3.2525 \(\int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right ) \, dx\)

Optimal. Leaf size=53 \[ -\frac{1}{40} (2 x+3)^{15/2}+\frac{47}{104} (2 x+3)^{13/2}-\frac{109}{88} (2 x+3)^{11/2}+\frac{65}{72} (2 x+3)^{9/2} \]

[Out]

(65*(3 + 2*x)^(9/2))/72 - (109*(3 + 2*x)^(11/2))/88 + (47*(3 + 2*x)^(13/2))/104
- (3 + 2*x)^(15/2)/40

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Rubi [A]  time = 0.0511704, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ -\frac{1}{40} (2 x+3)^{15/2}+\frac{47}{104} (2 x+3)^{13/2}-\frac{109}{88} (2 x+3)^{11/2}+\frac{65}{72} (2 x+3)^{9/2} \]

Antiderivative was successfully verified.

[In]  Int[(5 - x)*(3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2),x]

[Out]

(65*(3 + 2*x)^(9/2))/72 - (109*(3 + 2*x)^(11/2))/88 + (47*(3 + 2*x)^(13/2))/104
- (3 + 2*x)^(15/2)/40

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Rubi in Sympy [A]  time = 9.66255, size = 44, normalized size = 0.83 \[ - \frac{\left (2 x + 3\right )^{\frac{15}{2}}}{40} + \frac{47 \left (2 x + 3\right )^{\frac{13}{2}}}{104} - \frac{109 \left (2 x + 3\right )^{\frac{11}{2}}}{88} + \frac{65 \left (2 x + 3\right )^{\frac{9}{2}}}{72} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**(7/2)*(3*x**2+5*x+2),x)

[Out]

-(2*x + 3)**(15/2)/40 + 47*(2*x + 3)**(13/2)/104 - 109*(2*x + 3)**(11/2)/88 + 65
*(2*x + 3)**(9/2)/72

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Mathematica [A]  time = 0.0250556, size = 28, normalized size = 0.53 \[ -\frac{(2 x+3)^{9/2} \left (1287 x^3-5841 x^2-10269 x-3727\right )}{6435} \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - x)*(3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2),x]

[Out]

-((3 + 2*x)^(9/2)*(-3727 - 10269*x - 5841*x^2 + 1287*x^3))/6435

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Maple [A]  time = 0.006, size = 25, normalized size = 0.5 \[ -{\frac{1287\,{x}^{3}-5841\,{x}^{2}-10269\,x-3727}{6435} \left ( 3+2\,x \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2),x)

[Out]

-1/6435*(1287*x^3-5841*x^2-10269*x-3727)*(3+2*x)^(9/2)

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Maxima [A]  time = 0.707711, size = 50, normalized size = 0.94 \[ -\frac{1}{40} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} + \frac{47}{104} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - \frac{109}{88} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{65}{72} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)*(2*x + 3)^(7/2)*(x - 5),x, algorithm="maxima")

[Out]

-1/40*(2*x + 3)^(15/2) + 47/104*(2*x + 3)^(13/2) - 109/88*(2*x + 3)^(11/2) + 65/
72*(2*x + 3)^(9/2)

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Fricas [A]  time = 0.267035, size = 59, normalized size = 1.11 \[ -\frac{1}{6435} \,{\left (20592 \, x^{7} + 30096 \, x^{6} - 447048 \, x^{5} - 2029120 \, x^{4} - 3733305 \, x^{3} - 3496257 \, x^{2} - 1636821 \, x - 301887\right )} \sqrt{2 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)*(2*x + 3)^(7/2)*(x - 5),x, algorithm="fricas")

[Out]

-1/6435*(20592*x^7 + 30096*x^6 - 447048*x^5 - 2029120*x^4 - 3733305*x^3 - 349625
7*x^2 - 1636821*x - 301887)*sqrt(2*x + 3)

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Sympy [A]  time = 12.8811, size = 116, normalized size = 2.19 \[ - \frac{16 x^{7} \sqrt{2 x + 3}}{5} - \frac{304 x^{6} \sqrt{2 x + 3}}{65} + \frac{49672 x^{5} \sqrt{2 x + 3}}{715} + \frac{405824 x^{4} \sqrt{2 x + 3}}{1287} + \frac{248887 x^{3} \sqrt{2 x + 3}}{429} + \frac{388473 x^{2} \sqrt{2 x + 3}}{715} + \frac{181869 x \sqrt{2 x + 3}}{715} + \frac{33543 \sqrt{2 x + 3}}{715} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**(7/2)*(3*x**2+5*x+2),x)

[Out]

-16*x**7*sqrt(2*x + 3)/5 - 304*x**6*sqrt(2*x + 3)/65 + 49672*x**5*sqrt(2*x + 3)/
715 + 405824*x**4*sqrt(2*x + 3)/1287 + 248887*x**3*sqrt(2*x + 3)/429 + 388473*x*
*2*sqrt(2*x + 3)/715 + 181869*x*sqrt(2*x + 3)/715 + 33543*sqrt(2*x + 3)/715

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GIAC/XCAS [A]  time = 0.27088, size = 50, normalized size = 0.94 \[ -\frac{1}{40} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} + \frac{47}{104} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} - \frac{109}{88} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} + \frac{65}{72} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)*(2*x + 3)^(7/2)*(x - 5),x, algorithm="giac")

[Out]

-1/40*(2*x + 3)^(15/2) + 47/104*(2*x + 3)^(13/2) - 109/88*(2*x + 3)^(11/2) + 65/
72*(2*x + 3)^(9/2)