3.2494 \(\int \frac{2+3 x}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{206 \sqrt{1-2 x}}{1815 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{165 (5 x+3)^{3/2}} \]

[Out]

(-2*Sqrt[1 - 2*x])/(165*(3 + 5*x)^(3/2)) - (206*Sqrt[1 - 2*x])/(1815*Sqrt[3 + 5*
x])

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Rubi [A]  time = 0.0458088, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{206 \sqrt{1-2 x}}{1815 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{165 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x])/(165*(3 + 5*x)^(3/2)) - (206*Sqrt[1 - 2*x])/(1815*Sqrt[3 + 5*
x])

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Rubi in Sympy [A]  time = 5.25973, size = 41, normalized size = 0.91 \[ - \frac{206 \sqrt{- 2 x + 1}}{1815 \sqrt{5 x + 3}} - \frac{2 \sqrt{- 2 x + 1}}{165 \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-206*sqrt(-2*x + 1)/(1815*sqrt(5*x + 3)) - 2*sqrt(-2*x + 1)/(165*(5*x + 3)**(3/2
))

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Mathematica [A]  time = 0.0365117, size = 27, normalized size = 0.6 \[ -\frac{2 \sqrt{1-2 x} (103 x+64)}{363 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(64 + 103*x))/(363*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.006, size = 22, normalized size = 0.5 \[ -{\frac{206\,x+128}{363}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/363*(103*x+64)/(3+5*x)^(3/2)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.50417, size = 65, normalized size = 1.44 \[ -\frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{165 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{206 \, \sqrt{-10 \, x^{2} - x + 3}}{1815 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/165*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 206/1815*sqrt(-10*x^2 - x + 3
)/(5*x + 3)

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Fricas [A]  time = 0.224448, size = 45, normalized size = 1. \[ -\frac{2 \,{\left (103 \, x + 64\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{363 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/363*(103*x + 64)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(25*x^2 + 30*x + 9)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.244293, size = 170, normalized size = 3.78 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{145200 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{69 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{12100 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{207 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{9075 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/145200*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 69/1
2100*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/9075*(207*s
qrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)
^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3