3.2544 \(\int (5-x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^3 \, dx\)

Optimal. Leaf size=105 \[ -\frac{27 (2 x+3)^{17/2}}{2176}+\frac{189}{640} (2 x+3)^{15/2}-\frac{3519 (2 x+3)^{13/2}}{1664}+\frac{10475 (2 x+3)^{11/2}}{1408}-\frac{17201 (2 x+3)^{9/2}}{1152}+\frac{16005}{896} (2 x+3)^{7/2}-\frac{1585}{128} (2 x+3)^{5/2}+\frac{1625}{384} (2 x+3)^{3/2} \]

[Out]

(1625*(3 + 2*x)^(3/2))/384 - (1585*(3 + 2*x)^(5/2))/128 + (16005*(3 + 2*x)^(7/2)
)/896 - (17201*(3 + 2*x)^(9/2))/1152 + (10475*(3 + 2*x)^(11/2))/1408 - (3519*(3
+ 2*x)^(13/2))/1664 + (189*(3 + 2*x)^(15/2))/640 - (27*(3 + 2*x)^(17/2))/2176

_______________________________________________________________________________________

Rubi [A]  time = 0.0818549, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037 \[ -\frac{27 (2 x+3)^{17/2}}{2176}+\frac{189}{640} (2 x+3)^{15/2}-\frac{3519 (2 x+3)^{13/2}}{1664}+\frac{10475 (2 x+3)^{11/2}}{1408}-\frac{17201 (2 x+3)^{9/2}}{1152}+\frac{16005}{896} (2 x+3)^{7/2}-\frac{1585}{128} (2 x+3)^{5/2}+\frac{1625}{384} (2 x+3)^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(1625*(3 + 2*x)^(3/2))/384 - (1585*(3 + 2*x)^(5/2))/128 + (16005*(3 + 2*x)^(7/2)
)/896 - (17201*(3 + 2*x)^(9/2))/1152 + (10475*(3 + 2*x)^(11/2))/1408 - (3519*(3
+ 2*x)^(13/2))/1664 + (189*(3 + 2*x)^(15/2))/640 - (27*(3 + 2*x)^(17/2))/2176

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.8755, size = 94, normalized size = 0.9 \[ - \frac{27 \left (2 x + 3\right )^{\frac{17}{2}}}{2176} + \frac{189 \left (2 x + 3\right )^{\frac{15}{2}}}{640} - \frac{3519 \left (2 x + 3\right )^{\frac{13}{2}}}{1664} + \frac{10475 \left (2 x + 3\right )^{\frac{11}{2}}}{1408} - \frac{17201 \left (2 x + 3\right )^{\frac{9}{2}}}{1152} + \frac{16005 \left (2 x + 3\right )^{\frac{7}{2}}}{896} - \frac{1585 \left (2 x + 3\right )^{\frac{5}{2}}}{128} + \frac{1625 \left (2 x + 3\right )^{\frac{3}{2}}}{384} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27*(2*x + 3)**(17/2)/2176 + 189*(2*x + 3)**(15/2)/640 - 3519*(2*x + 3)**(13/2)/
1664 + 10475*(2*x + 3)**(11/2)/1408 - 17201*(2*x + 3)**(9/2)/1152 + 16005*(2*x +
3)**(7/2)/896 - 1585*(2*x + 3)**(5/2)/128 + 1625*(2*x + 3)**(3/2)/384

_______________________________________________________________________________________

Mathematica [A]  time = 0.0317174, size = 53, normalized size = 0.5 \[ -\frac{\sqrt{2 x+3} \left (2432430 x^8+243243 x^7-47045691 x^6-157479462 x^5-243619810 x^4-212153055 x^3-107295183 x^2-29433414 x-3591558\right )}{765765} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[3 + 2*x]*(-3591558 - 29433414*x - 107295183*x^2 - 212153055*x^3 - 2436198
10*x^4 - 157479462*x^5 - 47045691*x^6 + 243243*x^7 + 2432430*x^8))/765765

