3.2431 \(\int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=206 \[ -\frac{1}{33} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^4+\frac{41}{110} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^3+\frac{3298 \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^2}{4455}+\frac{(3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}}{1496880}+\frac{249299 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac{249299 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac{249299 (6 x+5) \sqrt{3 x^2+5 x+2}}{35831808}-\frac{249299 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{71663616 \sqrt{3}} \]

[Out]

(249299*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/35831808 - (249299*(5 + 6*x)*(2 + 5*x +
3*x^2)^(3/2))/4478976 + (249299*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/466560 + (32
98*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2))/4455 + (41*(3 + 2*x)^3*(2 + 5*x + 3*x^2)
^(7/2))/110 - ((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(7/2))/33 + ((7405817 + 3365726*x)*
(2 + 5*x + 3*x^2)^(7/2))/1496880 - (249299*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 +
5*x + 3*x^2])])/(71663616*Sqrt[3])

_______________________________________________________________________________________

Rubi [A]  time = 0.352802, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{1}{33} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^4+\frac{41}{110} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^3+\frac{3298 \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^2}{4455}+\frac{(3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}}{1496880}+\frac{249299 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac{249299 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac{249299 (6 x+5) \sqrt{3 x^2+5 x+2}}{35831808}-\frac{249299 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{71663616 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(249299*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/35831808 - (249299*(5 + 6*x)*(2 + 5*x +
3*x^2)^(3/2))/4478976 + (249299*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/466560 + (32
98*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2))/4455 + (41*(3 + 2*x)^3*(2 + 5*x + 3*x^2)
^(7/2))/110 - ((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(7/2))/33 + ((7405817 + 3365726*x)*
(2 + 5*x + 3*x^2)^(7/2))/1496880 - (249299*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 +
5*x + 3*x^2])])/(71663616*Sqrt[3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 36.2532, size = 190, normalized size = 0.92 \[ - \frac{\left (2 x + 3\right )^{4} \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{33} + \frac{41 \left (2 x + 3\right )^{3} \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{110} + \frac{3298 \left (2 x + 3\right )^{2} \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{4455} + \frac{249299 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{466560} - \frac{249299 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{4478976} + \frac{249299 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{35831808} + \frac{\left (30291534 x + 66652353\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{13471920} - \frac{249299 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{214990848} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)**4*(3*x**2 + 5*x + 2)**(7/2)/33 + 41*(2*x + 3)**3*(3*x**2 + 5*x + 2)*
*(7/2)/110 + 3298*(2*x + 3)**2*(3*x**2 + 5*x + 2)**(7/2)/4455 + 249299*(6*x + 5)
*(3*x**2 + 5*x + 2)**(5/2)/466560 - 249299*(6*x + 5)*(3*x**2 + 5*x + 2)**(3/2)/4
478976 + 249299*(6*x + 5)*sqrt(3*x**2 + 5*x + 2)/35831808 + (30291534*x + 666523
53)*(3*x**2 + 5*x + 2)**(7/2)/13471920 - 249299*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/
(6*sqrt(3*x**2 + 5*x + 2)))/214990848

_______________________________________________________________________________________

Mathematica [A]  time = 0.1765, size = 100, normalized size = 0.49 \[ \frac{-95980115 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (180592312320 x^{10}[phone] x^9-1932170526720 x^8-25759323039744 x^7-90095929758720 x^6-172473366866688 x^5-204855126595200 x^4-155155370878800 x^3-73069860056520 x^2-19521700361210 x-2261297826735\right )}{82771476480} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-2261297826735 - 19521700361210*x - 73069860056520*x^
2 - 155155370878800*x^3 - 204855126595200*x^4 - 172473366866688*x^5 - 9009592975
8720*x^6 - 25759323039744*x^7 - 1932170526720*x^8 + 875872714752*x^9 + 180592312
320*x^10) - 95980115*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/827714764
80

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 168, normalized size = 0.8 \[{\frac{1246495+1495794\,x}{466560} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{1246495+1495794\,x}{4478976} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{1246495+1495794\,x}{35831808}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{249299\,\sqrt{3}}{214990848}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{5753773}{299376} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{2642401\,x}{106920} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{8762\,{x}^{2}}{891} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{4\,{x}^{3}}{55} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{16\,{x}^{4}}{33} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

