3.2615 \(\int \frac{2+3 x}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{2960 \sqrt{1-2 x}}{43923 \sqrt{5 x+3}}+\frac{296}{3993 \sqrt{5 x+3} \sqrt{1-2 x}}+\frac{74}{1815 \sqrt{5 x+3} (1-2 x)^{3/2}}-\frac{2}{165 (5 x+3)^{3/2} (1-2 x)^{3/2}} \]

[Out]

-2/(165*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 74/(1815*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x
]) + 296/(3993*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (2960*Sqrt[1 - 2*x])/(43923*Sqrt[3
+ 5*x])

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Rubi [A]  time = 0.0866773, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2960 \sqrt{1-2 x}}{43923 \sqrt{5 x+3}}+\frac{296}{3993 \sqrt{5 x+3} \sqrt{1-2 x}}+\frac{74}{1815 \sqrt{5 x+3} (1-2 x)^{3/2}}-\frac{2}{165 (5 x+3)^{3/2} (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-2/(165*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 74/(1815*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x
]) + 296/(3993*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (2960*Sqrt[1 - 2*x])/(43923*Sqrt[3
+ 5*x])

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Rubi in Sympy [A]  time = 8.81924, size = 80, normalized size = 0.9 \[ - \frac{2960 \sqrt{- 2 x + 1}}{43923 \sqrt{5 x + 3}} - \frac{740 \sqrt{- 2 x + 1}}{3993 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{37}{121 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{7}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2960*sqrt(-2*x + 1)/(43923*sqrt(5*x + 3)) - 740*sqrt(-2*x + 1)/(3993*(5*x + 3)*
*(3/2)) + 37/(121*sqrt(-2*x + 1)*(5*x + 3)**(3/2)) + 7/(33*(-2*x + 1)**(3/2)*(5*
x + 3)**(3/2))

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Mathematica [A]  time = 0.0552422, size = 37, normalized size = 0.42 \[ \frac{-59200 x^3-8880 x^2+26418 x+5728}{43923 (1-2 x)^{3/2} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(5728 + 26418*x - 8880*x^2 - 59200*x^3)/(43923*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.004, size = 32, normalized size = 0.4 \[ -{\frac{59200\,{x}^{3}+8880\,{x}^{2}-26418\,x-5728}{43923} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/43923*(29600*x^3+4440*x^2-13209*x-2864)/(3+5*x)^(3/2)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.34377, size = 80, normalized size = 0.9 \[ \frac{5920 \, x}{43923 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{296}{43923 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{74 \, x}{363 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{40}{363 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5920/43923*x/sqrt(-10*x^2 - x + 3) + 296/43923/sqrt(-10*x^2 - x + 3) + 74/363*x/
(-10*x^2 - x + 3)^(3/2) + 40/363/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.228588, size = 72, normalized size = 0.81 \[ -\frac{2 \,{\left (29600 \, x^{3} + 4440 \, x^{2} - 13209 \, x - 2864\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{43923 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/43923*(29600*x^3 + 4440*x^2 - 13209*x - 2864)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(1
00*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.265601, size = 223, normalized size = 2.51 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{702768 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{7 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{5324 \, \sqrt{5 \, x + 3}} - \frac{8 \,{\left (181 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1188 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1098075 \,{\left (2 \, x - 1\right )}^{2}} + \frac{{\left (\frac{231 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{43923 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/702768*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 7/53
24*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 8/1098075*(181*
sqrt(5)*(5*x + 3) - 1188*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 1/
43923*(231*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