3.2432 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=138 \[ -\frac{76 (1-2 x)^{7/2}}{1815 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{7/2}}{825 (5 x+3)^{3/2}}+\frac{329 \sqrt{5 x+3} (1-2 x)^{5/2}}{45375}+\frac{329 \sqrt{5 x+3} (1-2 x)^{3/2}}{16500}+\frac{329 \sqrt{5 x+3} \sqrt{1-2 x}}{5000}+\frac{3619 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(7/2))/(825*(3 + 5*x)^(3/2)) - (76*(1 - 2*x)^(7/2))/(1815*Sqrt[3 +
5*x]) + (329*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5000 + (329*(1 - 2*x)^(3/2)*Sqrt[3 +
5*x])/16500 + (329*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/45375 + (3619*ArcSin[Sqrt[2/11
]*Sqrt[3 + 5*x]])/(5000*Sqrt[10])

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Rubi [A]  time = 0.16662, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{76 (1-2 x)^{7/2}}{1815 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{7/2}}{825 (5 x+3)^{3/2}}+\frac{329 \sqrt{5 x+3} (1-2 x)^{5/2}}{45375}+\frac{329 \sqrt{5 x+3} (1-2 x)^{3/2}}{16500}+\frac{329 \sqrt{5 x+3} \sqrt{1-2 x}}{5000}+\frac{3619 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(7/2))/(825*(3 + 5*x)^(3/2)) - (76*(1 - 2*x)^(7/2))/(1815*Sqrt[3 +
5*x]) + (329*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5000 + (329*(1 - 2*x)^(3/2)*Sqrt[3 +
5*x])/16500 + (329*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/45375 + (3619*ArcSin[Sqrt[2/11
]*Sqrt[3 + 5*x]])/(5000*Sqrt[10])

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Rubi in Sympy [A]  time = 14.3083, size = 126, normalized size = 0.91 \[ - \frac{76 \left (- 2 x + 1\right )^{\frac{7}{2}}}{1815 \sqrt{5 x + 3}} - \frac{2 \left (- 2 x + 1\right )^{\frac{7}{2}}}{825 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{329 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{45375} + \frac{329 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{16500} + \frac{329 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{5000} + \frac{3619 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{50000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-76*(-2*x + 1)**(7/2)/(1815*sqrt(5*x + 3)) - 2*(-2*x + 1)**(7/2)/(825*(5*x + 3)*
*(3/2)) + 329*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/45375 + 329*(-2*x + 1)**(3/2)*sqrt
(5*x + 3)/16500 + 329*sqrt(-2*x + 1)*sqrt(5*x + 3)/5000 + 3619*sqrt(10)*asin(sqr
t(22)*sqrt(5*x + 3)/11)/50000

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Mathematica [A]  time = 0.184499, size = 70, normalized size = 0.51 \[ \frac{\frac{10 \sqrt{1-2 x} \left (36000 x^4-35100 x^3+3585 x^2+40930 x+10633\right )}{(5 x+3)^{3/2}}-10857 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{150000} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*(10633 + 40930*x + 3585*x^2 - 35100*x^3 + 36000*x^4))/(3 + 5*
x)^(3/2) - 10857*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/150000

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Maple [A]  time = 0.018, size = 147, normalized size = 1.1 \[{\frac{1}{300000} \left ( 720000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+271425\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-702000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+325710\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+71700\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+97713\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +818600\,x\sqrt{-10\,{x}^{2}-x+3}+212660\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/300000*(720000*x^4*(-10*x^2-x+3)^(1/2)+271425*10^(1/2)*arcsin(20/11*x+1/11)*x^
2-702000*x^3*(-10*x^2-x+3)^(1/2)+325710*10^(1/2)*arcsin(20/11*x+1/11)*x+71700*x^
2*(-10*x^2-x+3)^(1/2)+97713*10^(1/2)*arcsin(20/11*x+1/11)+818600*x*(-10*x^2-x+3)
^(1/2)+212660*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/
2)

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Maxima [A]  time = 1.51489, size = 333, normalized size = 2.41 \[ \frac{3619}{100000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{125 \,{\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{125 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{1089}{5000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{11 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{750 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{500 \,{\left (5 \, x + 3\right )}} - \frac{121 \, \sqrt{-10 \, x^{2} - x + 3}}{3750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{3113 \, \sqrt{-10 \, x^{2} - x + 3}}{3750 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3619/100000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 1/125*(-10*x^2 - x + 3)^(5/
2)/(625*x^4 + 1500*x^3 + 1350*x^2 + 540*x + 81) + 3/125*(-10*x^2 - x + 3)^(5/2)/
(125*x^3 + 225*x^2 + 135*x + 27) + 3/125*(-10*x^2 - x + 3)^(5/2)/(25*x^2 + 30*x
+ 9) + 1089/5000*sqrt(-10*x^2 - x + 3) - 11/750*(-10*x^2 - x + 3)^(3/2)/(125*x^3
+ 225*x^2 + 135*x + 27) + 33/250*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) +
33/500*(-10*x^2 - x + 3)^(3/2)/(5*x + 3) - 121/3750*sqrt(-10*x^2 - x + 3)/(25*x^
2 + 30*x + 9) - 3113/3750*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.228599, size = 127, normalized size = 0.92 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (36000 \, x^{4} - 35100 \, x^{3} + 3585 \, x^{2} + 40930 \, x + 10633\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 10857 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{300000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/300000*sqrt(10)*(2*sqrt(10)*(36000*x^4 - 35100*x^3 + 3585*x^2 + 40930*x + 1063
3)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 10857*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)
*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(25*x^2 + 30*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((1-2*x)**(5/2)*(2+3*x)**2/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.329128, size = 255, normalized size = 1.85 \[ \frac{1}{125000} \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 135 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 9635 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{750000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{3619}{50000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{1353 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{62500 \, \sqrt{5 \, x + 3}} + \frac{11 \,{\left (\frac{369 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{46875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/125000*(12*(8*sqrt(5)*(5*x + 3) - 135*sqrt(5))*(5*x + 3) + 9635*sqrt(5))*sqrt(
5*x + 3)*sqrt(-10*x + 5) - 11/750000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22
))^3/(5*x + 3)^(3/2) + 3619/50000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) -
1353/62500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 11/468
75*(369*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