3.2347 \(\int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=72 \[ \frac{1}{10} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{33}{100} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{100 \sqrt{10}} \]

[Out]

(33*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/100 + ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/10 + (363
*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(100*Sqrt[10])

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Rubi [A]  time = 0.0618482, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{1}{10} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{33}{100} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{100 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(33*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/100 + ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/10 + (363
*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(100*Sqrt[10])

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Rubi in Sympy [A]  time = 6.16649, size = 63, normalized size = 0.88 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{10} + \frac{33 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{100} + \frac{363 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{1000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(3/2)*sqrt(5*x + 3)/10 + 33*sqrt(-2*x + 1)*sqrt(5*x + 3)/100 + 363*s
qrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/1000

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Mathematica [A]  time = 0.0485101, size = 55, normalized size = 0.76 \[ \frac{10 (43-20 x) \sqrt{1-2 x} \sqrt{5 x+3}-363 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1000} \]

Antiderivative was successfully verified.

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

[Out]

(10*(43 - 20*x)*Sqrt[1 - 2*x]*Sqrt[3 + 5*x] - 363*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqr
t[1 - 2*x]])/1000

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Maple [A]  time = 0.006, size = 72, normalized size = 1. \[{\frac{1}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}\sqrt{3+5\,x}}+{\frac{33}{100}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{363\,\sqrt{10}}{2000}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(1-2*x)^(3/2)*(3+5*x)^(1/2)+33/100*(1-2*x)^(1/2)*(3+5*x)^(1/2)+363/2000*((1
-2*x)*(3+5*x))^(1/2)/(3+5*x)^(1/2)/(1-2*x)^(1/2)*10^(1/2)*arcsin(20/11*x+1/11)

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Maxima [A]  time = 1.49399, size = 55, normalized size = 0.76 \[ -\frac{1}{5} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{363}{2000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{43}{100} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*sqrt(-10*x^2 - x + 3)*x - 363/2000*sqrt(10)*arcsin(-20/11*x - 1/11) + 43/10
0*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.214945, size = 77, normalized size = 1.07 \[ -\frac{1}{2000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (20 \, x - 43\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 363 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2000*sqrt(10)*(2*sqrt(10)*(20*x - 43)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 363*arct
an(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 5.33716, size = 184, normalized size = 2.56 \[ \begin{cases} - \frac{2 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{\sqrt{10 x - 5}} + \frac{77 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{10 \sqrt{10 x - 5}} - \frac{121 i \sqrt{x + \frac{3}{5}}}{20 \sqrt{10 x - 5}} - \frac{363 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{1000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{363 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{1000} + \frac{2 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{\sqrt{- 10 x + 5}} - \frac{77 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{10 \sqrt{- 10 x + 5}} + \frac{121 \sqrt{x + \frac{3}{5}}}{20 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*I*(x + 3/5)**(5/2)/sqrt(10*x - 5) + 77*I*(x + 3/5)**(3/2)/(10*sqrt
(10*x - 5)) - 121*I*sqrt(x + 3/5)/(20*sqrt(10*x - 5)) - 363*sqrt(10)*I*acosh(sqr
t(110)*sqrt(x + 3/5)/11)/1000, 10*Abs(x + 3/5)/11 > 1), (363*sqrt(10)*asin(sqrt(
110)*sqrt(x + 3/5)/11)/1000 + 2*(x + 3/5)**(5/2)/sqrt(-10*x + 5) - 77*(x + 3/5)*
*(3/2)/(10*sqrt(-10*x + 5)) + 121*sqrt(x + 3/5)/(20*sqrt(-10*x + 5)), True))

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GIAC/XCAS [A]  time = 0.237437, size = 116, normalized size = 1.61 \[ -\frac{1}{1000} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{50} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1000*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsi
n(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/50*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*
sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))