3.2423 \(\int \frac{(1-2 x)^{5/2}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{2 (1-2 x)^{5/2}}{5 \sqrt{5 x+3}}-\frac{1}{5} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{33}{50} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(5/2))/(5*Sqrt[3 + 5*x]) - (33*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/50 - (
(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/5 - (363*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(50*Sq
rt[10])

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Rubi [A]  time = 0.0857884, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{2 (1-2 x)^{5/2}}{5 \sqrt{5 x+3}}-\frac{1}{5} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{33}{50} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2))/(5*Sqrt[3 + 5*x]) - (33*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/50 - (
(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/5 - (363*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(50*Sq
rt[10])

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Rubi in Sympy [A]  time = 9.13439, size = 85, normalized size = 0.9 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{5 \sqrt{5 x + 3}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{5} - \frac{33 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{50} - \frac{363 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{500} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)/(5*sqrt(5*x + 3)) - (-2*x + 1)**(3/2)*sqrt(5*x + 3)/5 - 33*
sqrt(-2*x + 1)*sqrt(5*x + 3)/50 - 363*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/5
00

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Mathematica [A]  time = 0.102917, size = 60, normalized size = 0.64 \[ \frac{\sqrt{1-2 x} \left (20 x^2-75 x-149\right )}{50 \sqrt{5 x+3}}+\frac{363 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{50 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-149 - 75*x + 20*x^2))/(50*Sqrt[3 + 5*x]) + (363*ArcSin[Sqrt[5/1
1]*Sqrt[1 - 2*x]])/(50*Sqrt[10])

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{1 \left ( 1-2\,x \right ) ^{{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50461, size = 101, normalized size = 1.07 \[ -\frac{4 \, x^{3}}{5 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{17 \, x^{2}}{5 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{363}{1000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{223 \, x}{50 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{149}{50 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/5*x^3/sqrt(-10*x^2 - x + 3) + 17/5*x^2/sqrt(-10*x^2 - x + 3) + 363/1000*sqrt(
10)*arcsin(-20/11*x - 1/11) + 223/50*x/sqrt(-10*x^2 - x + 3) - 149/50/sqrt(-10*x
^2 - x + 3)

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Fricas [A]  time = 0.223257, size = 100, normalized size = 1.06 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (20 \, x^{2} - 75 \, x - 149\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 363 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1000 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1000*sqrt(10)*(2*sqrt(10)*(20*x^2 - 75*x - 149)*sqrt(5*x + 3)*sqrt(-2*x + 1) -
363*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/
(5*x + 3)

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Sympy [A]  time = 29.5776, size = 230, normalized size = 2.45 \[ \begin{cases} \frac{4 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{5 \sqrt{10 x - 5}} - \frac{121 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{25 \sqrt{10 x - 5}} + \frac{121 i \sqrt{x + \frac{3}{5}}}{250 \sqrt{10 x - 5}} + \frac{363 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{500} + \frac{2662 i}{625 \sqrt{x + \frac{3}{5}} \sqrt{10 x - 5}} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{363 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{500} - \frac{4 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{5 \sqrt{- 10 x + 5}} + \frac{121 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{25 \sqrt{- 10 x + 5}} - \frac{121 \sqrt{x + \frac{3}{5}}}{250 \sqrt{- 10 x + 5}} - \frac{2662}{625 \sqrt{- 10 x + 5} \sqrt{x + \frac{3}{5}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((4*I*(x + 3/5)**(5/2)/(5*sqrt(10*x - 5)) - 121*I*(x + 3/5)**(3/2)/(25*
sqrt(10*x - 5)) + 121*I*sqrt(x + 3/5)/(250*sqrt(10*x - 5)) + 363*sqrt(10)*I*acos
h(sqrt(110)*sqrt(x + 3/5)/11)/500 + 2662*I/(625*sqrt(x + 3/5)*sqrt(10*x - 5)), 1
0*Abs(x + 3/5)/11 > 1), (-363*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/500 - 4*
(x + 3/5)**(5/2)/(5*sqrt(-10*x + 5)) + 121*(x + 3/5)**(3/2)/(25*sqrt(-10*x + 5))
- 121*sqrt(x + 3/5)/(250*sqrt(-10*x + 5)) - 2662/(625*sqrt(-10*x + 5)*sqrt(x +
3/5)), True))

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GIAC/XCAS [A]  time = 0.268466, size = 150, normalized size = 1.6 \[ \frac{1}{1250} \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} - 99 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{363}{500} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{121 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{1250 \, \sqrt{5 \, x + 3}} + \frac{242 \, \sqrt{10} \sqrt{5 \, x + 3}}{625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1250*(4*sqrt(5)*(5*x + 3) - 99*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 363/50
0*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 121/1250*sqrt(10)*(sqrt(2)*sqrt
(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 242/625*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*
sqrt(-10*x + 5) - sqrt(22))