3.2313 \(\int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=94 \[ -\frac{1}{6} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{11}{120} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{121}{400} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1331 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]

[Out]

(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400 + (11*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/120 -
((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/6 + (1331*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(40
0*Sqrt[10])

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Rubi [A]  time = 0.084448, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{1}{6} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{11}{120} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{121}{400} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1331 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400 + (11*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/120 -
((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/6 + (1331*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(40
0*Sqrt[10])

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Rubi in Sympy [A]  time = 8.08384, size = 83, normalized size = 0.88 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{15} + \frac{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{100} - \frac{121 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{400} + \frac{1331 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{4000} \]

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)*(5*x + 3)**(3/2)/15 + 11*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/100 -
121*sqrt(-2*x + 1)*sqrt(5*x + 3)/400 + 1331*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3
)/11)/4000

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Mathematica [A]  time = 0.0795919, size = 60, normalized size = 0.64 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (-800 x^2+580 x+273\right )-3993 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{12000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(273 + 580*x - 800*x^2) - 3993*Sqrt[10]*ArcSin[S
qrt[5/11]*Sqrt[1 - 2*x]])/12000

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Maple [A]  time = 0.005, size = 88, normalized size = 0.9 \[{\frac{1}{15} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}}+{\frac{11}{100} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{121}{400}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{1331\,\sqrt{10}}{8000}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(1-2*x)^(3/2)*(3+5*x)^(3/2)+11/100*(3+5*x)^(3/2)*(1-2*x)^(1/2)-121/400*(1-2
*x)^(1/2)*(3+5*x)^(1/2)+1331/8000*((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 = 1.50415, size = 74, normalized size = 0.79 \[ \frac{1}{15} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{11}{20} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1331}{8000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{11}{400} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(-10*x^2 - x + 3)^(3/2) + 11/20*sqrt(-10*x^2 - x + 3)*x - 1331/8000*sqrt(10
)*arcsin(-20/11*x - 1/11) + 11/400*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.216999, size = 84, normalized size = 0.89 \[ -\frac{1}{24000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (800 \, x^{2} - 580 \, x - 273\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 3993 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24000*sqrt(10)*(2*sqrt(10)*(800*x^2 - 580*x - 273)*sqrt(5*x + 3)*sqrt(-2*x +
1) - 3993*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 9.01193, size = 230, normalized size = 2.45 \[ \begin{cases} - \frac{20 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{3 \sqrt{10 x - 5}} + \frac{121 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{6 \sqrt{10 x - 5}} - \frac{2057 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{120 \sqrt{10 x - 5}} + \frac{1331 i \sqrt{x + \frac{3}{5}}}{400 \sqrt{10 x - 5}} - \frac{1331 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{4000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{1331 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{4000} + \frac{20 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{3 \sqrt{- 10 x + 5}} - \frac{121 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{6 \sqrt{- 10 x + 5}} + \frac{2057 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{120 \sqrt{- 10 x + 5}} - \frac{1331 \sqrt{x + \frac{3}{5}}}{400 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

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((-20*I*(x + 3/5)**(7/2)/(3*sqrt(10*x - 5)) + 121*I*(x + 3/5)**(5/2)/(6
*sqrt(10*x - 5)) - 2057*I*(x + 3/5)**(3/2)/(120*sqrt(10*x - 5)) + 1331*I*sqrt(x
+ 3/5)/(400*sqrt(10*x - 5)) - 1331*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/
4000, 10*Abs(x + 3/5)/11 > 1), (1331*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/4
000 + 20*(x + 3/5)**(7/2)/(3*sqrt(-10*x + 5)) - 121*(x + 3/5)**(5/2)/(6*sqrt(-10
*x + 5)) + 2057*(x + 3/5)**(3/2)/(120*sqrt(-10*x + 5)) - 1331*sqrt(x + 3/5)/(400
*sqrt(-10*x + 5)), True))

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GIAC/XCAS [A]  time = 0.224231, size = 135, normalized size = 1.44 \[ -\frac{1}{12000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-10*x + 5)
- 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/400*sqrt(5)*(2*(20*x + 1
)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)
))