3.2564 \(\int \frac{(2+3 x)^3 \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{\sqrt{5 x+3} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{233 \sqrt{5 x+3} (3 x+2)^2}{66 \sqrt{1-2 x}}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (69780 x+168157)}{3520}+\frac{126513 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320 \sqrt{10}} \]

[Out]

(-233*(2 + 3*x)^2*Sqrt[3 + 5*x])/(66*Sqrt[1 - 2*x]) + ((2 + 3*x)^3*Sqrt[3 + 5*x]
)/(3*(1 - 2*x)^(3/2)) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(168157 + 69780*x))/3520 +
(126513*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(320*Sqrt[10])

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Rubi [A]  time = 0.190855, antiderivative size = 113, 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 \[ \frac{\sqrt{5 x+3} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{233 \sqrt{5 x+3} (3 x+2)^2}{66 \sqrt{1-2 x}}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (69780 x+168157)}{3520}+\frac{126513 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-233*(2 + 3*x)^2*Sqrt[3 + 5*x])/(66*Sqrt[1 - 2*x]) + ((2 + 3*x)^3*Sqrt[3 + 5*x]
)/(3*(1 - 2*x)^(3/2)) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(168157 + 69780*x))/3520 +
(126513*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(320*Sqrt[10])

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Rubi in Sympy [A]  time = 19.6097, size = 104, normalized size = 0.92 \[ - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{261675 x}{2} + \frac{2522355}{8}\right )}{6600} + \frac{126513 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{3200} - \frac{233 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{66 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

-sqrt(-2*x + 1)*sqrt(5*x + 3)*(261675*x/2 + 2522355/8)/6600 + 126513*sqrt(10)*as
in(sqrt(22)*sqrt(5*x + 3)/11)/3200 - 233*(3*x + 2)**2*sqrt(5*x + 3)/(66*sqrt(-2*
x + 1)) + (3*x + 2)**3*sqrt(5*x + 3)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.160605, size = 74, normalized size = 0.65 \[ \frac{4174929 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (71280 x^3+431244 x^2-1786144 x+625431\right )}{105600 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(625431 - 1786144*x + 431244*x^2 + 71280*x^3) + 4174929*Sqrt[
10 - 20*x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(105600*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.018, size = 137, normalized size = 1.2 \[{\frac{1}{211200\, \left ( -1+2\,x \right ) ^{2}} \left ( 16699716\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-1425600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-16699716\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-8624880\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+4174929\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-12508620\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/211200*(16699716*10^(1/2)*arcsin(20/11*x+1/11)*x^2-1425600*x^3*(-10*x^2-x+3)^(
1/2)-16699716*10^(1/2)*arcsin(20/11*x+1/11)*x-8624880*x^2*(-10*x^2-x+3)^(1/2)+41
74929*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-12508620*(-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)^3/(-2*x + 1)^(5/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0.221795, size = 120, normalized size = 1.06 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (71280 \, x^{3} + 431244 \, x^{2} - 1786144 \, x + 625431\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 4174929 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{211200 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/211200*sqrt(10)*(2*sqrt(10)*(71280*x^3 + 431244*x^2 - 1786144*x + 625431)*sqr
t(5*x + 3)*sqrt(-2*x + 1) - 4174929*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x
+ 1)/(sqrt(5*x + 3)*sqrt(-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)**3*(3+5*x)**(1/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230856, size = 113, normalized size = 1. \[ \frac{126513}{3200} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (891 \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} + 85 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 2783318 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 45924219 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1320000 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

126513/3200*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/1320000*(4*(891*(4*
sqrt(5)*(5*x + 3) + 85*sqrt(5))*(5*x + 3) - 2783318*sqrt(5))*(5*x + 3) + 4592421
9*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2