3.2545 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^3}{\sqrt{3+2 x}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{9}{640} (2 x+3)^{15/2}+\frac{567 (2 x+3)^{13/2}}{1664}-\frac{3519 (2 x+3)^{11/2}}{1408}+\frac{10475 (2 x+3)^{9/2}}{1152}-\frac{17201}{896} (2 x+3)^{7/2}+\frac{3201}{128} (2 x+3)^{5/2}-\frac{7925}{384} (2 x+3)^{3/2}+\frac{1625}{128} \sqrt{2 x+3} \]

[Out]

(1625*Sqrt[3 + 2*x])/128 - (7925*(3 + 2*x)^(3/2))/384 + (3201*(3 + 2*x)^(5/2))/1
28 - (17201*(3 + 2*x)^(7/2))/896 + (10475*(3 + 2*x)^(9/2))/1152 - (3519*(3 + 2*x
)^(11/2))/1408 + (567*(3 + 2*x)^(13/2))/1664 - (9*(3 + 2*x)^(15/2))/640

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Rubi [A]  time = 0.0854419, 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 \[ -\frac{9}{640} (2 x+3)^{15/2}+\frac{567 (2 x+3)^{13/2}}{1664}-\frac{3519 (2 x+3)^{11/2}}{1408}+\frac{10475 (2 x+3)^{9/2}}{1152}-\frac{17201}{896} (2 x+3)^{7/2}+\frac{3201}{128} (2 x+3)^{5/2}-\frac{7925}{384} (2 x+3)^{3/2}+\frac{1625}{128} \sqrt{2 x+3} \]

Antiderivative was successfully verified.

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

[Out]

(1625*Sqrt[3 + 2*x])/128 - (7925*(3 + 2*x)^(3/2))/384 + (3201*(3 + 2*x)^(5/2))/1
28 - (17201*(3 + 2*x)^(7/2))/896 + (10475*(3 + 2*x)^(9/2))/1152 - (3519*(3 + 2*x
)^(11/2))/1408 + (567*(3 + 2*x)^(13/2))/1664 - (9*(3 + 2*x)^(15/2))/640

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Rubi in Sympy [A]  time = 15.9693, size = 94, normalized size = 0.9 \[ - \frac{9 \left (2 x + 3\right )^{\frac{15}{2}}}{640} + \frac{567 \left (2 x + 3\right )^{\frac{13}{2}}}{1664} - \frac{3519 \left (2 x + 3\right )^{\frac{11}{2}}}{1408} + \frac{10475 \left (2 x + 3\right )^{\frac{9}{2}}}{1152} - \frac{17201 \left (2 x + 3\right )^{\frac{7}{2}}}{896} + \frac{3201 \left (2 x + 3\right )^{\frac{5}{2}}}{128} - \frac{7925 \left (2 x + 3\right )^{\frac{3}{2}}}{384} + \frac{1625 \sqrt{2 x + 3}}{128} \]

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]

-9*(2*x + 3)**(15/2)/640 + 567*(2*x + 3)**(13/2)/1664 - 3519*(2*x + 3)**(11/2)/1
408 + 10475*(2*x + 3)**(9/2)/1152 - 17201*(2*x + 3)**(7/2)/896 + 3201*(2*x + 3)*
*(5/2)/128 - 7925*(2*x + 3)**(3/2)/384 + 1625*sqrt(2*x + 3)/128

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Mathematica [A]  time = 0.0375522, size = 48, normalized size = 0.46 \[ -\frac{\sqrt{2 x+3} \left (81081 x^7-130977 x^6-1407294 x^5-3109960 x^4-3285105 x^3-1924641 x^2-535098 x-196506\right )}{45045} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[3 + 2*x]*(-196506 - 535098*x - 1924641*x^2 - 3285105*x^3 - 3109960*x^4 -
1407294*x^5 - 130977*x^6 + 81081*x^7))/45045

