3.2575 \(\int \frac{(2+3 x) (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ \frac{7 (5 x+3)^{5/2}}{33 (1-2 x)^{3/2}}-\frac{169 (5 x+3)^{3/2}}{66 \sqrt{1-2 x}}-\frac{845}{88} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{169}{8} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-845*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/88 - (169*(3 + 5*x)^(3/2))/(66*Sqrt[1 - 2*x])
+ (7*(3 + 5*x)^(5/2))/(33*(1 - 2*x)^(3/2)) + (169*Sqrt[5/2]*ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]])/8

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Rubi [A]  time = 0.0932571, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{7 (5 x+3)^{5/2}}{33 (1-2 x)^{3/2}}-\frac{169 (5 x+3)^{3/2}}{66 \sqrt{1-2 x}}-\frac{845}{88} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{169}{8} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-845*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/88 - (169*(3 + 5*x)^(3/2))/(66*Sqrt[1 - 2*x])
+ (7*(3 + 5*x)^(5/2))/(33*(1 - 2*x)^(3/2)) + (169*Sqrt[5/2]*ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]])/8

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Rubi in Sympy [A]  time = 9.4227, size = 85, normalized size = 0.89 \[ - \frac{845 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{88} + \frac{169 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16} - \frac{169 \left (5 x + 3\right )^{\frac{3}{2}}}{66 \sqrt{- 2 x + 1}} + \frac{7 \left (5 x + 3\right )^{\frac{5}{2}}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-845*sqrt(-2*x + 1)*sqrt(5*x + 3)/88 + 169*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/
11)/16 - 169*(5*x + 3)**(3/2)/(66*sqrt(-2*x + 1)) + 7*(5*x + 3)**(5/2)/(33*(-2*x
+ 1)**(3/2))

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Mathematica [A]  time = 0.109347, size = 69, normalized size = 0.72 \[ \frac{507 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-2 \sqrt{5 x+3} \left (180 x^2-1136 x+369\right )}{48 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[3 + 5*x]*(369 - 1136*x + 180*x^2) + 507*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcS
in[Sqrt[5/11]*Sqrt[1 - 2*x]])/(48*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.016, size = 120, normalized size = 1.3 \[{\frac{1}{96\, \left ( -1+2\,x \right ) ^{2}} \left ( 2028\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-2028\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-720\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+507\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +4544\,x\sqrt{-10\,{x}^{2}-x+3}-1476\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/96*(2028*10^(1/2)*arcsin(20/11*x+1/11)*x^2-2028*10^(1/2)*arcsin(20/11*x+1/11)*
x-720*x^2*(-10*x^2-x+3)^(1/2)+507*10^(1/2)*arcsin(20/11*x+1/11)+4544*x*(-10*x^2-
x+3)^(1/2)-1476*(-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 [A]  time = 1.51011, size = 161, normalized size = 1.68 \[ \frac{169}{32} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{12 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{77 \, \sqrt{-10 \, x^{2} - x + 3}}{24 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{271 \, \sqrt{-10 \, x^{2} - x + 3}}{12 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

169/32*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 7/12*(-10*x^2 - x + 3)^(3/2)/(8*
x^3 - 12*x^2 + 6*x - 1) + 3/4*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 77/24*
sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 271/12*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.228521, size = 122, normalized size = 1.27 \[ -\frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (180 \, x^{2} - 1136 \, x + 369\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 507 \, \sqrt{5}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{96 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/96*sqrt(2)*(2*sqrt(2)*(180*x^2 - 1136*x + 369)*sqrt(5*x + 3)*sqrt(-2*x + 1) -
507*sqrt(5)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(5)*sqrt(2)*(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+5*x)**(3/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229413, size = 96, normalized size = 1. \[ \frac{169}{16} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (9 \, \sqrt{5}{\left (5 \, x + 3\right )} - 338 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 5577 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{600 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

169/16*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/600*(4*(9*sqrt(5)*(5*x +
3) - 338*sqrt(5))*(5*x + 3) + 5577*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x
- 1)^2