3.2348 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x) \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=86 \[ -\frac{2}{15} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{103}{45} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{14}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/15 - (103*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/45 - (14*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9

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Rubi [A]  time = 0.171327, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2}{15} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{103}{45} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{14}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/15 - (103*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/45 - (14*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9

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Rubi in Sympy [A]  time = 15.194, size = 80, normalized size = 0.93 \[ - \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{15} - \frac{103 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{225} - \frac{14 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*sqrt(5*x + 3)/15 - 103*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11
)/225 - 14*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/9

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Mathematica [A]  time = 0.117096, size = 95, normalized size = 1.1 \[ \frac{1}{450} \left (-60 \sqrt{1-2 x} \sqrt{5 x+3}-350 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-103 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-60*Sqrt[1 - 2*x]*Sqrt[3 + 5*x] - 350*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 1
4*x]*Sqrt[3 + 5*x])] - 103*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 +
50*x])])/450

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Maple [A]  time = 0.017, size = 83, normalized size = 1. \[{\frac{1}{450}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 350\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -103\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -60\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/450*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(350*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))-103*10^(1/2)*arcsin(20/11*x+1/11)-60*(-10*x^2-x+3)^(1/2))/(-10
*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50517, size = 73, normalized size = 0.85 \[ -\frac{103}{450} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{7}{9} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2}{15} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-103/450*sqrt(10)*arcsin(20/11*x + 1/11) + 7/9*sqrt(7)*arcsin(37/11*x/abs(3*x +
2) + 20/11/abs(3*x + 2)) - 2/15*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.238913, size = 123, normalized size = 1.43 \[ \frac{1}{450} \, \sqrt{5}{\left (70 \, \sqrt{7} \sqrt{5} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 12 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 103 \, \sqrt{2} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/450*sqrt(5)*(70*sqrt(7)*sqrt(5)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)
*sqrt(-2*x + 1))) - 12*sqrt(5)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 103*sqrt(2)*arctan
(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.259391, size = 216, normalized size = 2.51 \[ \frac{7}{90} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{103}{450} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{2}{75} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/90*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))
) - 103/450*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)))) - 2/75*sqrt
(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)