/995328

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Mathematica [A]  time = 0.103882, size = 85, normalized size = 0.54 \[ \frac{429695 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (17418240 x^7+25297920 x^6-368236800 x^5-1650151296 x^4-2993047920 x^3-2762417688 x^2-1276112350 x-233137461\right )}{34836480} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-233137461 - 1276112350*x - 2762417688*x^2 - 29930479
20*x^3 - 1650151296*x^4 - 368236800*x^5 + 25297920*x^6 + 17418240*x^7) + 429695*
Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/34836480

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Maple [A]  time = 0.01, size = 132, normalized size = 0.8 \[{\frac{61385+73662\,x}{20736} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{61385+73662\,x}{165888}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{12277\,\sqrt{3}}{995328}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{130801}{15120} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{1063\,x}{168} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{79\,{x}^{2}}{126} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{{x}^{3}}{3} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12277/20736*(5+6*x)*(3*x^2+5*x+2)^(3/2)-12277/165888*(5+6*x)*(3*x^2+5*x+2)^(1/2)
+12277/995328*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+130801/15120
*(3*x^2+5*x+2)^(5/2)+1063/168*x*(3*x^2+5*x+2)^(5/2)+79/126*x^2*(3*x^2+5*x+2)^(5/
2)-1/3*x^3*(3*x^2+5*x+2)^(5/2)

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Maxima [A]  time = 0.779197, size = 203, normalized size = 1.28 \[ -\frac{1}{3} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{3} + \frac{79}{126} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{2} + \frac{1063}{168} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{130801}{15120} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{12277}{3456} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{61385}{20736} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{12277}{27648} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{12277}{995328} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{61385}{165888} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*x^2 + 5*x + 2)^(5/2)*x^3 + 79/126*(3*x^2 + 5*x + 2)^(5/2)*x^2 + 1063/168
*(3*x^2 + 5*x + 2)^(5/2)*x + 130801/15120*(3*x^2 + 5*x + 2)^(5/2) + 12277/3456*(
3*x^2 + 5*x + 2)^(3/2)*x + 61385/20736*(3*x^2 + 5*x + 2)^(3/2) - 12277/27648*sqr
t(3*x^2 + 5*x + 2)*x + 12277/995328*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2)
+ 6*x + 5) - 61385/165888*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.276166, size = 128, normalized size = 0.81 \[ -\frac{1}{69672960} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (17418240 \, x^{7} + 25297920 \, x^{6} - 368236800 \, x^{5} - 1650151296 \, x^{4} - 2993047920 \, x^{3} - 2762417688 \, x^{2} - 1276112350 \, x - 233137461\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 429695 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/69672960*sqrt(3)*(4*sqrt(3)*(17418240*x^7 + 25297920*x^6 - 368236800*x^5 - 16
50151296*x^4 - 2993047920*x^3 - 2762417688*x^2 - 1276112350*x - 233137461)*sqrt(
3*x^2 + 5*x + 2) - 429695*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. \[ - \int \left (- 1161 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1872 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1367 x^{3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 382 x^{4} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 28 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 24 x^{6} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 270 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-1161*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1872*x**2*sqrt(3*x**2 +
5*x + 2), x) - Integral(-1367*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-382*x
**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(28*x**5*sqrt(3*x**2 + 5*x + 2), x) - I
ntegral(24*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(-270*sqrt(3*x**2 + 5*x + 2
), x)

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GIAC/XCAS [A]  time = 0.268377, size = 113, normalized size = 0.72 \[ -\frac{1}{5806080} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (30 \,{\left (12 \,{\left (42 \, x + 61\right )} x - 10655\right )} x - 1432423\right )} x - 20785055\right )} x - 115100737\right )} x - 638056175\right )} x - 233137461\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{12277}{995328} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5806080*(2*(12*(6*(8*(30*(12*(42*x + 61)*x - 10655)*x - 1432423)*x - 20785055
)*x - 115100737)*x - 638056175)*x - 233137461)*sqrt(3*x^2 + 5*x + 2) - 12277/995
328*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))