3.2247 \(\int \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=121 \[ -\frac{3}{40} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{37}{160} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{1313 \sqrt{5 x+3} (1-2 x)^{3/2}}{1280}+\frac{14443 \sqrt{5 x+3} \sqrt{1-2 x}}{12800}+\frac{158873 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{12800 \sqrt{10}} \]

[Out]

(14443*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/12800 - (1313*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])
/1280 - (37*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/160 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)*
(3 + 5*x)^(3/2))/40 + (158873*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(12800*Sqrt[10])

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Rubi [A]  time = 0.137332, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3}{40} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{37}{160} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{1313 \sqrt{5 x+3} (1-2 x)^{3/2}}{1280}+\frac{14443 \sqrt{5 x+3} \sqrt{1-2 x}}{12800}+\frac{158873 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{12800 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(14443*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/12800 - (1313*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])
/1280 - (37*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/160 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)*
(3 + 5*x)^(3/2))/40 + (158873*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(12800*Sqrt[10])

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Rubi in Sympy [A]  time = 11.0918, size = 109, normalized size = 0.9 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}} \left (9 x + 6\right )}{40} - \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{160} + \frac{1313 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{3200} - \frac{14443 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{12800} + \frac{158873 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{128000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)*(9*x + 6)/40 - 37*(-2*x + 1)**(3/2)*(5*x + 3
)**(3/2)/160 + 1313*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/3200 - 14443*sqrt(-2*x + 1)*
sqrt(5*x + 3)/12800 + 158873*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/128000

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Mathematica [A]  time = 0.100339, size = 65, normalized size = 0.54 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (28800 x^3+51680 x^2+22500 x-13327\right )-158873 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{128000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-13327 + 22500*x + 51680*x^2 + 28800*x^3) - 158
873*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/128000

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Maple [A]  time = 0.013, size = 104, normalized size = 0.9 \[{\frac{1}{256000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 576000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+1033600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+158873\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +450000\,x\sqrt{-10\,{x}^{2}-x+3}-266540\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/256000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(576000*x^3*(-10*x^2-x+3)^(1/2)+1033600*x^2
*(-10*x^2-x+3)^(1/2)+158873*10^(1/2)*arcsin(20/11*x+1/11)+450000*x*(-10*x^2-x+3)
^(1/2)-266540*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50142, size = 95, normalized size = 0.79 \[ -\frac{9}{40} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{61}{160} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{1313}{640} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{158873}{256000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1313}{12800} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/40*(-10*x^2 - x + 3)^(3/2)*x - 61/160*(-10*x^2 - x + 3)^(3/2) + 1313/640*sqrt
(-10*x^2 - x + 3)*x - 158873/256000*sqrt(10)*arcsin(-20/11*x - 1/11) + 1313/1280
0*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.218289, size = 90, normalized size = 0.74 \[ \frac{1}{256000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (28800 \, x^{3} + 51680 \, x^{2} + 22500 \, x - 13327\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 158873 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/256000*sqrt(10)*(2*sqrt(10)*(28800*x^3 + 51680*x^2 + 22500*x - 13327)*sqrt(5*x
+ 3)*sqrt(-2*x + 1) + 158873*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqr
t(-2*x + 1))))

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Sympy [A]  time = 114.398, size = 314, normalized size = 2.6 \[ - \frac{49 \sqrt{2} \left (\begin{cases} \frac{121 \sqrt{5} \left (- \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{121} + \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\right )}{200} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{8} + \frac{21 \sqrt{2} \left (\begin{cases} \frac{1331 \sqrt{5} \left (- \frac{5 \sqrt{5} \left (- 2 x + 1\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{7986} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{1936} + \frac{\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{16}\right )}{125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{4} - \frac{9 \sqrt{2} \left (\begin{cases} \frac{14641 \sqrt{5} \left (- \frac{5 \sqrt{5} \left (- 2 x + 1\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{7986} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{3872} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (12100 x - 2000 \left (- 2 x + 1\right )^{3} + 6600 \left (- 2 x + 1\right )^{2} - 4719\right )}{1874048} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{128}\right )}{625} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-49*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)))/8 +
21*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(sqrt(55)
*sqrt(-2*x + 1)/11)/16)/125, (x <= 1/2) & (x > -3/5)))/4 - 9*sqrt(2)*Piecewise((
14641*sqrt(5)*(-5*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqr
t(-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(s
qrt(55)*sqrt(-2*x + 1)/11)/128)/625, (x <= 1/2) & (x > -3/5)))/8

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GIAC/XCAS [A]  time = 0.255179, size = 220, normalized size = 1.82 \[ \frac{3}{640000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{2000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{100} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/640000*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
/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))) + 1/100*sqrt(5)*(2*(20*x + 1)*s
qrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))