3.2442 \(\int \frac{(2+3 x)^3 \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{3}{40} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (408 x+865)}{1280}-\frac{61547 \sqrt{1-2 x} \sqrt{5 x+3}}{5120}+\frac{677017 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5120 \sqrt{10}} \]

[Out]

(-61547*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5120 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*
x)^(3/2))/40 - (3*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(865 + 408*x))/1280 + (677017*Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5120*Sqrt[10])

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Rubi [A]  time = 0.136643, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3}{40} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (408 x+865)}{1280}-\frac{61547 \sqrt{1-2 x} \sqrt{5 x+3}}{5120}+\frac{677017 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5120 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-61547*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5120 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*
x)^(3/2))/40 - (3*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(865 + 408*x))/1280 + (677017*Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5120*Sqrt[10])

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Rubi in Sympy [A]  time = 13.3589, size = 97, normalized size = 0.92 \[ - \frac{3 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}}{40} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}} \left (22950 x + \frac{194625}{4}\right )}{24000} - \frac{61547 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{5120} + \frac{677017 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{51200} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(-2*x + 1)*(3*x + 2)**2*(5*x + 3)**(3/2)/40 - sqrt(-2*x + 1)*(5*x + 3)**(
3/2)*(22950*x + 194625/4)/24000 - 61547*sqrt(-2*x + 1)*sqrt(5*x + 3)/5120 + 6770
17*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/51200

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Mathematica [A]  time = 0.092676, size = 65, normalized size = 0.61 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (17280 x^3+57888 x^2+88092 x+97295\right )-677017 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{51200} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(97295 + 88092*x + 57888*x^2 + 17280*x^3) - 677
017*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/51200

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Maple [A]  time = 0.019, size = 104, normalized size = 1. \[{\frac{1}{102400}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -345600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-1157760\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+677017\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1761840\,x\sqrt{-10\,{x}^{2}-x+3}-1945900\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/102400*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-345600*x^3*(-10*x^2-x+3)^(1/2)-1157760*x^
2*(-10*x^2-x+3)^(1/2)+677017*10^(1/2)*arcsin(20/11*x+1/11)-1761840*x*(-10*x^2-x+
3)^(1/2)-1945900*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.51126, size = 99, normalized size = 0.93 \[ \frac{27}{80} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{677017}{102400} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{351}{320} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{4383}{256} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{114143}{5120} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/80*(-10*x^2 - x + 3)^(3/2)*x + 677017/102400*sqrt(5)*sqrt(2)*arcsin(20/11*x +
1/11) + 351/320*(-10*x^2 - x + 3)^(3/2) - 4383/256*sqrt(-10*x^2 - x + 3)*x - 11
4143/5120*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.218087, size = 90, normalized size = 0.85 \[ -\frac{1}{102400} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (17280 \, x^{3} + 57888 \, x^{2} + 88092 \, x + 97295\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 677017 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/102400*sqrt(10)*(2*sqrt(10)*(17280*x^3 + 57888*x^2 + 88092*x + 97295)*sqrt(5*
x + 3)*sqrt(-2*x + 1) - 677017*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sq
rt(-2*x + 1))))

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Sympy [A]  time = 15.3609, size = 466, normalized size = 4.4 \[ \frac{2 \sqrt{5} \left (\begin{cases} \frac{11 \sqrt{2} \left (- \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{\operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2}\right )}{4} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} + \frac{18 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (\frac{\sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{968} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} + \frac{54 \sqrt{5} \left (\begin{cases} \frac{1331 \sqrt{2} \left (\frac{3 \sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{1936} + \frac{\sqrt{2} \left (- 10 x + 5\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16}\right )}{16} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} + \frac{54 \sqrt{5} \left (\begin{cases} \frac{14641 \sqrt{2} \left (\frac{7 \sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{3872} + \frac{2 \sqrt{2} \left (- 10 x + 5\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} + \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{35 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{128}\right )}{32} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(5)*Piecewise((11*sqrt(2)*(-sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)/22 + asi
n(sqrt(22)*sqrt(5*x + 3)/11)/2)/4, (x >= -3/5) & (x < 1/2)))/625 + 18*sqrt(5)*Pi
ecewise((121*sqrt(2)*(sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/968 - sq
rt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)/22 + 3*asin(sqrt(22)*sqrt(5*x + 3)/11)/8)/8,
(x >= -3/5) & (x < 1/2)))/625 + 54*sqrt(5)*Piecewise((1331*sqrt(2)*(3*sqrt(2)*(
-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/1936 + sqrt(2)*(-10*x + 5)**(3/2)*(5*x
+ 3)**(3/2)/3993 - sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)/22 + 5*asin(sqrt(22)*sq
rt(5*x + 3)/11)/16)/16, (x >= -3/5) & (x < 1/2)))/625 + 54*sqrt(5)*Piecewise((14
641*sqrt(2)*(7*sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/3872 + 2*sqrt(2
)*(-10*x + 5)**(3/2)*(5*x + 3)**(3/2)/3993 + sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x +
3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 - sqrt(2)*sq
rt(-10*x + 5)*sqrt(5*x + 3)/22 + 35*asin(sqrt(22)*sqrt(5*x + 3)/11)/128)/32, (x
>= -3/5) & (x < 1/2)))/625

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GIAC/XCAS [A]  time = 0.231878, size = 85, normalized size = 0.8 \[ -\frac{1}{1280000} \, \sqrt{5}{\left (2 \,{\left (36 \,{\left (24 \,{\left (20 \, x + 43\right )}{\left (5 \, x + 3\right )} + 5179\right )}{\left (5 \, x + 3\right )} + 1538675\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 16925425 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1280000*sqrt(5)*(2*(36*(24*(20*x + 43)*(5*x + 3) + 5179)*(5*x + 3) + 1538675)
*sqrt(5*x + 3)*sqrt(-10*x + 5) - 16925425*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x
+ 3)))