3.2383 \(\int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=187 \[ -\frac{3}{70} (3 x+2) (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{263 (5 x+3)^{5/2} (1-2 x)^{7/2}}{2800}-\frac{2287 (5 x+3)^{3/2} (1-2 x)^{7/2}}{8000}-\frac{75471 \sqrt{5 x+3} (1-2 x)^{7/2}}{128000}+\frac{276727 \sqrt{5 x+3} (1-2 x)^{5/2}}{1280000}+\frac{3043997 \sqrt{5 x+3} (1-2 x)^{3/2}}{5120000}+\frac{100451901 \sqrt{5 x+3} \sqrt{1-2 x}}{51200000}+\frac{1104970911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200000 \sqrt{10}} \]

[Out]

(100451901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200000 + (3043997*(1 - 2*x)^(3/2)*Sqrt
[3 + 5*x])/5120000 + (276727*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/1280000 - (75471*(1
- 2*x)^(7/2)*Sqrt[3 + 5*x])/128000 - (2287*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/8000
- (263*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/2800 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*(3
+ 5*x)^(5/2))/70 + (1104970911*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(51200000*Sqrt[
10])

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Rubi [A]  time = 0.220777, 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 \[ -\frac{3}{70} (3 x+2) (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{263 (5 x+3)^{5/2} (1-2 x)^{7/2}}{2800}-\frac{2287 (5 x+3)^{3/2} (1-2 x)^{7/2}}{8000}-\frac{75471 \sqrt{5 x+3} (1-2 x)^{7/2}}{128000}+\frac{276727 \sqrt{5 x+3} (1-2 x)^{5/2}}{1280000}+\frac{3043997 \sqrt{5 x+3} (1-2 x)^{3/2}}{5120000}+\frac{100451901 \sqrt{5 x+3} \sqrt{1-2 x}}{51200000}+\frac{1104970911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(100451901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200000 + (3043997*(1 - 2*x)^(3/2)*Sqrt
[3 + 5*x])/5120000 + (276727*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/1280000 - (75471*(1
- 2*x)^(7/2)*Sqrt[3 + 5*x])/128000 - (2287*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/8000
- (263*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/2800 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*(3
+ 5*x)^(5/2))/70 + (1104970911*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(51200000*Sqrt[
10])

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Rubi in Sympy [A]  time = 17.5224, size = 170, normalized size = 0.91 \[ - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{5}{2}} \left (9 x + 6\right )}{70} - \frac{263 \left (- 2 x + 1\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{2800} + \frac{2287 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{20000} - \frac{25157 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{64000} - \frac{276727 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{256000} + \frac{3043997 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{5120000} + \frac{100451901 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{51200000} + \frac{1104970911 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{512000000} \]

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)**(3/2),x)

[Out]

-(-2*x + 1)**(7/2)*(5*x + 3)**(5/2)*(9*x + 6)/70 - 263*(-2*x + 1)**(7/2)*(5*x +
3)**(5/2)/2800 + 2287*(-2*x + 1)**(5/2)*(5*x + 3)**(5/2)/20000 - 25157*(-2*x + 1
)**(5/2)*(5*x + 3)**(3/2)/64000 - 276727*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/256000
+ 3043997*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/5120000 + 100451901*sqrt(-2*x + 1)*sqr
t(5*x + 3)/51200000 + 1104970911*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/512000
000

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Mathematica [A]  time = 0.145692, size = 80, normalized size = 0.43 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (9216000000 x^6+10112000000 x^5-6123776000 x^4-8717155200 x^3+1291331040 x^2+2994263780 x-104420943\right )-7734796377 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3584000000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-104420943 + 2994263780*x + 1291331040*x^2 - 87
17155200*x^3 - 6123776000*x^4 + 10112000000*x^5 + 9216000000*x^6) - 7734796377*S
qrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/3584000000

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Maple [A]  time = 0.013, size = 155, normalized size = 0.8 \[{\frac{1}{7168000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 184320000000\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-122475520000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-174343104000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-2088418860\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7168000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(184320000000*x^6*(-10*x^2-x+3)^(1/2)+2
02240000000*x^5*(-10*x^2-x+3)^(1/2)-122475520000*x^4*(-10*x^2-x+3)^(1/2)-1743431
04000*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)-2088418860*(-10*x^2
-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50258, size = 157, normalized size = 0.84 \[ \frac{9}{35} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{323}{1400} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{9141}{140000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{25157}{32000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{25157}{640000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{9131991}{2560000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1104970911}{1024000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{9131991}{51200000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/35*(-10*x^2 - x + 3)^(5/2)*x^2 + 323/1400*(-10*x^2 - x + 3)^(5/2)*x - 9141/140
000*(-10*x^2 - x + 3)^(5/2) + 25157/32000*(-10*x^2 - x + 3)^(3/2)*x + 25157/6400
00*(-10*x^2 - x + 3)^(3/2) + 9131991/2560000*sqrt(-10*x^2 - x + 3)*x - 110497091
1/1024000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 9131991/51200000*sqrt(-10*x^2 -
x + 3)

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Fricas [A]  time = 0.219136, size = 111, normalized size = 0.59 \[ \frac{1}{7168000000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (9216000000 \, x^{6} + 10112000000 \, x^{5} - 6123776000 \, x^{4} - 8717155200 \, x^{3} + 1291331040 \, x^{2} + 2994263780 \, x - 104420943\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 7734796377 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7168000000*sqrt(10)*(2*sqrt(10)*(9216000000*x^6 + 10112000000*x^5 - 6123776000
*x^4 - 8717155200*x^3 + 1291331040*x^2 + 2994263780*x - 104420943)*sqrt(5*x + 3)
*sqrt(-2*x + 1) + 7734796377*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)**(5/2)*(2+3*x)**2*(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287411, 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)^(3/2)*(3*x + 2)^2*(-2*x + 1)^(5/2),x, algorithm="giac")

[Out]

3/17920000000*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))) + 7/320000000*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)*sqr
t(-10*x + 5) + 29223315*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 79/192000
000*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(2
2)*sqrt(5*x + 3))) - 89/1920000*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))) + 1/3000*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))) + 3/100*
sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sq
rt(22)*sqrt(5*x + 3)))