3.2516 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=71 \[ \frac{(5 x+3)^{3/2}}{\sqrt{1-2 x}}+\frac{15}{4} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{33}{4} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(15*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4 + (3 + 5*x)^(3/2)/Sqrt[1 - 2*x] - (33*Sqrt[5/
2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/4

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Rubi [A]  time = 0.0631282, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{(5 x+3)^{3/2}}{\sqrt{1-2 x}}+\frac{15}{4} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{33}{4} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4 + (3 + 5*x)^(3/2)/Sqrt[1 - 2*x] - (33*Sqrt[5/
2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/4

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Rubi in Sympy [A]  time = 6.55448, size = 61, normalized size = 0.86 \[ \frac{15 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{4} - \frac{33 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8} + \frac{\left (5 x + 3\right )^{\frac{3}{2}}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*sqrt(-2*x + 1)*sqrt(5*x + 3)/4 - 33*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/
8 + (5*x + 3)**(3/2)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.0551798, size = 59, normalized size = 0.83 \[ \frac{2 \sqrt{5 x+3} (27-10 x)+33 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{8 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(27 - 10*x)*Sqrt[3 + 5*x] + 33*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x
]])/(8*Sqrt[1 - 2*x])

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{1 \left ( 3+5\,x \right ) ^{{\frac{3}{2}}} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.49862, size = 84, normalized size = 1.18 \[ -\frac{33}{16} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{33 \, \sqrt{-10 \, x^{2} - x + 3}}{4 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-33/16*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/2*(-10*x^2 - x + 3)^(3/2)/(4*x
^2 - 4*x + 1) - 33/4*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.230411, size = 101, normalized size = 1.42 \[ \frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (10 \, x - 27\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 33 \, \sqrt{5}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{16 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*sqrt(2)*(2*sqrt(2)*(10*x - 27)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 33*sqrt(5)*(2
*x - 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/
(2*x - 1)

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Sympy [A]  time = 6.08687, size = 144, normalized size = 2.03 \[ \begin{cases} \frac{25 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{2 \sqrt{10 x - 5}} - \frac{165 i \sqrt{x + \frac{3}{5}}}{4 \sqrt{10 x - 5}} + \frac{33 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{8} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{33 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{8} - \frac{25 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{2 \sqrt{- 10 x + 5}} + \frac{165 \sqrt{x + \frac{3}{5}}}{4 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((25*I*(x + 3/5)**(3/2)/(2*sqrt(10*x - 5)) - 165*I*sqrt(x + 3/5)/(4*sqr
t(10*x - 5)) + 33*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/8, 10*Abs(x + 3/5
)/11 > 1), (-33*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/8 - 25*(x + 3/5)**(3/2
)/(2*sqrt(-10*x + 5)) + 165*sqrt(x + 3/5)/(4*sqrt(-10*x + 5)), True))

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GIAC/XCAS [A]  time = 0.228936, size = 78, normalized size = 1.1 \[ -\frac{33}{8} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \, \sqrt{5}{\left (5 \, x + 3\right )} - 33 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{20 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-33/8*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/20*(2*sqrt(5)*(5*x + 3) -
33*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)