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Maple [A]  time = 0.007, size = 45, normalized size = 0.4 \[ -{\frac{81081\,{x}^{7}-130977\,{x}^{6}-1407294\,{x}^{5}-3109960\,{x}^{4}-3285105\,{x}^{3}-1924641\,{x}^{2}-535098\,x-196506}{45045}\sqrt{3+2\,x}} \]

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]

-1/45045*(81081*x^7-130977*x^6-1407294*x^5-3109960*x^4-3285105*x^3-1924641*x^2-5
35098*x-196506)*(3+2*x)^(1/2)

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Maxima [A]  time = 0.705514, size = 99, normalized size = 0.94 \[ -\frac{9}{640} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} + \frac{567}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - \frac{3519}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{10475}{1152} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - \frac{17201}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{3201}{128} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - \frac{7925}{384} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{1625}{128} \, \sqrt{2 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/640*(2*x + 3)^(15/2) + 567/1664*(2*x + 3)^(13/2) - 3519/1408*(2*x + 3)^(11/2)
+ 10475/1152*(2*x + 3)^(9/2) - 17201/896*(2*x + 3)^(7/2) + 3201/128*(2*x + 3)^(
5/2) - 7925/384*(2*x + 3)^(3/2) + 1625/128*sqrt(2*x + 3)

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Fricas [A]  time = 0.276635, size = 59, normalized size = 0.56 \[ -\frac{1}{45045} \,{\left (81081 \, x^{7} - 130977 \, x^{6} - 1407294 \, x^{5} - 3109960 \, x^{4} - 3285105 \, x^{3} - 1924641 \, x^{2} - 535098 \, x - 196506\right )} \sqrt{2 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/45045*(81081*x^7 - 130977*x^6 - 1407294*x^5 - 3109960*x^4 - 3285105*x^3 - 192
4641*x^2 - 535098*x - 196506)*sqrt(2*x + 3)

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Sympy [A]  time = 69.0343, size = 94, normalized size = 0.9 \[ - \frac{9 \left (2 x + 3\right )^{\frac{15}{2}}}{640} + \frac{567 \left (2 x + 3\right )^{\frac{13}{2}}}{1664} - \frac{3519 \left (2 x + 3\right )^{\frac{11}{2}}}{1408} + \frac{10475 \left (2 x + 3\right )^{\frac{9}{2}}}{1152} - \frac{17201 \left (2 x + 3\right )^{\frac{7}{2}}}{896} + \frac{3201 \left (2 x + 3\right )^{\frac{5}{2}}}{128} - \frac{7925 \left (2 x + 3\right )^{\frac{3}{2}}}{384} + \frac{1625 \sqrt{2 x + 3}}{128} \]

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]

-9*(2*x + 3)**(15/2)/640 + 567*(2*x + 3)**(13/2)/1664 - 3519*(2*x + 3)**(11/2)/1
408 + 10475*(2*x + 3)**(9/2)/1152 - 17201*(2*x + 3)**(7/2)/896 + 3201*(2*x + 3)*
*(5/2)/128 - 7925*(2*x + 3)**(3/2)/384 + 1625*sqrt(2*x + 3)/128

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GIAC/XCAS [A]  time = 0.275205, size = 99, normalized size = 0.94 \[ -\frac{9}{640} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} + \frac{567}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - \frac{3519}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{10475}{1152} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - \frac{17201}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{3201}{128} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - \frac{7925}{384} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{1625}{128} \, \sqrt{2 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/640*(2*x + 3)^(15/2) + 567/1664*(2*x + 3)^(13/2) - 3519/1408*(2*x + 3)^(11/2)
+ 10475/1152*(2*x + 3)^(9/2) - 17201/896*(2*x + 3)^(7/2) + 3201/128*(2*x + 3)^(
5/2) - 7925/384*(2*x + 3)^(3/2) + 1625/128*sqrt(2*x + 3)