3.2614 \(\int \frac{(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{3298 \sqrt{1-2 x}}{43923 \sqrt{5 x+3}}-\frac{1649 \sqrt{1-2 x}}{7986 (5 x+3)^{3/2}}+\frac{14}{121 (5 x+3)^{3/2} \sqrt{1-2 x}}+\frac{49}{66 (5 x+3)^{3/2} (1-2 x)^{3/2}} \]

[Out]

49/(66*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 14/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))
- (1649*Sqrt[1 - 2*x])/(7986*(3 + 5*x)^(3/2)) - (3298*Sqrt[1 - 2*x])/(43923*Sqr
t[3 + 5*x])

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Rubi [A]  time = 0.110757, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{3298 \sqrt{1-2 x}}{43923 \sqrt{5 x+3}}-\frac{1649 \sqrt{1-2 x}}{7986 (5 x+3)^{3/2}}+\frac{14}{121 (5 x+3)^{3/2} \sqrt{1-2 x}}+\frac{49}{66 (5 x+3)^{3/2} (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

49/(66*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 14/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))
- (1649*Sqrt[1 - 2*x])/(7986*(3 + 5*x)^(3/2)) - (3298*Sqrt[1 - 2*x])/(43923*Sqr
t[3 + 5*x])

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Rubi in Sympy [A]  time = 10.1688, size = 80, normalized size = 0.9 \[ \frac{6596 \sqrt{5 x + 3}}{219615 \sqrt{- 2 x + 1}} + \frac{3298 \sqrt{5 x + 3}}{99825 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{28}{605 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} - \frac{2}{825 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6596*sqrt(5*x + 3)/(219615*sqrt(-2*x + 1)) + 3298*sqrt(5*x + 3)/(99825*(-2*x + 1
)**(3/2)) - 28/(605*(-2*x + 1)**(3/2)*sqrt(5*x + 3)) - 2/(825*(-2*x + 1)**(3/2)*
(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.0549251, size = 37, normalized size = 0.42 \[ \frac{-65960 x^3-9894 x^2+49200 x+18728}{43923 (1-2 x)^{3/2} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(18728 + 49200*x - 9894*x^2 - 65960*x^3)/(43923*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.006, size = 32, normalized size = 0.4 \[ -{\frac{65960\,{x}^{3}+9894\,{x}^{2}-49200\,x-18728}{43923} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/43923*(32980*x^3+4947*x^2-24600*x-9364)/(3+5*x)^(3/2)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.33944, size = 80, normalized size = 0.9 \[ \frac{6596 \, x}{43923 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1649}{219615 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1229 \, x}{1815 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{733}{1815 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6596/43923*x/sqrt(-10*x^2 - x + 3) + 1649/219615/sqrt(-10*x^2 - x + 3) + 1229/18
15*x/(-10*x^2 - x + 3)^(3/2) + 733/1815/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.219463, size = 72, normalized size = 0.81 \[ -\frac{2 \,{\left (32980 \, x^{3} + 4947 \, x^{2} - 24600 \, x - 9364\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{43923 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/43923*(32980*x^3 + 4947*x^2 - 24600*x - 9364)*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)**2/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.270794, size = 223, normalized size = 2.51 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{3513840 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{13 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{26620 \, \sqrt{5 \, x + 3}} - \frac{14 \,{\left (164 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1287 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1098075 \,{\left (2 \, x - 1\right )}^{2}} + \frac{{\left (\frac{429 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{219615 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3513840*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 13/
26620*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 14/1098075*(
164*sqrt(5)*(5*x + 3) - 1287*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2
+ 1/219615*(429*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sq
rt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3