3.2558 \(\int \frac{2+3 x}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{428 \sqrt{1-2 x}}{3993 \sqrt{5 x+3}}-\frac{107 \sqrt{1-2 x}}{363 (5 x+3)^{3/2}}+\frac{7}{11 (5 x+3)^{3/2} \sqrt{1-2 x}} \]

[Out]

7/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (107*Sqrt[1 - 2*x])/(363*(3 + 5*x)^(3/2))
- (428*Sqrt[1 - 2*x])/(3993*Sqrt[3 + 5*x])

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Rubi [A]  time = 0.068454, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{428 \sqrt{1-2 x}}{3993 \sqrt{5 x+3}}-\frac{107 \sqrt{1-2 x}}{363 (5 x+3)^{3/2}}+\frac{7}{11 (5 x+3)^{3/2} \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

7/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (107*Sqrt[1 - 2*x])/(363*(3 + 5*x)^(3/2))
- (428*Sqrt[1 - 2*x])/(3993*Sqrt[3 + 5*x])

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Rubi in Sympy [A]  time = 6.90168, size = 60, normalized size = 0.9 \[ - \frac{428 \sqrt{- 2 x + 1}}{3993 \sqrt{5 x + 3}} - \frac{107 \sqrt{- 2 x + 1}}{363 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{7}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-428*sqrt(-2*x + 1)/(3993*sqrt(5*x + 3)) - 107*sqrt(-2*x + 1)/(363*(5*x + 3)**(3
/2)) + 7/(11*sqrt(-2*x + 1)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.0435868, size = 32, normalized size = 0.48 \[ \frac{2 \left (2140 x^2+1391 x+40\right )}{3993 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(40 + 1391*x + 2140*x^2))/(3993*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.005, size = 27, normalized size = 0.4 \[{\frac{4280\,{x}^{2}+2782\,x+80}{3993} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3993*(2140*x^2+1391*x+40)/(3+5*x)^(3/2)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.33945, size = 86, normalized size = 1.28 \[ \frac{856 \, x}{3993 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{214}{19965 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2}{165 \,{\left (5 \, \sqrt{-10 \, x^{2} - x + 3} x + 3 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

856/3993*x/sqrt(-10*x^2 - x + 3) + 214/19965/sqrt(-10*x^2 - x + 3) - 2/165/(5*sq
rt(-10*x^2 - x + 3)*x + 3*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.219199, size = 58, normalized size = 0.87 \[ -\frac{2 \,{\left (2140 \, x^{2} + 1391 \, x + 40\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3993 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3993*(2140*x^2 + 1391*x + 40)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(50*x^3 + 35*x^2 -
12*x - 9)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.255708, size = 205, normalized size = 3.06 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{319440 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{73 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{26620 \, \sqrt{5 \, x + 3}} - \frac{28 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{6655 \,{\left (2 \, x - 1\right )}} + \frac{{\left (\frac{219 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{19965 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/319440*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 73/2
6620*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 28/6655*sqrt(
5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 1/19965*(219*sqrt(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