3.2144 \(\int \frac{(2+3 x) (3+5 x)^3}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{75}{16} (1-2 x)^{5/2}+\frac{1675}{24} (1-2 x)^{3/2}-\frac{2805}{4} \sqrt{1-2 x}-\frac{8349}{8 \sqrt{1-2 x}}+\frac{9317}{48 (1-2 x)^{3/2}} \]

[Out]

9317/(48*(1 - 2*x)^(3/2)) - 8349/(8*Sqrt[1 - 2*x]) - (2805*Sqrt[1 - 2*x])/4 + (1
675*(1 - 2*x)^(3/2))/24 - (75*(1 - 2*x)^(5/2))/16

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Rubi [A]  time = 0.0593476, antiderivative size = 66, 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 \[ -\frac{75}{16} (1-2 x)^{5/2}+\frac{1675}{24} (1-2 x)^{3/2}-\frac{2805}{4} \sqrt{1-2 x}-\frac{8349}{8 \sqrt{1-2 x}}+\frac{9317}{48 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

9317/(48*(1 - 2*x)^(3/2)) - 8349/(8*Sqrt[1 - 2*x]) - (2805*Sqrt[1 - 2*x])/4 + (1
675*(1 - 2*x)^(3/2))/24 - (75*(1 - 2*x)^(5/2))/16

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Rubi in Sympy [A]  time = 8.08477, size = 58, normalized size = 0.88 \[ - \frac{75 \left (- 2 x + 1\right )^{\frac{5}{2}}}{16} + \frac{1675 \left (- 2 x + 1\right )^{\frac{3}{2}}}{24} - \frac{2805 \sqrt{- 2 x + 1}}{4} - \frac{8349}{8 \sqrt{- 2 x + 1}} + \frac{9317}{48 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-75*(-2*x + 1)**(5/2)/16 + 1675*(-2*x + 1)**(3/2)/24 - 2805*sqrt(-2*x + 1)/4 - 8
349/(8*sqrt(-2*x + 1)) + 9317/(48*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.0448725, size = 33, normalized size = 0.5 \[ -\frac{225 x^4+1225 x^3+6240 x^2-13533 x+4457}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(4457 - 13533*x + 6240*x^2 + 1225*x^3 + 225*x^4)/(3*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.005, size = 30, normalized size = 0.5 \[ -{\frac{225\,{x}^{4}+1225\,{x}^{3}+6240\,{x}^{2}-13533\,x+4457}{3} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(225*x^4+1225*x^3+6240*x^2-13533*x+4457)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.36182, size = 57, normalized size = 0.86 \[ -\frac{75}{16} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{1675}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{2805}{4} \, \sqrt{-2 \, x + 1} + \frac{121 \,{\left (828 \, x - 337\right )}}{48 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-75/16*(-2*x + 1)^(5/2) + 1675/24*(-2*x + 1)^(3/2) - 2805/4*sqrt(-2*x + 1) + 121
/48*(828*x - 337)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.212817, size = 49, normalized size = 0.74 \[ \frac{225 \, x^{4} + 1225 \, x^{3} + 6240 \, x^{2} - 13533 \, x + 4457}{3 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(225*x^4 + 1225*x^3 + 6240*x^2 - 13533*x + 4457)/((2*x - 1)*sqrt(-2*x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214628, size = 76, normalized size = 1.15 \[ -\frac{75}{16} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{1675}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{2805}{4} \, \sqrt{-2 \, x + 1} - \frac{121 \,{\left (828 \, x - 337\right )}}{48 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-75/16*(2*x - 1)^2*sqrt(-2*x + 1) + 1675/24*(-2*x + 1)^(3/2) - 2805/4*sqrt(-2*x
+ 1) - 121/48*(828*x - 337)/((2*x - 1)*sqrt(-2*x + 1))