3.2271 \(\int \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=165 \[ -\frac{13}{80} (1-2 x)^{3/2} (5 x+3)^{7/2}-\frac{1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}-\frac{1069 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1280}-\frac{11759 (1-2 x)^{3/2} (5 x+3)^{3/2}}{3072}-\frac{129349 (1-2 x)^{3/2} \sqrt{5 x+3}}{8192}+\frac{1422839 \sqrt{1-2 x} \sqrt{5 x+3}}{81920}+\frac{15651229 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{81920 \sqrt{10}} \]

[Out]

(1422839*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/81920 - (129349*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x])/8192 - (11759*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/3072 - (1069*(1 - 2*x)^(3/2)
*(3 + 5*x)^(5/2))/1280 - (13*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/80 - ((1 - 2*x)^(3
/2)*(2 + 3*x)*(3 + 5*x)^(7/2))/20 + (15651229*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/
(81920*Sqrt[10])

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Rubi [A]  time = 0.194294, antiderivative size = 165, 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 \[ -\frac{13}{80} (1-2 x)^{3/2} (5 x+3)^{7/2}-\frac{1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}-\frac{1069 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1280}-\frac{11759 (1-2 x)^{3/2} (5 x+3)^{3/2}}{3072}-\frac{129349 (1-2 x)^{3/2} \sqrt{5 x+3}}{8192}+\frac{1422839 \sqrt{1-2 x} \sqrt{5 x+3}}{81920}+\frac{15651229 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{81920 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(1422839*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/81920 - (129349*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x])/8192 - (11759*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/3072 - (1069*(1 - 2*x)^(3/2)
*(3 + 5*x)^(5/2))/1280 - (13*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/80 - ((1 - 2*x)^(3
/2)*(2 + 3*x)*(3 + 5*x)^(7/2))/20 + (15651229*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/
(81920*Sqrt[10])

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Rubi in Sympy [A]  time = 15.2061, size = 150, normalized size = 0.91 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}} \left (9 x + 6\right )}{60} - \frac{13 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{80} + \frac{1069 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{3200} - \frac{11759 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{38400} - \frac{129349 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{61440} - \frac{1422839 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{81920} + \frac{15651229 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{819200} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)*(5*x + 3)**(7/2)*(9*x + 6)/60 - 13*(-2*x + 1)**(3/2)*(5*x + 3
)**(7/2)/80 + 1069*sqrt(-2*x + 1)*(5*x + 3)**(7/2)/3200 - 11759*sqrt(-2*x + 1)*(
5*x + 3)**(5/2)/38400 - 129349*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/61440 - 1422839*s
qrt(-2*x + 1)*sqrt(5*x + 3)/81920 + 15651229*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3
)/11)/819200

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Mathematica [A]  time = 0.113877, size = 75, normalized size = 0.45 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (9216000 x^5+28108800 x^4+32887680 x^3+16507936 x^2+17884 x-6023169\right )-46953687 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2457600} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-6023169 + 17884*x + 16507936*x^2 + 32887680*x^
3 + 28108800*x^4 + 9216000*x^5) - 46953687*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2
*x]])/2457600

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Maple [A]  time = 0.013, size = 138, normalized size = 0.8 \[{\frac{1}{4915200}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 184320000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +357680\,x\sqrt{-10\,{x}^{2}-x+3}-120463380\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4915200*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(184320000*x^5*(-10*x^2-x+3)^(1/2)+5621760
00*x^4*(-10*x^2-x+3)^(1/2)[phone]*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)+357680*x*(-10*x^2-x+3)^(1/
2)-120463380*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50612, size = 140, normalized size = 0.85 \[ -\frac{15}{4} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{177}{16} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{17153}{1280} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{133567}{15360} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{129349}{4096} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{15651229}{1638400} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{129349}{81920} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15/4*(-10*x^2 - x + 3)^(3/2)*x^3 - 177/16*(-10*x^2 - x + 3)^(3/2)*x^2 - 17153/1
280*(-10*x^2 - x + 3)^(3/2)*x - 133567/15360*(-10*x^2 - x + 3)^(3/2) + 129349/40
96*sqrt(-10*x^2 - x + 3)*x - 15651229/1638400*sqrt(10)*arcsin(-20/11*x - 1/11) +
129349/81920*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.217268, size = 104, normalized size = 0.63 \[ \frac{1}{4915200} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (9216000 \, x^{5} + 28108800 \, x^{4} + 32887680 \, x^{3} + 16507936 \, x^{2} + 17884 \, x - 6023169\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 46953687 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4915200*sqrt(10)*(2*sqrt(10)*(9216000*x^5 + 28108800*x^4 + 32887680*x^3 + 1650
7936*x^2 + 17884*x - 6023169)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 46953687*arctan(1/2
0*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 166.953, size = 694, normalized size = 4.21 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5929*sqrt(2)*Piecewise((121*sqrt(5)*(-sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20
*x + 1)/121 + asin(sqrt(55)*sqrt(-2*x + 1)/11))/200, (x <= 1/2) & (x > -3/5)))/3
2 + 1309*sqrt(2)*Piecewise((1331*sqrt(5)*(-5*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6
)**(3/2)/7986 - sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20*x + 1)/1936 + asin(sqr
t(55)*sqrt(-2*x + 1)/11)/16)/125, (x <= 1/2) & (x > -3/5)))/4 - 3467*sqrt(2)*Pie
cewise((14641*sqrt(5)*(-5*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6)**(3/2)/7986 - sqr
t(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20*x + 1)/3872 - sqrt(5)*sqrt(-2*x + 1)*sqrt
(10*x + 6)*(12100*x - 2000*(-2*x + 1)**3 + 6600*(-2*x + 1)**2 - 4719)/1874048 +
5*asin(sqrt(55)*sqrt(-2*x + 1)/11)/128)/625, (x <= 1/2) & (x > -3/5)))/16 + 255*
sqrt(2)*Piecewise((161051*sqrt(5)*(5*sqrt(5)*(-2*x + 1)**(5/2)*(10*x + 6)**(5/2)
/322102 - 5*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(-2*x
+ 1)*sqrt(10*x + 6)*(20*x + 1)/7744 - 3*sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(
12100*x - 2000*(-2*x + 1)**3 + 6600*(-2*x + 1)**2 - 4719)/3748096 + 7*asin(sqrt(
55)*sqrt(-2*x + 1)/11)/256)/3125, (x <= 1/2) & (x > -3/5)))/4 - 225*sqrt(2)*Piec
ewise((1771561*sqrt(5)*(5*sqrt(5)*(-2*x + 1)**(5/2)*(10*x + 6)**(5/2)/161051 + 5
*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6)**(3/2)*(20*x + 1)**3/170069856 - 5*sqrt(5)
*(-2*x + 1)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6
)*(20*x + 1)/15488 - 13*sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(12100*x - 2000*(-
2*x + 1)**3 + 6600*(-2*x + 1)**2 - 4719)/14992384 + 21*asin(sqrt(55)*sqrt(-2*x +
1)/11)/1024)/15625, (x <= 1/2) & (x > -3/5)))/32

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GIAC/XCAS [A]  time = 0.272583, size = 427, normalized size = 2.59 \[ \frac{3}{102400000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{19}{6400000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{541}{1920000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{19}{2000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{100} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/102400000*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))) + 19/6400000*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))) + 541/1920000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 41
25)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x
+ 3))) + 19/2000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-1
0*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/100*sqrt(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)))