3.2270 \(\int \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=172 \[ -\frac{3}{70} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac{3 (1-2 x)^{3/2} (1140 x+1963) (5 x+3)^{7/2}}{8000}-\frac{296633 (1-2 x)^{3/2} (5 x+3)^{5/2}}{128000}-\frac{3262963 (1-2 x)^{3/2} (5 x+3)^{3/2}}{307200}-\frac{35892593 (1-2 x)^{3/2} \sqrt{5 x+3}}{819200}+\frac{394818523 \sqrt{1-2 x} \sqrt{5 x+3}}{8192000}+\frac{4343003753 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8192000 \sqrt{10}} \]

[Out]

(394818523*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8192000 - (35892593*(1 - 2*x)^(3/2)*Sqrt
[3 + 5*x])/819200 - (3262963*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/307200 - (296633*(
1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/128000 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x
)^(7/2))/70 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2)*(1963 + 1140*x))/8000 + (434300
3753*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8192000*Sqrt[10])

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Rubi [A]  time = 0.206583, 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 \[ -\frac{3}{70} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac{3 (1-2 x)^{3/2} (1140 x+1963) (5 x+3)^{7/2}}{8000}-\frac{296633 (1-2 x)^{3/2} (5 x+3)^{5/2}}{128000}-\frac{3262963 (1-2 x)^{3/2} (5 x+3)^{3/2}}{307200}-\frac{35892593 (1-2 x)^{3/2} \sqrt{5 x+3}}{819200}+\frac{394818523 \sqrt{1-2 x} \sqrt{5 x+3}}{8192000}+\frac{4343003753 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8192000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(394818523*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8192000 - (35892593*(1 - 2*x)^(3/2)*Sqrt
[3 + 5*x])/819200 - (3262963*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/307200 - (296633*(
1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/128000 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x
)^(7/2))/70 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2)*(1963 + 1140*x))/8000 + (434300
3753*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8192000*Sqrt[10])

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Rubi in Sympy [A]  time = 19.2492, size = 158, normalized size = 0.92 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{7}{2}}}{70} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}} \left (89775 x + \frac{618345}{4}\right )}{210000} + \frac{296633 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{320000} - \frac{3262963 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{3840000} - \frac{35892593 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{6144000} - \frac{394818523 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{8192000} + \frac{4343003753 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{81920000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(3/2)*(3*x + 2)**2*(5*x + 3)**(7/2)/70 - (-2*x + 1)**(3/2)*(5*x +
3)**(7/2)*(89775*x + 618345/4)/210000 + 296633*sqrt(-2*x + 1)*(5*x + 3)**(7/2)/
320000 - 3262963*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/3840000 - 35892593*sqrt(-2*x +
1)*(5*x + 3)**(3/2)/6144000 - 394818523*sqrt(-2*x + 1)*sqrt(5*x + 3)/8192000 + 4
343003753*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/81920000

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Mathematica [A]  time = 0.13993, size = 80, normalized size = 0.47 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (16588800000 x^6+62069760000 x^5+94673664000 x^4+72591427200 x^3+24336990560 x^2-4902803980 x-12531569067\right )-91203078813 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1720320000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-12531569067 - 4902803980*x + 24336990560*x^2 +
72591427200*x^3 + 94673664000*x^4 + 62069760000*x^5 + 16588800000*x^6) - 912030
78813*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/1720320000

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Maple [A]  time = 0.015, size = 155, normalized size = 0.9 \[{\frac{1}{3440640000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 331776000000\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -98056079600\,x\sqrt{-10\,{x}^{2}-x+3}-250631381340\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3440640000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(331776000000*x^6*(-10*x^2-x+3)^(1/2)+1
241395200000*x^5*(-10*x^2-x+3)^(1/2)[phone]*x^4*(-10*x^2-x+3)^(1/2)+14518
28544000*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)[phone]
3*10^(1/2)*arcsin(20/11*x+1/11)-98056079600*x*(-10*x^2-x+3)^(1/2)-250631381340*(
-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49205, size = 163, normalized size = 0.95 \[ -\frac{135}{14} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} - \frac{3933}{112} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{121887}{2240} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{8474351}{179200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{55355473}{2150400} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{35892593}{409600} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{4343003753}{163840000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{35892593}{8192000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-135/14*(-10*x^2 - x + 3)^(3/2)*x^4 - 3933/112*(-10*x^2 - x + 3)^(3/2)*x^3 - 121
887/2240*(-10*x^2 - x + 3)^(3/2)*x^2 - 8474351/179200*(-10*x^2 - x + 3)^(3/2)*x
- 55355473/2150400*(-10*x^2 - x + 3)^(3/2) + 35892593/409600*sqrt(-10*x^2 - x +
3)*x - 4343003753/163840000*sqrt(10)*arcsin(-20/11*x - 1/11) + 35892593/8192000*
sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.219394, size = 111, normalized size = 0.65 \[ \frac{1}{3440640000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (16588800000 \, x^{6} + 62069760000 \, x^{5} + 94673664000 \, x^{4} + 72591427200 \, x^{3} + 24336990560 \, x^{2} - 4902803980 \, x - 12531569067\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 91203078813 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3440640000*sqrt(10)*(2*sqrt(10)*(16588800000*x^6 + 62069760000*x^5 + 946736640
00*x^4 + 72591427200*x^3 + 24336990560*x^2 - 4902803980*x - 12531569067)*sqrt(5*
x + 3)*sqrt(-2*x + 1) + 91203078813*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((2+3*x)**3*(3+5*x)**(5/2)*(1-2*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278815, 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)^(5/2)*(3*x + 2)^3*sqrt(-2*x + 1),x, algorithm="giac")

[Out]

9/14336000000*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))) + 9/32000000*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)*sqrt
(-10*x + 5) + 29223315*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 921/640000
00*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))) + 883/960000*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*sqrt(2
2)*sqrt(5*x + 3))) + 47/2000*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))) + 9/50*s
qrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqr
t(22)*sqrt(5*x + 3)))