3.2443 \(\int \frac{(2+3 x)^2 \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=99 \[ -\frac{1}{10} \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}-\frac{181}{400} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{6269 \sqrt{1-2 x} \sqrt{5 x+3}}{1600}+\frac{68959 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]

[Out]

(-6269*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1600 - (181*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/4
00 - (Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(3/2))/10 + (68959*ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]])/(1600*Sqrt[10])

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Rubi [A]  time = 0.114035, antiderivative size = 99, 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 \[ -\frac{1}{10} \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}-\frac{181}{400} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{6269 \sqrt{1-2 x} \sqrt{5 x+3}}{1600}+\frac{68959 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-6269*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1600 - (181*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/4
00 - (Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(3/2))/10 + (68959*ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]])/(1600*Sqrt[10])

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Rubi in Sympy [A]  time = 9.47344, size = 88, normalized size = 0.89 \[ - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}} \left (9 x + 6\right )}{30} - \frac{181 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{400} - \frac{6269 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1600} + \frac{68959 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(5*x + 3)**(3/2)*(9*x + 6)/30 - 181*sqrt(-2*x + 1)*(5*x + 3)**(3
/2)/400 - 6269*sqrt(-2*x + 1)*sqrt(5*x + 3)/1600 + 68959*sqrt(10)*asin(sqrt(22)*
sqrt(5*x + 3)/11)/16000

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Mathematica [A]  time = 0.0807525, size = 60, normalized size = 0.61 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (2400 x^2+6660 x+9401\right )-68959 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{16000} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(9401 + 6660*x + 2400*x^2) - 68959*Sqrt[10]*Arc
Sin[Sqrt[5/11]*Sqrt[1 - 2*x]])/16000

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Maple [A]  time = 0.015, size = 87, normalized size = 0.9 \[{\frac{1}{32000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -48000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+68959\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -133200\,x\sqrt{-10\,{x}^{2}-x+3}-188020\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32000*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-48000*x^2*(-10*x^2-x+3)^(1/2)+68959*10^(1/
2)*arcsin(20/11*x+1/11)-133200*x*(-10*x^2-x+3)^(1/2)-188020*(-10*x^2-x+3)^(1/2))
/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50221, size = 78, normalized size = 0.79 \[ \frac{68959}{32000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3}{20} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{321}{80} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{10121}{1600} \, \sqrt{-10 \, x^{2} - x + 3} \]

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]

68959/32000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 3/20*(-10*x^2 - x + 3)^(3/2
) - 321/80*sqrt(-10*x^2 - x + 3)*x - 10121/1600*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.213823, size = 84, normalized size = 0.85 \[ -\frac{1}{32000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (2400 \, x^{2} + 6660 \, x + 9401\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 68959 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

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]

-1/32000*sqrt(10)*(2*sqrt(10)*(2400*x^2 + 6660*x + 9401)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) - 68959*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 10.8309, size = 292, normalized size = 2.95 \[ \frac{2 \sqrt{5} \left (\begin{cases} \frac{11 \sqrt{2} \left (- \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{\operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2}\right )}{4} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{125} + \frac{12 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (\frac{\sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{968} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{125} + \frac{18 \sqrt{5} \left (\begin{cases} \frac{1331 \sqrt{2} \left (\frac{3 \sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{1936} + \frac{\sqrt{2} \left (- 10 x + 5\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16}\right )}{16} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**2*(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)))/125 + 12*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)))/125 + 18*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)))/125

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GIAC/XCAS [A]  time = 0.23897, size = 73, normalized size = 0.74 \[ -\frac{1}{16000} \, \sqrt{5}{\left (2 \,{\left (12 \,{\left (40 \, x + 87\right )}{\left (5 \, x + 3\right )} + 6269\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 68959 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

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]

-1/16000*sqrt(5)*(2*(12*(40*x + 87)*(5*x + 3) + 6269)*sqrt(5*x + 3)*sqrt(-10*x +
5) - 68959*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))