3.2515 \(\int \frac{(2+3 x) (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ \frac{7 (5 x+3)^{5/2}}{11 \sqrt{1-2 x}}+\frac{173}{88} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{519}{32} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{5709 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{32 \sqrt{10}} \]

[Out]

(519*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/32 + (173*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/88 +
(7*(3 + 5*x)^(5/2))/(11*Sqrt[1 - 2*x]) - (5709*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])
/(32*Sqrt[10])

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Rubi [A]  time = 0.0994635, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{7 (5 x+3)^{5/2}}{11 \sqrt{1-2 x}}+\frac{173}{88} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{519}{32} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{5709 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{32 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(519*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/32 + (173*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/88 +
(7*(3 + 5*x)^(5/2))/(11*Sqrt[1 - 2*x]) - (5709*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])
/(32*Sqrt[10])

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Rubi in Sympy [A]  time = 8.87354, size = 85, normalized size = 0.9 \[ \frac{173 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{88} + \frac{519 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{32} - \frac{5709 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{320} + \frac{7 \left (5 x + 3\right )^{\frac{5}{2}}}{11 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

173*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/88 + 519*sqrt(-2*x + 1)*sqrt(5*x + 3)/32 - 5
709*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/320 + 7*(5*x + 3)**(5/2)/(11*sqrt(-
2*x + 1))

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Mathematica [A]  time = 0.0680364, size = 64, normalized size = 0.68 \[ \frac{5709 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (120 x^2+490 x-891\right )}{320 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(-891 + 490*x + 120*x^2) + 5709*Sqrt[10 - 20*x]*ArcSin[Sqrt[5
/11]*Sqrt[1 - 2*x]])/(320*Sqrt[1 - 2*x])

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Maple [A]  time = 0.016, size = 106, normalized size = 1.1 \[ -{\frac{1}{-640+1280\,x} \left ( 11418\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-2400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-5709\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -9800\,x\sqrt{-10\,{x}^{2}-x+3}+17820\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/640*(11418*10^(1/2)*arcsin(20/11*x+1/11)*x-2400*x^2*(-10*x^2-x+3)^(1/2)-5709*
10^(1/2)*arcsin(20/11*x+1/11)-9800*x*(-10*x^2-x+3)^(1/2)+17820*(-10*x^2-x+3)^(1/
2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50843, size = 131, normalized size = 1.39 \[ -\frac{5709}{640} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{99}{32} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{8 \,{\left (2 \, x - 1\right )}} - \frac{231 \, \sqrt{-10 \, x^{2} - x + 3}}{8 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5709/640*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 99/32*sqrt(-10*x^2 - x + 3) -
7/4*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) - 3/8*(-10*x^2 - x + 3)^(3/2)/(2*
x - 1) - 231/8*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.233754, size = 100, normalized size = 1.06 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (120 \, x^{2} + 490 \, x - 891\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 5709 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{640 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/640*sqrt(10)*(2*sqrt(10)*(120*x^2 + 490*x - 891)*sqrt(5*x + 3)*sqrt(-2*x + 1)
- 5709*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1)))
)/(2*x - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}}{\left (- 2 x + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)*(5*x + 3)**(3/2)/(-2*x + 1)**(3/2), x)

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GIAC/XCAS [A]  time = 0.231876, size = 96, normalized size = 1.02 \[ -\frac{5709}{320} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 173 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 5709 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{800 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5709/320*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/800*(2*(12*sqrt(5)*(5
*x + 3) + 173*sqrt(5))*(5*x + 3) - 5709*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(
2*x - 1)