3.2320 \(\int (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=172 \[ -\frac{3}{70} (3 x+2)^2 (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{141599 (5 x+3)^{3/2} (1-2 x)^{5/2}}{128000}-\frac{3 (5 x+3)^{5/2} (33300 x+49829) (1-2 x)^{5/2}}{280000}-\frac{1557589 \sqrt{5 x+3} (1-2 x)^{5/2}}{512000}+\frac{17133479 \sqrt{5 x+3} (1-2 x)^{3/2}}{10240000}+\frac{565404807 \sqrt{5 x+3} \sqrt{1-2 x}}{102400000}+\frac{6219452877 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400000 \sqrt{10}} \]

[Out]

(565404807*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/102400000 + (17133479*(1 - 2*x)^(3/2)*Sq
rt[3 + 5*x])/10240000 - (1557589*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/512000 - (141599
*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/128000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5
*x)^(5/2))/70 - (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)*(49829 + 33300*x))/280000 + (
6219452877*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(102400000*Sqrt[10])

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Rubi [A]  time = 0.211975, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3}{70} (3 x+2)^2 (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{141599 (5 x+3)^{3/2} (1-2 x)^{5/2}}{128000}-\frac{3 (5 x+3)^{5/2} (33300 x+49829) (1-2 x)^{5/2}}{280000}-\frac{1557589 \sqrt{5 x+3} (1-2 x)^{5/2}}{512000}+\frac{17133479 \sqrt{5 x+3} (1-2 x)^{3/2}}{10240000}+\frac{565404807 \sqrt{5 x+3} \sqrt{1-2 x}}{102400000}+\frac{6219452877 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(565404807*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/102400000 + (17133479*(1 - 2*x)^(3/2)*Sq
rt[3 + 5*x])/10240000 - (1557589*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/512000 - (141599
*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/128000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5
*x)^(5/2))/70 - (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)*(49829 + 33300*x))/280000 + (
6219452877*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(102400000*Sqrt[10])

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Rubi in Sympy [A]  time = 19.3634, size = 158, normalized size = 0.92 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{5}{2}}}{70} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}} \left (74925 x + \frac{448461}{4}\right )}{210000} + \frac{141599 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{320000} + \frac{1557589 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{3200000} - \frac{17133479 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{25600000} - \frac{565404807 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{102400000} + \frac{6219452877 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{1024000000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(5/2)*(3*x + 2)**2*(5*x + 3)**(5/2)/70 - (-2*x + 1)**(5/2)*(5*x +
3)**(5/2)*(74925*x + 448461/4)/210000 + 141599*(-2*x + 1)**(3/2)*(5*x + 3)**(5/
2)/320000 + 1557589*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/3200000 - 17133479*sqrt(-2*x
+ 1)*(5*x + 3)**(3/2)/25600000 - 565404807*sqrt(-2*x + 1)*sqrt(5*x + 3)/1024000
00 + 6219452877*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/1024000000

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Mathematica [A]  time = 0.144569, size = 80, normalized size = 0.47 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (27648000000 x^6+67968000000 x^5+46732032000 x^4-12527113600 x^3-28707557280 x^2-9288436460 x+3952411101\right )-43536170139 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{7168000000} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3952411101 - 9288436460*x - 28707557280*x^2 -
12527113600*x^3 + 46732032000*x^4 + 67968000000*x^5 + 27648000000*x^6) - 4353617
0139*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/7168000000

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Maple [A]  time = 0.014, size = 155, normalized size = 0.9 \[{\frac{1}{14336000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -552960000000\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}-1359360000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-934640640000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-79048222020\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14336000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-552960000000*x^6*(-10*x^2-x+3)^(1/2)
-1359360000000*x^5*(-10*x^2-x+3)^(1/2)-934640640000*x^4*(-10*x^2-x+3)^(1/2)+2505
42272000*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)[phone]
9*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-79048222020*(
-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.5078, size = 157, normalized size = 0.91 \[ -\frac{27}{70} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} - \frac{2439}{2800} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{197487}{280000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{141599}{64000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{141599}{1280000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{51400437}{5120000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{6219452877}{2048000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{51400437}{102400000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/70*(-10*x^2 - x + 3)^(5/2)*x^2 - 2439/2800*(-10*x^2 - x + 3)^(5/2)*x - 19748
7/280000*(-10*x^2 - x + 3)^(5/2) + 141599/64000*(-10*x^2 - x + 3)^(3/2)*x + 1415
99/1280000*(-10*x^2 - x + 3)^(3/2) + 51400437/5120000*sqrt(-10*x^2 - x + 3)*x -
6219452877/2048000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 51400437/102400000*sqrt
(-10*x^2 - x + 3)

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Fricas [A]  time = 0.225127, size = 111, normalized size = 0.65 \[ -\frac{1}{14336000000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (27648000000 \, x^{6} + 67968000000 \, x^{5} + 46732032000 \, x^{4} - 12527113600 \, x^{3} - 28707557280 \, x^{2} - 9288436460 \, x + 3952411101\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 43536170139 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/14336000000*sqrt(10)*(2*sqrt(10)*(27648000000*x^6 + 67968000000*x^5 + 4673203
2000*x^4 - 12527113600*x^3 - 28707557280*x^2 - 9288436460*x + 3952411101)*sqrt(5
*x + 3)*sqrt(-2*x + 1) - 43536170139*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x +
3)*sqrt(-2*x + 1))))

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275537, size = 548, normalized size = 3.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/35840000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 359)*(5*x + 3) + 63769)*(5*x
+ 3) - 3968469)*(5*x + 3) + 33617829)*(5*x + 3) - 276044685)*(5*x + 3) + 873561
15)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 960917265*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(
5*x + 3))) - 189/2560000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 239)*(5*x + 3) + 27
999)*(5*x + 3) - 318159)*(5*x + 3) + 3237255)*(5*x + 3) - 2656665)*sqrt(5*x + 3)
*sqrt(-10*x + 5) + 29223315*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 111/6
4000000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 136405)*
(5*x + 3) + 60555)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sq
rt(22)*sqrt(5*x + 3))) + 23/960000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179
)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sq
rt(22)*sqrt(5*x + 3))) + 1/240*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*
x + 3)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 3/50
*sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*s
qrt(22)*sqrt(5*x + 3)))