3.2057 \(\int \frac{(2+3 x)^2 (3+5 x)}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{9}{8} (1-2 x)^{5/2}-\frac{103}{8} (1-2 x)^{3/2}+\frac{707}{8} \sqrt{1-2 x}+\frac{539}{8 \sqrt{1-2 x}} \]

[Out]

539/(8*Sqrt[1 - 2*x]) + (707*Sqrt[1 - 2*x])/8 - (103*(1 - 2*x)^(3/2))/8 + (9*(1
- 2*x)^(5/2))/8

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Rubi [A]  time = 0.0536013, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{9}{8} (1-2 x)^{5/2}-\frac{103}{8} (1-2 x)^{3/2}+\frac{707}{8} \sqrt{1-2 x}+\frac{539}{8 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

539/(8*Sqrt[1 - 2*x]) + (707*Sqrt[1 - 2*x])/8 - (103*(1 - 2*x)^(3/2))/8 + (9*(1
- 2*x)^(5/2))/8

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Rubi in Sympy [A]  time = 7.33099, size = 46, normalized size = 0.87 \[ \frac{9 \left (- 2 x + 1\right )^{\frac{5}{2}}}{8} - \frac{103 \left (- 2 x + 1\right )^{\frac{3}{2}}}{8} + \frac{707 \sqrt{- 2 x + 1}}{8} + \frac{539}{8 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*(-2*x + 1)**(5/2)/8 - 103*(-2*x + 1)**(3/2)/8 + 707*sqrt(-2*x + 1)/8 + 539/(8*
sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0379074, size = 25, normalized size = 0.47 \[ \frac{-9 x^3-38 x^2-132 x+144}{\sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(144 - 132*x - 38*x^2 - 9*x^3)/Sqrt[1 - 2*x]

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Maple [A]  time = 0.006, size = 25, normalized size = 0.5 \[ -{(9\,{x}^{3}+38\,{x}^{2}+132\,x-144){\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(9*x^3+38*x^2+132*x-144)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.34059, size = 50, normalized size = 0.94 \[ \frac{9}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{103}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{707}{8} \, \sqrt{-2 \, x + 1} + \frac{539}{8 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(3*x + 2)^2/(-2*x + 1)^(3/2),x, algorithm="maxima")

[Out]

9/8*(-2*x + 1)^(5/2) - 103/8*(-2*x + 1)^(3/2) + 707/8*sqrt(-2*x + 1) + 539/8/sqr
t(-2*x + 1)

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Fricas [A]  time = 0.255252, size = 32, normalized size = 0.6 \[ -\frac{9 \, x^{3} + 38 \, x^{2} + 132 \, x - 144}{\sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(3*x + 2)^2/(-2*x + 1)^(3/2),x, algorithm="fricas")

[Out]

-(9*x^3 + 38*x^2 + 132*x - 144)/sqrt(-2*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{2} \left (5 x + 3\right )}{\left (- 2 x + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)**2*(5*x + 3)/(-2*x + 1)**(3/2), x)

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GIAC/XCAS [A]  time = 0.209103, size = 59, normalized size = 1.11 \[ \frac{9}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{103}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{707}{8} \, \sqrt{-2 \, x + 1} + \frac{539}{8 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(3*x + 2)^2/(-2*x + 1)^(3/2),x, algorithm="giac")

[Out]

9/8*(2*x - 1)^2*sqrt(-2*x + 1) - 103/8*(-2*x + 1)^(3/2) + 707/8*sqrt(-2*x + 1) +
539/8/sqrt(-2*x + 1)