3.2122 \(\int \frac{(2+3 x) (3+5 x)}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=38 \[ -\frac{15}{4} \sqrt{1-2 x}-\frac{17}{\sqrt{1-2 x}}+\frac{77}{12 (1-2 x)^{3/2}} \]

[Out]

77/(12*(1 - 2*x)^(3/2)) - 17/Sqrt[1 - 2*x] - (15*Sqrt[1 - 2*x])/4

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Rubi [A]  time = 0.0399739, antiderivative size = 38, 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 \[ -\frac{15}{4} \sqrt{1-2 x}-\frac{17}{\sqrt{1-2 x}}+\frac{77}{12 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

77/(12*(1 - 2*x)^(3/2)) - 17/Sqrt[1 - 2*x] - (15*Sqrt[1 - 2*x])/4

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Rubi in Sympy [A]  time = 5.82606, size = 32, normalized size = 0.84 \[ - \frac{15 \sqrt{- 2 x + 1}}{4} - \frac{17}{\sqrt{- 2 x + 1}} + \frac{77}{12 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*sqrt(-2*x + 1)/4 - 17/sqrt(-2*x + 1) + 77/(12*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.0111421, size = 23, normalized size = 0.61 \[ -\frac{45 x^2-147 x+43}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(43 - 147*x + 45*x^2)/(3*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.003, size = 20, normalized size = 0.5 \[ -{\frac{45\,{x}^{2}-147\,x+43}{3} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(45*x^2-147*x+43)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.33948, size = 32, normalized size = 0.84 \[ -\frac{15}{4} \, \sqrt{-2 \, x + 1} + \frac{408 \, x - 127}{12 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15/4*sqrt(-2*x + 1) + 1/12*(408*x - 127)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.211577, size = 35, normalized size = 0.92 \[ \frac{45 \, x^{2} - 147 \, x + 43}{3 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(45*x^2 - 147*x + 43)/((2*x - 1)*sqrt(-2*x + 1))

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Sympy [A]  time = 1.13503, size = 75, normalized size = 1.97 \[ \frac{45 x^{2}}{6 x \sqrt{- 2 x + 1} - 3 \sqrt{- 2 x + 1}} - \frac{147 x}{6 x \sqrt{- 2 x + 1} - 3 \sqrt{- 2 x + 1}} + \frac{43}{6 x \sqrt{- 2 x + 1} - 3 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

45*x**2/(6*x*sqrt(-2*x + 1) - 3*sqrt(-2*x + 1)) - 147*x/(6*x*sqrt(-2*x + 1) - 3*
sqrt(-2*x + 1)) + 43/(6*x*sqrt(-2*x + 1) - 3*sqrt(-2*x + 1))

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GIAC/XCAS [A]  time = 0.208689, size = 42, normalized size = 1.11 \[ -\frac{15}{4} \, \sqrt{-2 \, x + 1} - \frac{408 \, x - 127}{12 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15/4*sqrt(-2*x + 1) - 1/12*(408*x - 127)/((2*x - 1)*sqrt(-2*x + 1))