249299/466560*(5+6*x)*(3*x^2+5*x+2)^(5/2)-249299/4478976*(5+6*x)*(3*x^2+5*x+2)^(
3/2)+249299/35831808*(5+6*x)*(3*x^2+5*x+2)^(1/2)-249299/214990848*ln(1/3*(5/2+3*
x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+5753773/299376*(3*x^2+5*x+2)^(7/2)+26424
01/106920*x*(3*x^2+5*x+2)^(7/2)+8762/891*x^2*(3*x^2+5*x+2)^(7/2)+4/55*x^3*(3*x^2
+5*x+2)^(7/2)-16/33*x^4*(3*x^2+5*x+2)^(7/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.771547, size = 265, normalized size = 1.29 \[ -\frac{16}{33} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x^{4} + \frac{4}{55} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x^{3} + \frac{8762}{891} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x^{2} + \frac{2642401}{106920} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{5753773}{299376} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} + \frac{249299}{77760} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{249299}{93312} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{249299}{746496} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{1246495}{4478976} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{249299}{5971968} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{249299}{214990848} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{1246495}{35831808} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16/33*(3*x^2 + 5*x + 2)^(7/2)*x^4 + 4/55*(3*x^2 + 5*x + 2)^(7/2)*x^3 + 8762/891
*(3*x^2 + 5*x + 2)^(7/2)*x^2 + 2642401/106920*(3*x^2 + 5*x + 2)^(7/2)*x + 575377
3/299376*(3*x^2 + 5*x + 2)^(7/2) + 249299/77760*(3*x^2 + 5*x + 2)^(5/2)*x + 2492
99/93312*(3*x^2 + 5*x + 2)^(5/2) - 249299/746496*(3*x^2 + 5*x + 2)^(3/2)*x - 124
6495/4478976*(3*x^2 + 5*x + 2)^(3/2) + 249299/5971968*sqrt(3*x^2 + 5*x + 2)*x -
249299/214990848*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 124649
5/35831808*sqrt(3*x^2 + 5*x + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.282625, size = 149, normalized size = 0.72 \[ -\frac{1}{165542952960} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (180592312320 \, x^{10} + 875872714752 \, x^{9} - 1932170526720 \, x^{8} - 25759323039744 \, x^{7} - 90095929758720 \, x^{6} - 172473366866688 \, x^{5} - 204855126595200 \, x^{4} - 155155370878800 \, x^{3} - 73069860056520 \, x^{2} - 19521700361210 \, x - 2261297826735\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 95980115 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/165542952960*sqrt(3)*(4*sqrt(3)*(180592312320*x^10 + 875872714752*x^9 - 19321
70526720*x^8 - 25759323039744*x^7 - 90095929758720*x^6 - 172473366866688*x^5 - 2
04855126595200*x^4 - 155155370878800*x^3 - 73069860056520*x^2 - 19521700361210*x
- 2261297826735)*sqrt(3*x^2 + 5*x + 2) - 95980115*log(sqrt(3)*(72*x^2 + 120*x +
49) - 12*sqrt(3*x^2 + 5*x + 2)*(6*x + 5)))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- 12096 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 38421 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 67449 x^{3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 70799 x^{4} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 44295 x^{5} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 14784 x^{6} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1304 x^{7} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 624 x^{8} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 144 x^{9} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 1620 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-12096*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-38421*x**2*sqrt(3*x**2
+ 5*x + 2), x) - Integral(-67449*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-70
799*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-44295*x**5*sqrt(3*x**2 + 5*x + 2
), x) - Integral(-14784*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1304*x**7*sq
rt(3*x**2 + 5*x + 2), x) - Integral(624*x**8*sqrt(3*x**2 + 5*x + 2), x) - Integr
al(144*x**9*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1620*sqrt(3*x**2 + 5*x + 2),
x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269036, size = 134, normalized size = 0.65 \[ -\frac{1}{13795246080} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (14 \,{\left (48 \,{\left (54 \,{\left (20 \, x + 97\right )} x - 11555\right )} x - 7394353\right )} x - 362075335\right )} x - 24952744049\right )} x - 177825630725\right )} x - 1077467853325\right )} x - 3044577502355\right )} x - 9760850180605\right )} x - 2261297826735\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{249299}{214990848} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/13795246080*(2*(12*(6*(8*(6*(36*(14*(48*(54*(20*x + 97)*x - 11555)*x - 739435
3)*x - 362075335)*x - 24952744049)*x - 177825630725)*x - 1077467853325)*x - 3044
577502355)*x - 9760850180605)*x - 2261297826735)*sqrt(3*x^2 + 5*x + 2) + 249299/
214990848*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))