3.2417 \(\int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=183 \[ -\frac{1}{27} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^4+\frac{299}{648} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3+\frac{487}{486} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^2+\frac{(188910 x+420721) \left (3 x^2+5 x+2\right )^{5/2}}{58320}+\frac{454969 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{559872}-\frac{454969 (6 x+5) \sqrt{3 x^2+5 x+2}}{4478976}+\frac{454969 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{8957952 \sqrt{3}} \]

[Out]

(-454969*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/4478976 + (454969*(5 + 6*x)*(2 + 5*x +
3*x^2)^(3/2))/559872 + (487*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(5/2))/486 + (299*(3
+ 2*x)^3*(2 + 5*x + 3*x^2)^(5/2))/648 - ((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(5/2))/27
+ ((420721 + 188910*x)*(2 + 5*x + 3*x^2)^(5/2))/58320 + (454969*ArcTanh[(5 + 6*
x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(8957952*Sqrt[3])

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Rubi [A]  time = 0.298853, antiderivative size = 183, 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 \[ -\frac{1}{27} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^4+\frac{299}{648} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3+\frac{487}{486} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^2+\frac{(188910 x+420721) \left (3 x^2+5 x+2\right )^{5/2}}{58320}+\frac{454969 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{559872}-\frac{454969 (6 x+5) \sqrt{3 x^2+5 x+2}}{4478976}+\frac{454969 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{8957952 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-454969*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/4478976 + (454969*(5 + 6*x)*(2 + 5*x +
3*x^2)^(3/2))/559872 + (487*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(5/2))/486 + (299*(3
+ 2*x)^3*(2 + 5*x + 3*x^2)^(5/2))/648 - ((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(5/2))/27
+ ((420721 + 188910*x)*(2 + 5*x + 3*x^2)^(5/2))/58320 + (454969*ArcTanh[(5 + 6*
x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(8957952*Sqrt[3])

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Rubi in Sympy [A]  time = 35.0816, size = 168, normalized size = 0.92 \[ - \frac{\left (2 x + 3\right )^{4} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{27} + \frac{299 \left (2 x + 3\right )^{3} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{648} + \frac{487 \left (2 x + 3\right )^{2} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{486} + \frac{454969 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{559872} - \frac{454969 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{4478976} + \frac{\left (11901330 x + 26505423\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{3674160} + \frac{454969 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{26873856} \]

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)**(3/2),x)

[Out]

-(2*x + 3)**4*(3*x**2 + 5*x + 2)**(5/2)/27 + 299*(2*x + 3)**3*(3*x**2 + 5*x + 2)
**(5/2)/648 + 487*(2*x + 3)**2*(3*x**2 + 5*x + 2)**(5/2)/486 + 454969*(6*x + 5)*
(3*x**2 + 5*x + 2)**(3/2)/559872 - 454969*(6*x + 5)*sqrt(3*x**2 + 5*x + 2)/44789
76 + (11901330*x + 26505423)*(3*x**2 + 5*x + 2)**(5/2)/3674160 + 454969*sqrt(3)*
atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/26873856

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Mathematica [A]  time = 0.129438, size = 90, normalized size = 0.49 \[ \frac{2274845 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (119439360 x^8+370759680 x^7-2143687680 x^6-14811482880 x^5-37262745216 x^4-49917376080 x^3-37650690888 x^2-15049298650 x-2471988351\right )}{134369280} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-2471988351 - 15049298650*x - 37650690888*x^2 - 49917
376080*x^3 - 37262745216*x^4 - 14811482880*x^5 - 2143687680*x^6 + 370759680*x^7
+ 119439360*x^8) + 2274845*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/134
369280

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Maple [A]  time = 0.02, size = 149, normalized size = 0.8 \[{\frac{2274845+2729814\,x}{559872} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{2274845+2729814\,x}{4478976}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{454969\,\sqrt{3}}{26873856}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{1498291}{58320} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{2317\,x}{72} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{6133\,{x}^{2}}{486} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{11\,{x}^{3}}{81} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{16\,{x}^{4}}{27} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

454969/559872*(5+6*x)*(3*x^2+5*x+2)^(3/2)-454969/4478976*(5+6*x)*(3*x^2+5*x+2)^(
1/2)+454969/26873856*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+14982
91/58320*(3*x^2+5*x+2)^(5/2)+2317/72*x*(3*x^2+5*x+2)^(5/2)+6133/486*x^2*(3*x^2+5
*x+2)^(5/2)+11/81*x^3*(3*x^2+5*x+2)^(5/2)-16/27*x^4*(3*x^2+5*x+2)^(5/2)

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Maxima [A]  time = 0.776981, size = 225, normalized size = 1.23 \[ -\frac{16}{27} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{4} + \frac{11}{81} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{3} + \frac{6133}{486} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{2} + \frac{2317}{72} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{1498291}{58320} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{454969}{93312} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{2274845}{559872} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{454969}{746496} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{454969}{26873856} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{2274845}{4478976} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16/27*(3*x^2 + 5*x + 2)^(5/2)*x^4 + 11/81*(3*x^2 + 5*x + 2)^(5/2)*x^3 + 6133/48
6*(3*x^2 + 5*x + 2)^(5/2)*x^2 + 2317/72*(3*x^2 + 5*x + 2)^(5/2)*x + 1498291/5832
0*(3*x^2 + 5*x + 2)^(5/2) + 454969/93312*(3*x^2 + 5*x + 2)^(3/2)*x + 2274845/559
872*(3*x^2 + 5*x + 2)^(3/2) - 454969/746496*sqrt(3*x^2 + 5*x + 2)*x + 454969/268
73856*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 2274845/4478976*s
qrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.276641, size = 135, normalized size = 0.74 \[ -\frac{1}{268738560} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (119439360 \, x^{8} + 370759680 \, x^{7} - 2143687680 \, x^{6} - 14811482880 \, x^{5} - 37262745216 \, x^{4} - 49917376080 \, x^{3} - 37650690888 \, x^{2} - 15049298650 \, x - 2471988351\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 2274845 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/268738560*sqrt(3)*(4*sqrt(3)*(119439360*x^8 + 370759680*x^7 - 2143687680*x^6
- 14811482880*x^5 - 37262745216*x^4 - 49917376080*x^3 - 37650690888*x^2 - 150492
98650*x - 2471988351)*sqrt(3*x^2 + 5*x + 2) - 2274845*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. \[ - \int \left (- 4023 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 7938 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 7845 x^{3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3880 x^{4} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 680 x^{5} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 128 x^{6} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 48 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 810 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-4023*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-7938*x**2*sqrt(3*x**2 +
5*x + 2), x) - Integral(-7845*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3880*
x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-680*x**5*sqrt(3*x**2 + 5*x + 2), x)
- Integral(128*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(48*x**7*sqrt(3*x**2 +
5*x + 2), x) - Integral(-810*sqrt(3*x**2 + 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.27028, size = 120, normalized size = 0.66 \[ -\frac{1}{22394880} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (30 \,{\left (36 \,{\left (2 \,{\left (48 \, x + 149\right )} x - 1723\right )} x - 428573\right )} x - 32346133\right )} x - 346648445\right )} x - 1568778787\right )} x - 7524649325\right )} x - 2471988351\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{454969}{26873856} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/22394880*(2*(12*(6*(8*(30*(36*(2*(48*x + 149)*x - 1723)*x - 428573)*x - 32346
133)*x - 346648445)*x - 1568778787)*x - 7524649325)*x - 2471988351)*sqrt(3*x^2 +
5*x + 2) - 454969/26873856*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2)) - 5))