3.2366 \(\int \frac{(1-2 x)^{3/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{2 (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac{4 \sqrt{1-2 x}}{25 \sqrt{5 x+3}}+\frac{4}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-2*(1 - 2*x)^(3/2))/(15*(3 + 5*x)^(3/2)) + (4*Sqrt[1 - 2*x])/(25*Sqrt[3 + 5*x])
+ (4*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/25

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Rubi [A]  time = 0.0627282, antiderivative size = 74, 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 \[ -\frac{2 (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac{4 \sqrt{1-2 x}}{25 \sqrt{5 x+3}}+\frac{4}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2))/(15*(3 + 5*x)^(3/2)) + (4*Sqrt[1 - 2*x])/(25*Sqrt[3 + 5*x])
+ (4*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/25

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Rubi in Sympy [A]  time = 6.91068, size = 65, normalized size = 0.88 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{4 \sqrt{- 2 x + 1}}{25 \sqrt{5 x + 3}} + \frac{4 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)/(15*(5*x + 3)**(3/2)) + 4*sqrt(-2*x + 1)/(25*sqrt(5*x + 3))
+ 4*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/125

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Mathematica [A]  time = 0.130437, size = 55, normalized size = 0.74 \[ \frac{2}{375} \left (\frac{5 \sqrt{1-2 x} (40 x+13)}{(5 x+3)^{3/2}}-6 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((5*Sqrt[1 - 2*x]*(13 + 40*x))/(3 + 5*x)^(3/2) - 6*Sqrt[10]*ArcSin[Sqrt[5/11]
*Sqrt[1 - 2*x]]))/375

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{1 \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.48783, size = 126, normalized size = 1.7 \[ \frac{2}{125} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{15 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} - \frac{11 \, \sqrt{-10 \, x^{2} - x + 3}}{75 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{14 \, \sqrt{-10 \, x^{2} - x + 3}}{75 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/125*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/15*(-10*x^2 - x + 3)^(3/2)/(125
*x^3 + 225*x^2 + 135*x + 27) - 11/75*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) +
14/75*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.218221, size = 113, normalized size = 1.53 \[ \frac{2 \, \sqrt{5}{\left (\sqrt{5}{\left (40 \, x + 13\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3 \, \sqrt{2}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{375 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/375*sqrt(5)*(sqrt(5)*(40*x + 13)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 3*sqrt(2)*(25*
x^2 + 30*x + 9)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x
+ 1))))/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 9.87256, size = 206, normalized size = 2.78 \[ \begin{cases} \frac{16 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{375} - \frac{22 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1875 \left (x + \frac{3}{5}\right )} + \frac{2 \sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{125} + \frac{2 \sqrt{10} i \log{\left (x + \frac{3}{5} \right )}}{125} + \frac{4 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{125} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\\frac{16 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{375} - \frac{22 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1875 \left (x + \frac{3}{5}\right )} + \frac{2 \sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{125} - \frac{4 \sqrt{10} i \log{\left (\sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} + 1 \right )}}{125} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((16*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/375 - 22*sqrt(10)*sqrt(-1 +
11/(10*(x + 3/5)))/(1875*(x + 3/5)) + 2*sqrt(10)*I*log(1/(x + 3/5))/125 + 2*sqrt
(10)*I*log(x + 3/5)/125 + 4*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/125, 11*Ab
s(1/(x + 3/5))/10 > 1), (16*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/375 - 22*sqrt
(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/(1875*(x + 3/5)) + 2*sqrt(10)*I*log(1/(x + 3/
5))/125 - 4*sqrt(10)*I*log(sqrt(1 - 11/(10*(x + 3/5))) + 1)/125, True))

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GIAC/XCAS [A]  time = 0.259711, size = 194, normalized size = 2.62 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{6000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{4}{125} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{100 \, \sqrt{5 \, x + 3}} - \frac{{\left (\frac{15 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 4/125*
sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/100*sqrt(10)*(sqrt(2)*sqrt(-10*
x + 5) - sqrt(22))/sqrt(5*x + 3) - 1/375*(15*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))^2/(5*x + 3) - 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))^3