3.2411 \(\int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=94 \[ \frac{1}{15} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{11}{60} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{121}{200} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1331 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

[Out]

(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/200 + (11*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/60 +
((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/15 + (1331*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(20
0*Sqrt[10])

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Rubi [A]  time = 0.0847475, 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 \[ \frac{1}{15} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{11}{60} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{121}{200} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1331 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/200 + (11*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/60 +
((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/15 + (1331*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(20
0*Sqrt[10])

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Rubi in Sympy [A]  time = 7.92392, size = 83, normalized size = 0.88 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{15} + \frac{11 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{60} + \frac{121 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{200} + \frac{1331 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(5/2)*sqrt(5*x + 3)/15 + 11*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/60 + 121
*sqrt(-2*x + 1)*sqrt(5*x + 3)/200 + 1331*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11
)/2000

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Mathematica [A]  time = 0.0560981, size = 60, normalized size = 0.64 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (160 x^2-380 x+513\right )-3993 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(513 - 380*x + 160*x^2) - 3993*Sqrt[10]*ArcSin[S
qrt[5/11]*Sqrt[1 - 2*x]])/6000

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Maple [A]  time = 0.006, size = 88, normalized size = 0.9 \[{\frac{1}{15} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}\sqrt{3+5\,x}}+{\frac{11}{60} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}\sqrt{3+5\,x}}+{\frac{121}{200}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{1331\,\sqrt{10}}{4000}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(1-2*x)^(5/2)*(3+5*x)^(1/2)+11/60*(1-2*x)^(3/2)*(3+5*x)^(1/2)+121/200*(1-2*
x)^(1/2)*(3+5*x)^(1/2)+1331/4000*((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 = 1.51504, size = 78, normalized size = 0.83 \[ \frac{4}{15} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{19}{30} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1331}{4000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{171}{200} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/15*sqrt(-10*x^2 - x + 3)*x^2 - 19/30*sqrt(-10*x^2 - x + 3)*x - 1331/4000*sqrt(
10)*arcsin(-20/11*x - 1/11) + 171/200*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.224608, size = 84, normalized size = 0.89 \[ \frac{1}{12000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (160 \, x^{2} - 380 \, x + 513\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3993 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12000*sqrt(10)*(2*sqrt(10)*(160*x^2 - 380*x + 513)*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 = 22.3242, size = 230, normalized size = 2.45 \[ \begin{cases} \frac{8 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{3 \sqrt{10 x - 5}} - \frac{187 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{15 \sqrt{10 x - 5}} + \frac{7139 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{300 \sqrt{10 x - 5}} - \frac{14641 i \sqrt{x + \frac{3}{5}}}{1000 \sqrt{10 x - 5}} - \frac{1331 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{2000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{1331 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{2000} - \frac{8 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{3 \sqrt{- 10 x + 5}} + \frac{187 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{15 \sqrt{- 10 x + 5}} - \frac{7139 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{300 \sqrt{- 10 x + 5}} + \frac{14641 \sqrt{x + \frac{3}{5}}}{1000 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((8*I*(x + 3/5)**(7/2)/(3*sqrt(10*x - 5)) - 187*I*(x + 3/5)**(5/2)/(15*
sqrt(10*x - 5)) + 7139*I*(x + 3/5)**(3/2)/(300*sqrt(10*x - 5)) - 14641*I*sqrt(x
+ 3/5)/(1000*sqrt(10*x - 5)) - 1331*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)
/2000, 10*Abs(x + 3/5)/11 > 1), (1331*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/
2000 - 8*(x + 3/5)**(7/2)/(3*sqrt(-10*x + 5)) + 187*(x + 3/5)**(5/2)/(15*sqrt(-1
0*x + 5)) - 7139*(x + 3/5)**(3/2)/(300*sqrt(-10*x + 5)) + 14641*sqrt(x + 3/5)/(1
000*sqrt(-10*x + 5)), True))

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GIAC/XCAS [A]  time = 0.240128, size = 189, normalized size = 2.01 \[ \frac{1}{30000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 59\right )}{\left (5 \, x + 3\right )} + 1293\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 4785 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{500} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{50} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5
) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1/500*sqrt(5)*(2*(20*x -
23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(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))