3.2333 \(\int (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=187 \[ -\frac{47}{400} (1-2 x)^{5/2} (5 x+3)^{7/2}-\frac{3}{70} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{7/2}-\frac{783 (1-2 x)^{5/2} (5 x+3)^{5/2}}{1600}-\frac{8613 (1-2 x)^{5/2} (5 x+3)^{3/2}}{5120}-\frac{94743 (1-2 x)^{5/2} \sqrt{5 x+3}}{20480}+\frac{1042173 (1-2 x)^{3/2} \sqrt{5 x+3}}{409600}+\frac{34391709 \sqrt{1-2 x} \sqrt{5 x+3}}{4096000}+\frac{378308799 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4096000 \sqrt{10}} \]

[Out]

(34391709*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4096000 + (1042173*(1 - 2*x)^(3/2)*Sqrt[3
+ 5*x])/409600 - (94743*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/20480 - (8613*(1 - 2*x)^
(5/2)*(3 + 5*x)^(3/2))/5120 - (783*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/1600 - (47*(
1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/400 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(7/
2))/70 + (378308799*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4096000*Sqrt[10])

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Rubi [A]  time = 0.221702, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{47}{400} (1-2 x)^{5/2} (5 x+3)^{7/2}-\frac{3}{70} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{7/2}-\frac{783 (1-2 x)^{5/2} (5 x+3)^{5/2}}{1600}-\frac{8613 (1-2 x)^{5/2} (5 x+3)^{3/2}}{5120}-\frac{94743 (1-2 x)^{5/2} \sqrt{5 x+3}}{20480}+\frac{1042173 (1-2 x)^{3/2} \sqrt{5 x+3}}{409600}+\frac{34391709 \sqrt{1-2 x} \sqrt{5 x+3}}{4096000}+\frac{378308799 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4096000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(34391709*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4096000 + (1042173*(1 - 2*x)^(3/2)*Sqrt[3
+ 5*x])/409600 - (94743*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/20480 - (8613*(1 - 2*x)^
(5/2)*(3 + 5*x)^(3/2))/5120 - (783*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/1600 - (47*(
1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/400 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(7/
2))/70 + (378308799*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4096000*Sqrt[10])

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Rubi in Sympy [A]  time = 17.5526, size = 170, normalized size = 0.91 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{7}{2}} \left (9 x + 6\right )}{70} - \frac{47 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{400} + \frac{783 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{4000} - \frac{25839 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{64000} - \frac{94743 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{51200} - \frac{3126519 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{409600} + \frac{34391709 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{4096000} + \frac{378308799 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{40960000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)*(5*x + 3)**(7/2)*(9*x + 6)/70 - 47*(-2*x + 1)**(5/2)*(5*x + 3
)**(7/2)/400 + 783*(-2*x + 1)**(3/2)*(5*x + 3)**(7/2)/4000 - 25839*(-2*x + 1)**(
3/2)*(5*x + 3)**(5/2)/64000 - 94743*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/51200 - 3
126519*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/409600 + 34391709*sqrt(-2*x + 1)*sqrt(5*x
+ 3)/4096000 + 378308799*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/40960000

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Mathematica [A]  time = 0.132873, size = 80, normalized size = 0.43 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (1843200000 x^6+4387840000 x^5+2867456000 x^4-887043200 x^3-1789716960 x^2-549624420 x+247243887\right )-2648161593 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{286720000} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(247243887 - 549624420*x - 1789716960*x^2 - 887
043200*x^3 + 2867456000*x^4 + 4387840000*x^5 + 1843200000*x^6) - 2648161593*Sqrt
[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/286720000

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Maple [A]  time = 0.014, size = 155, normalized size = 0.8 \[{\frac{1}{573440000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -36864000000\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}-87756800000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-57349120000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-4944877740\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/573440000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-36864000000*x^6*(-10*x^2-x+3)^(1/2)-87
756800000*x^5*(-10*x^2-x+3)^(1/2)-57349120000*x^4*(-10*x^2-x+3)^(1/2)[phone]
0*x^3*(-10*x^2-x+3)^(1/2)[phone]*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)-4944877740*(-10*x^2-x+3
)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50493, size = 157, normalized size = 0.84 \[ -\frac{9}{14} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} - \frac{157}{112} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{12309}{11200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{8613}{2560} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{8613}{51200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3126519}{204800} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{378308799}{81920000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{3126519}{4096000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/14*(-10*x^2 - x + 3)^(5/2)*x^2 - 157/112*(-10*x^2 - x + 3)^(5/2)*x - 12309/11
200*(-10*x^2 - x + 3)^(5/2) + 8613/2560*(-10*x^2 - x + 3)^(3/2)*x + 8613/51200*(
-10*x^2 - x + 3)^(3/2) + 3126519/204800*sqrt(-10*x^2 - x + 3)*x - 378308799/8192
0000*sqrt(10)*arcsin(-20/11*x - 1/11) + 3126519/4096000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.217464, size = 111, normalized size = 0.59 \[ -\frac{1}{573440000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (1843200000 \, x^{6} + 4387840000 \, x^{5} + 2867456000 \, x^{4} - 887043200 \, x^{3} - 1789716960 \, x^{2} - 549624420 \, x + 247243887\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 2648161593 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/573440000*sqrt(10)*(2*sqrt(10)*(1843200000*x^6 + 4387840000*x^5 + 2867456000*
x^4 - 887043200*x^3 - 1789716960*x^2 - 549624420*x + 247243887)*sqrt(5*x + 3)*sq
rt(-2*x + 1) - 2648161593*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)**2*(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.272355, size = 548, normalized size = 2.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/7168000000*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) + 8735611
5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 960917265*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5
*x + 3))) - 61/512000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 239)*(5*x + 3) + 27999
)*(5*x + 3) - 318159)*(5*x + 3) + 3237255)*(5*x + 3) - 2656665)*sqrt(5*x + 3)*sq
rt(-10*x + 5) + 29223315*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1/375000
*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*sqrt(22)*
sqrt(5*x + 3))) + 17/384000*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*sqrt(22)*
sqrt(5*x + 3))) + 13/2000*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))) + 9/100*sqr
t(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(
22)*sqrt(5*x + 3)))