3.2548 \(\int \frac{2+3 x}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ \frac{74 \sqrt{5 x+3}}{605 \sqrt{1-2 x}}-\frac{2}{55 \sqrt{1-2 x} \sqrt{5 x+3}} \]

[Out]

-2/(55*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (74*Sqrt[3 + 5*x])/(605*Sqrt[1 - 2*x])

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Rubi [A]  time = 0.0484819, 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 \[ \frac{74 \sqrt{5 x+3}}{605 \sqrt{1-2 x}}-\frac{2}{55 \sqrt{1-2 x} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

-2/(55*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (74*Sqrt[3 + 5*x])/(605*Sqrt[1 - 2*x])

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Rubi in Sympy [A]  time = 5.14134, size = 39, normalized size = 0.87 \[ - \frac{37 \sqrt{- 2 x + 1}}{121 \sqrt{5 x + 3}} + \frac{7}{11 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-37*sqrt(-2*x + 1)/(121*sqrt(5*x + 3)) + 7/(11*sqrt(-2*x + 1)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.0399492, size = 27, normalized size = 0.6 \[ \frac{2 (37 x+20)}{121 \sqrt{1-2 x} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(20 + 37*x))/(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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Maple [A]  time = 0.004, size = 22, normalized size = 0.5 \[{\frac{74\,x+40}{121}{\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/121*(37*x+20)/(3+5*x)^(1/2)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.32476, size = 41, normalized size = 0.91 \[ \frac{74 \, x}{121 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{40}{121 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

74/121*x/sqrt(-10*x^2 - x + 3) + 40/121/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.227332, size = 42, normalized size = 0.93 \[ -\frac{2 \,{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{121 \,{\left (10 \, x^{2} + x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/121*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.238433, size = 117, normalized size = 2.6 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{1210 \, \sqrt{5 \, x + 3}} - \frac{14 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{605 \,{\left (2 \, x - 1\right )}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{605 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1210*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 14/605*sqr
t(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 2/605*sqrt(10)*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))