3.2432 \(\int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=181 \[ -\frac{1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}+\frac{169}{405} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac{(213878 x+477101) \left (3 x^2+5 x+2\right )^{7/2}}{136080}+\frac{182917 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac{182917 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac{182917 (6 x+5) \sqrt{3 x^2+5 x+2}}{35831808}-\frac{182917 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{71663616 \sqrt{3}} \]

[Out]

(182917*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/35831808 - (182917*(5 + 6*x)*(2 + 5*x +
3*x^2)^(3/2))/4478976 + (182917*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/466560 + (16
9*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2))/405 - ((3 + 2*x)^3*(2 + 5*x + 3*x^2)^(7/2
))/30 + ((477101 + 213878*x)*(2 + 5*x + 3*x^2)^(7/2))/136080 - (182917*ArcTanh[(
5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(71663616*Sqrt[3])

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Rubi [A]  time = 0.268542, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}+\frac{169}{405} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac{(213878 x+477101) \left (3 x^2+5 x+2\right )^{7/2}}{136080}+\frac{182917 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac{182917 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac{182917 (6 x+5) \sqrt{3 x^2+5 x+2}}{35831808}-\frac{182917 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{71663616 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(182917*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/35831808 - (182917*(5 + 6*x)*(2 + 5*x +
3*x^2)^(3/2))/4478976 + (182917*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/466560 + (16
9*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2))/405 - ((3 + 2*x)^3*(2 + 5*x + 3*x^2)^(7/2
))/30 + ((477101 + 213878*x)*(2 + 5*x + 3*x^2)^(7/2))/136080 - (182917*ArcTanh[(
5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(71663616*Sqrt[3])

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Rubi in Sympy [A]  time = 27.6911, size = 167, normalized size = 0.92 \[ - \frac{\left (2 x + 3\right )^{3} \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{30} + \frac{169 \left (2 x + 3\right )^{2} \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{405} + \frac{182917 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{466560} - \frac{182917 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{4478976} + \frac{182917 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{35831808} + \frac{\left (641634 x + 1431303\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{408240} - \frac{182917 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{214990848} \]

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)**(5/2),x)

[Out]

-(2*x + 3)**3*(3*x**2 + 5*x + 2)**(7/2)/30 + 169*(2*x + 3)**2*(3*x**2 + 5*x + 2)
**(7/2)/405 + 182917*(6*x + 5)*(3*x**2 + 5*x + 2)**(5/2)/466560 - 182917*(6*x +
5)*(3*x**2 + 5*x + 2)**(3/2)/4478976 + 182917*(6*x + 5)*sqrt(3*x**2 + 5*x + 2)/3
5831808 + (641634*x + 1431303)*(3*x**2 + 5*x + 2)**(7/2)/408240 - 182917*sqrt(3)
*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/214990848

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Mathematica [A]  time = 0.126018, size = 95, normalized size = 0.52 \[ \frac{-6402095 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (9029615616 x^9+29262643200 x^8-147947046912 x^7-1086687912960 x^6-2893044950784 x^5-4253933381760 x^4-3762746217360 x^3-1995914277480 x^2-585749416130 x-73178684475\right )}{7524679680} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-73178684475 - 585749416130*x - 1995914277480*x^2 - 3
762746217360*x^3 - 4253933381760*x^4 - 2893044950784*x^5 - 1086687912960*x^6 - 1
47947046912*x^7 + 29262643200*x^8 + 9029615616*x^9) - 6402095*Sqrt[3]*Log[-5 - 6
*x - 2*Sqrt[6 + 15*x + 9*x^2]])/7524679680

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Maple [A]  time = 0.01, size = 151, normalized size = 0.8 \[{\frac{914585+1097502\,x}{466560} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{914585+1097502\,x}{4478976} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{914585+1097502\,x}{35831808}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{182917\,\sqrt{3}}{214990848}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{173137}{27216} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{46453\,x}{9720} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{38\,{x}^{2}}{81} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{4\,{x}^{3}}{15} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

