3.2412 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x) \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{1}{15} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{239}{450} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{17687 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1350 \sqrt{10}}-\frac{98}{27} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-239*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/450 - ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/15 - (1
7687*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1350*Sqrt[10]) - (98*Sqrt[7]*ArcTan[Sqrt
[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/27

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Rubi [A]  time = 0.235767, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{1}{15} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{239}{450} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{17687 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1350 \sqrt{10}}-\frac{98}{27} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-239*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/450 - ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/15 - (1
7687*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1350*Sqrt[10]) - (98*Sqrt[7]*ArcTan[Sqrt
[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/27

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Rubi in Sympy [A]  time = 22.7331, size = 99, normalized size = 0.93 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{15} - \frac{239 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{450} - \frac{17687 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{13500} - \frac{98 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)*sqrt(5*x + 3)/15 - 239*sqrt(-2*x + 1)*sqrt(5*x + 3)/450 - 176
87*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/13500 - 98*sqrt(7)*atan(sqrt(7)*sqrt
(-2*x + 1)/(7*sqrt(5*x + 3)))/27

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Mathematica [A]  time = 0.176451, size = 100, normalized size = 0.94 \[ \frac{60 \sqrt{1-2 x} \sqrt{5 x+3} (60 x-269)-49000 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-17687 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{27000} \]

Antiderivative was successfully verified.

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

[Out]

(60*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-269 + 60*x) - 49000*Sqrt[7]*ArcTan[(-20 - 37*x
)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] - 17687*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1
- 2*x]*Sqrt[30 + 50*x])])/27000

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Maple [A]  time = 0.017, size = 98, normalized size = 0.9 \[{\frac{1}{27000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 49000\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -17687\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +3600\,x\sqrt{-10\,{x}^{2}-x+3}-16140\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/27000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(49000*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))-17687*10^(1/2)*arcsin(20/11*x+1/11)+3600*x*(-10*x^2-x+3)^(
1/2)-16140*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.52743, size = 93, normalized size = 0.88 \[ \frac{2}{15} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{17687}{27000} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{49}{27} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{269}{450} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*sqrt(-10*x^2 - x + 3)*x - 17687/27000*sqrt(10)*arcsin(20/11*x + 1/11) + 49/
27*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 269/450*sqrt(-10*
x^2 - x + 3)

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Fricas [A]  time = 0.239316, size = 122, normalized size = 1.15 \[ \frac{1}{27000} \, \sqrt{10}{\left (6 \, \sqrt{10}{\left (60 \, x - 269\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 4900 \, \sqrt{10} \sqrt{7} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 17687 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/27000*sqrt(10)*(6*sqrt(10)*(60*x - 269)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 4900*sq
rt(10)*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) -
17687*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((-2*x + 1)**(5/2)/((3*x + 2)*sqrt(5*x + 3)), x)

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GIAC/XCAS [A]  time = 0.282723, size = 234, normalized size = 2.21 \[ \frac{49}{270} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{2250} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 305 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{17687}{27000} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

49/270*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*
sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)
))) + 1/2250*(12*sqrt(5)*(5*x + 3) - 305*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)
- 17687/27000*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x +
5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))