3.2595 \(\int \frac{2+3 x}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=45 \[ \frac{7 \sqrt{5 x+3}}{33 (1-2 x)^{3/2}}-\frac{29 \sqrt{5 x+3}}{363 \sqrt{1-2 x}} \]

[Out]

(7*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) - (29*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x])

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Rubi [A]  time = 0.046817, 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 \[ \frac{7 \sqrt{5 x+3}}{33 (1-2 x)^{3/2}}-\frac{29 \sqrt{5 x+3}}{363 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) - (29*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x])

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Rubi in Sympy [A]  time = 5.12597, size = 39, normalized size = 0.87 \[ - \frac{29 \sqrt{5 x + 3}}{363 \sqrt{- 2 x + 1}} + \frac{7 \sqrt{5 x + 3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-29*sqrt(5*x + 3)/(363*sqrt(-2*x + 1)) + 7*sqrt(5*x + 3)/(33*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.037447, size = 27, normalized size = 0.6 \[ \frac{2 \sqrt{5 x+3} (29 x+24)}{363 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x]*(24 + 29*x))/(363*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.004, size = 22, normalized size = 0.5 \[{\frac{58\,x+48}{363}\sqrt{3+5\,x} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/363*(3+5*x)^(1/2)*(29*x+24)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.49072, size = 65, normalized size = 1.44 \[ \frac{7 \, \sqrt{-10 \, x^{2} - x + 3}}{33 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{29 \, \sqrt{-10 \, x^{2} - x + 3}}{363 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/33*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 29/363*sqrt(-10*x^2 - x + 3)/(2*x
- 1)

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Fricas [A]  time = 0.21652, size = 45, normalized size = 1. \[ \frac{2 \,{\left (29 \, x + 24\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{363 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/363*(29*x + 24)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.251923, size = 53, normalized size = 1.18 \[ \frac{2 \,{\left (29 \, \sqrt{5}{\left (5 \, x + 3\right )} + 33 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{9075 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9075*(29*sqrt(5)*(5*x + 3) + 33*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x -
1)^2