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 45, normalized size = 0.4 \[ -{\frac{1216215\,{x}^{7}-1702701\,{x}^{6}-20968794\,{x}^{5}-47286540\,{x}^{4}-50880095\,{x}^{3}-29756385\,{x}^{2}-9013014\,x-1197186}{765765} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/765765*(1216215*x^7-1702701*x^6-20968794*x^5-47286540*x^4-50880095*x^3-297563
85*x^2-9013014*x-1197186)*(3+2*x)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.707832, size = 99, normalized size = 0.94 \[ -\frac{27}{2176} \,{\left (2 \, x + 3\right )}^{\frac{17}{2}} + \frac{189}{640} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} - \frac{3519}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{10475}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{17201}{1152} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{16005}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{1585}{128} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{1625}{384} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/2176*(2*x + 3)^(17/2) + 189/640*(2*x + 3)^(15/2) - 3519/1664*(2*x + 3)^(13/2
) + 10475/1408*(2*x + 3)^(11/2) - 17201/1152*(2*x + 3)^(9/2) + 16005/896*(2*x +
3)^(7/2) - 1585/128*(2*x + 3)^(5/2) + 1625/384*(2*x + 3)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.277998, size = 66, normalized size = 0.63 \[ -\frac{1}{765765} \,{\left (2432430 \, x^{8} + 243243 \, x^{7} - 47045691 \, x^{6} - 157479462 \, x^{5} - 243619810 \, x^{4} - 212153055 \, x^{3} - 107295183 \, x^{2} - 29433414 \, x - 3591558\right )} \sqrt{2 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/765765*(2432430*x^8 + 243243*x^7 - 47045691*x^6 - 157479462*x^5 - 243619810*x
^4 - 212153055*x^3 - 107295183*x^2 - 29433414*x - 3591558)*sqrt(2*x + 3)

_______________________________________________________________________________________

Sympy [A]  time = 4.45232, size = 94, normalized size = 0.9 \[ - \frac{27 \left (2 x + 3\right )^{\frac{17}{2}}}{2176} + \frac{189 \left (2 x + 3\right )^{\frac{15}{2}}}{640} - \frac{3519 \left (2 x + 3\right )^{\frac{13}{2}}}{1664} + \frac{10475 \left (2 x + 3\right )^{\frac{11}{2}}}{1408} - \frac{17201 \left (2 x + 3\right )^{\frac{9}{2}}}{1152} + \frac{16005 \left (2 x + 3\right )^{\frac{7}{2}}}{896} - \frac{1585 \left (2 x + 3\right )^{\frac{5}{2}}}{128} + \frac{1625 \left (2 x + 3\right )^{\frac{3}{2}}}{384} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27*(2*x + 3)**(17/2)/2176 + 189*(2*x + 3)**(15/2)/640 - 3519*(2*x + 3)**(13/2)/
1664 + 10475*(2*x + 3)**(11/2)/1408 - 17201*(2*x + 3)**(9/2)/1152 + 16005*(2*x +
3)**(7/2)/896 - 1585*(2*x + 3)**(5/2)/128 + 1625*(2*x + 3)**(3/2)/384

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.268, size = 99, normalized size = 0.94 \[ -\frac{27}{2176} \,{\left (2 \, x + 3\right )}^{\frac{17}{2}} + \frac{189}{640} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} - \frac{3519}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{10475}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{17201}{1152} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{16005}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{1585}{128} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{1625}{384} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/2176*(2*x + 3)^(17/2) + 189/640*(2*x + 3)^(15/2) - 3519/1664*(2*x + 3)^(13/2
) + 10475/1408*(2*x + 3)^(11/2) - 17201/1152*(2*x + 3)^(9/2) + 16005/896*(2*x +
3)^(7/2) - 1585/128*(2*x + 3)^(5/2) + 1625/384*(2*x + 3)^(3/2)