182917/466560*(5+6*x)*(3*x^2+5*x+2)^(5/2)-182917/4478976*(5+6*x)*(3*x^2+5*x+2)^(
3/2)+182917/35831808*(5+6*x)*(3*x^2+5*x+2)^(1/2)-182917/214990848*ln(1/3*(5/2+3*
x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+173137/27216*(3*x^2+5*x+2)^(7/2)+46453/9
720*x*(3*x^2+5*x+2)^(7/2)+38/81*x^2*(3*x^2+5*x+2)^(7/2)-4/15*x^3*(3*x^2+5*x+2)^(
7/2)

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Maxima [A]  time = 0.769872, size = 242, normalized size = 1.34 \[ -\frac{4}{15} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x^{3} + \frac{38}{81} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x^{2} + \frac{46453}{9720} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{173137}{27216} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} + \frac{182917}{77760} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{182917}{93312} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{182917}{746496} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{914585}{4478976} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{182917}{5971968} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{182917}{214990848} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{914585}{35831808} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/15*(3*x^2 + 5*x + 2)^(7/2)*x^3 + 38/81*(3*x^2 + 5*x + 2)^(7/2)*x^2 + 46453/97
20*(3*x^2 + 5*x + 2)^(7/2)*x + 173137/27216*(3*x^2 + 5*x + 2)^(7/2) + 182917/777
60*(3*x^2 + 5*x + 2)^(5/2)*x + 182917/93312*(3*x^2 + 5*x + 2)^(5/2) - 182917/746
496*(3*x^2 + 5*x + 2)^(3/2)*x - 914585/4478976*(3*x^2 + 5*x + 2)^(3/2) + 182917/
5971968*sqrt(3*x^2 + 5*x + 2)*x - 182917/214990848*sqrt(3)*log(2*sqrt(3)*sqrt(3*
x^2 + 5*x + 2) + 6*x + 5) + 914585/35831808*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.278186, size = 142, normalized size = 0.78 \[ -\frac{1}{15049359360} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (9029615616 \, x^{9} + 29262643200 \, x^{8} - 147947046912 \, x^{7} - 1086687912960 \, x^{6} - 2893044950784 \, x^{5} - 4253933381760 \, x^{4} - 3762746217360 \, x^{3} - 1995914277480 \, x^{2} - 585749416130 \, x - 73178684475\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 6402095 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15049359360*sqrt(3)*(4*sqrt(3)*(9029615616*x^9 + 29262643200*x^8 - 1479470469
12*x^7 - 1086687912960*x^6 - 2893044950784*x^5 - 4253933381760*x^4 - 37627462173
60*x^3 - 1995914277480*x^2 - 585749416130*x - 73178684475)*sqrt(3*x^2 + 5*x + 2)
- 6402095*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. \[ - \int \left (- 3672 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 10359 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 15577 x^{3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 13215 x^{4} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 5955 x^{5} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 958 x^{6} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 204 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 72 x^{8} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 540 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-3672*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-10359*x**2*sqrt(3*x**2
+ 5*x + 2), x) - Integral(-15577*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-132
15*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-5955*x**5*sqrt(3*x**2 + 5*x + 2),
x) - Integral(-958*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(204*x**7*sqrt(3*x
**2 + 5*x + 2), x) - Integral(72*x**8*sqrt(3*x**2 + 5*x + 2), x) - Integral(-540
*sqrt(3*x**2 + 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.269524, size = 127, normalized size = 0.7 \[ -\frac{1}{1254113280} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (14 \,{\left (48 \,{\left (54 \, x + 175\right )} x - 42469\right )} x - 4367155\right )} x - 418553957\right )} x - 3692650505\right )} x - 26130182065\right )} x - 83163094895\right )} x - 292874708065\right )} x - 73178684475\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{182917}{214990848} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1254113280*(2*(12*(6*(8*(6*(36*(14*(48*(54*x + 175)*x - 42469)*x - 4367155)*x
- 418553957)*x - 3692650505)*x - 26130182065)*x - 83163094895)*x - 292874708065
)*x - 73178684475)*sqrt(3*x^2 + 5*x + 2) + 182917/214990848*sqrt(3)*ln(abs(-2*sq
rt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))