3.1988 \(\int \frac{(2+3 x) (3+5 x)}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{3}{4} (1-2 x)^{5/2}+\frac{17}{3} (1-2 x)^{3/2}-\frac{77}{4} \sqrt{1-2 x} \]

[Out]

(-77*Sqrt[1 - 2*x])/4 + (17*(1 - 2*x)^(3/2))/3 - (3*(1 - 2*x)^(5/2))/4

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Rubi [A]  time = 0.0378761, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{3}{4} (1-2 x)^{5/2}+\frac{17}{3} (1-2 x)^{3/2}-\frac{77}{4} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

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

[Out]

(-77*Sqrt[1 - 2*x])/4 + (17*(1 - 2*x)^(3/2))/3 - (3*(1 - 2*x)^(5/2))/4

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Rubi in Sympy [A]  time = 5.6273, size = 34, normalized size = 0.85 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{5}{2}}}{4} + \frac{17 \left (- 2 x + 1\right )^{\frac{3}{2}}}{3} - \frac{77 \sqrt{- 2 x + 1}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(5/2)/4 + 17*(-2*x + 1)**(3/2)/3 - 77*sqrt(-2*x + 1)/4

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Mathematica [A]  time = 0.00886065, size = 23, normalized size = 0.57 \[ -\frac{1}{3} \sqrt{1-2 x} \left (9 x^2+25 x+43\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(43 + 25*x + 9*x^2))/3

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Maple [A]  time = 0.003, size = 20, normalized size = 0.5 \[ -{\frac{9\,{x}^{2}+25\,x+43}{3}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(9*x^2+25*x+43)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.34589, size = 38, normalized size = 0.95 \[ -\frac{3}{4} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{17}{3} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{77}{4} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/4*(-2*x + 1)^(5/2) + 17/3*(-2*x + 1)^(3/2) - 77/4*sqrt(-2*x + 1)

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Fricas [A]  time = 0.209956, size = 26, normalized size = 0.65 \[ -\frac{1}{3} \,{\left (9 \, x^{2} + 25 \, x + 43\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(9*x^2 + 25*x + 43)*sqrt(-2*x + 1)

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Sympy [A]  time = 4.17247, size = 34, normalized size = 0.85 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{5}{2}}}{4} + \frac{17 \left (- 2 x + 1\right )^{\frac{3}{2}}}{3} - \frac{77 \sqrt{- 2 x + 1}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(5/2)/4 + 17*(-2*x + 1)**(3/2)/3 - 77*sqrt(-2*x + 1)/4

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GIAC/XCAS [A]  time = 0.206517, size = 47, normalized size = 1.18 \[ -\frac{3}{4} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{17}{3} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{77}{4} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/4*(2*x - 1)^2*sqrt(-2*x + 1) + 17/3*(-2*x + 1)^(3/2) - 77/4*sqrt(-2*x + 1)