3.2420 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=157 \[ -\frac{2 (1-2 x)^{5/2} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{33}{125} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^2-\frac{9 (2127-460 x) (1-2 x)^{5/2} \sqrt{5 x+3}}{200000}+\frac{66997 (1-2 x)^{3/2} \sqrt{5 x+3}}{800000}+\frac{2210901 \sqrt{1-2 x} \sqrt{5 x+3}}{8000000}+\frac{24319911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^3)/(5*Sqrt[3 + 5*x]) + (2210901*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/8000000 + (66997*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/800000 - (9*(2127 - 4
60*x)*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/200000 + (33*(1 - 2*x)^(5/2)*(2 + 3*x)^2*Sq
rt[3 + 5*x])/125 + (24319911*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8000000*Sqrt[10]
)

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Rubi [A]  time = 0.235221, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2 (1-2 x)^{5/2} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{33}{125} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^2-\frac{9 (2127-460 x) (1-2 x)^{5/2} \sqrt{5 x+3}}{200000}+\frac{66997 (1-2 x)^{3/2} \sqrt{5 x+3}}{800000}+\frac{2210901 \sqrt{1-2 x} \sqrt{5 x+3}}{8000000}+\frac{24319911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^3)/(5*Sqrt[3 + 5*x]) + (2210901*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/8000000 + (66997*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/800000 - (9*(2127 - 4
60*x)*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/200000 + (33*(1 - 2*x)^(5/2)*(2 + 3*x)^2*Sq
rt[3 + 5*x])/125 + (24319911*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8000000*Sqrt[10]
)

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Rubi in Sympy [A]  time = 24.8715, size = 144, normalized size = 0.92 \[ - \frac{\left (- 3105 x + \frac{57429}{4}\right ) \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{150000} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{3}}{5 \sqrt{5 x + 3}} + \frac{33 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{125} + \frac{66997 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{800000} + \frac{2210901 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{8000000} + \frac{24319911 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{80000000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-3105*x + 57429/4)*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/150000 - 2*(-2*x + 1)**(5/2
)*(3*x + 2)**3/(5*sqrt(5*x + 3)) + 33*(-2*x + 1)**(5/2)*(3*x + 2)**2*sqrt(5*x +
3)/125 + 66997*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/800000 + 2210901*sqrt(-2*x + 1)*s
qrt(5*x + 3)/8000000 + 24319911*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/8000000
0

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Mathematica [A]  time = 0.19687, size = 75, normalized size = 0.48 \[ \frac{\frac{10 \sqrt{1-2 x} \left (34560000 x^5+12528000 x^4-39487200 x^3-4101140 x^2+20337375 x+6089453\right )}{\sqrt{5 x+3}}-24319911 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{80000000} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*(6089453 + 20337375*x - 4101140*x^2 - 39487200*x^3 + 12528000
*x^4 + 34560000*x^5))/Sqrt[3 + 5*x] - 24319911*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1
- 2*x]])/80000000

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Maple [A]  time = 0.023, size = 150, normalized size = 1. \[{\frac{1}{160000000} \left ( 691200000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-789744000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-82022800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/160000000*(691200000*x^5*(-10*x^2-x+3)^(1/2)[phone]*x^4*(-10*x^2-x+3)^(1/2)
-789744000*x^3*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x-820
22800*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x
*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^
(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.50577, size = 170, normalized size = 1.08 \[ -\frac{216 \, x^{6}}{25 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{297 \, x^{5}}{250 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{57189 \, x^{4}}{5000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{782123 \, x^{3}}{200000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{4477589 \, x^{2}}{800000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{24319911}{160000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{8158469 \, x}{8000000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{6089453}{8000000 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-216/25*x^6/sqrt(-10*x^2 - x + 3) + 297/250*x^5/sqrt(-10*x^2 - x + 3) + 57189/50
00*x^4/sqrt(-10*x^2 - x + 3) - 782123/200000*x^3/sqrt(-10*x^2 - x + 3) - 4477589
/800000*x^2/sqrt(-10*x^2 - x + 3) - 24319911/160000000*sqrt(10)*arcsin(-20/11*x
- 1/11) + 8158469/8000000*x/sqrt(-10*x^2 - x + 3) + 6089453/8000000/sqrt(-10*x^2
- x + 3)

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Fricas [A]  time = 0.229431, size = 120, normalized size = 0.76 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (34560000 \, x^{5} + 12528000 \, x^{4} - 39487200 \, x^{3} - 4101140 \, x^{2} + 20337375 \, x + 6089453\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 24319911 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{160000000 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/160000000*sqrt(10)*(2*sqrt(10)*(34560000*x^5 + 12528000*x^4 - 39487200*x^3 - 4
101140*x^2 + 20337375*x + 6089453)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 24319911*(5*x
+ 3)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(5*x + 3)

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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)**3/(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.322291, size = 203, normalized size = 1.29 \[ \frac{1}{200000000} \,{\left (4 \,{\left (24 \,{\left (36 \,{\left (16 \, \sqrt{5}{\left (5 \, x + 3\right )} - 211 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 22859 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 969335 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 5816745 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{24319911}{80000000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{121 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{156250 \, \sqrt{5 \, x + 3}} + \frac{242 \, \sqrt{10} \sqrt{5 \, x + 3}}{78125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/200000000*(4*(24*(36*(16*sqrt(5)*(5*x + 3) - 211*sqrt(5))*(5*x + 3) + 22859*sq
rt(5))*(5*x + 3) + 969335*sqrt(5))*(5*x + 3) - 5816745*sqrt(5))*sqrt(5*x + 3)*sq
rt(-10*x + 5) + 24319911/80000000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) -
121/156250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 242/78
125*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))