3.2565 \(\int \frac{(2+3 x)^2 \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{21 (5 x+3)^{3/2}}{11 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{3/2}}{66 (1-2 x)^{3/2}}-\frac{519}{88} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{519 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8 \sqrt{10}} \]

[Out]

(-519*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/88 + (49*(3 + 5*x)^(3/2))/(66*(1 - 2*x)^(3/2)
) - (21*(3 + 5*x)^(3/2))/(11*Sqrt[1 - 2*x]) + (519*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]])/(8*Sqrt[10])

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Rubi [A]  time = 0.11762, antiderivative size = 94, 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 \[ -\frac{21 (5 x+3)^{3/2}}{11 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{3/2}}{66 (1-2 x)^{3/2}}-\frac{519}{88} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{519 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-519*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/88 + (49*(3 + 5*x)^(3/2))/(66*(1 - 2*x)^(3/2)
) - (21*(3 + 5*x)^(3/2))/(11*Sqrt[1 - 2*x]) + (519*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]])/(8*Sqrt[10])

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Rubi in Sympy [A]  time = 10.3761, size = 85, normalized size = 0.9 \[ - \frac{519 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{88} + \frac{519 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{80} - \frac{21 \left (5 x + 3\right )^{\frac{3}{2}}}{11 \sqrt{- 2 x + 1}} + \frac{49 \left (5 x + 3\right )^{\frac{3}{2}}}{66 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

-519*sqrt(-2*x + 1)*sqrt(5*x + 3)/88 + 519*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/
11)/80 - 21*(5*x + 3)**(3/2)/(11*sqrt(-2*x + 1)) + 49*(5*x + 3)**(3/2)/(66*(-2*x
+ 1)**(3/2))

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Mathematica [A]  time = 0.132494, size = 69, normalized size = 0.73 \[ \frac{17127 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (1188 x^2-7712 x+2481\right )}{2640 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(2481 - 7712*x + 1188*x^2) + 17127*Sqrt[10 - 20*x]*(-1 + 2*x)
*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(2640*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.019, size = 120, normalized size = 1.3 \[{\frac{1}{5280\, \left ( -1+2\,x \right ) ^{2}} \left ( 68508\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-68508\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-23760\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+17127\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +154240\,x\sqrt{-10\,{x}^{2}-x+3}-49620\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5280*(68508*10^(1/2)*arcsin(20/11*x+1/11)*x^2-68508*10^(1/2)*arcsin(20/11*x+1/
11)*x-23760*x^2*(-10*x^2-x+3)^(1/2)+17127*10^(1/2)*arcsin(20/11*x+1/11)+154240*x
*(-10*x^2-x+3)^(1/2)-49620*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+
2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Fricas [A]  time = 0.228309, size = 113, normalized size = 1.2 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (1188 \, x^{2} - 7712 \, x + 2481\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 17127 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{5280 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5280*sqrt(10)*(2*sqrt(10)*(1188*x^2 - 7712*x + 2481)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) - 17127*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sq
rt(-2*x + 1))))/(4*x^2 - 4*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((2+3*x)**2*(3+5*x)**(1/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.234647, size = 96, normalized size = 1.02 \[ \frac{519}{80} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (297 \, \sqrt{5}{\left (5 \, x + 3\right )} - 11422 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 188397 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{33000 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

519/80*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/33000*(4*(297*sqrt(5)*(5
*x + 3) - 11422*sqrt(5))*(5*x + 3) + 188397*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x +
5)/(2*x - 1)^2