3.2306 \(\int \frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=75 \[ -\frac{10 (1-2 x)^{3/2}}{33 (5 x+3)^{3/2}}+\frac{6 \sqrt{1-2 x}}{\sqrt{5 x+3}}-6 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-10*(1 - 2*x)^(3/2))/(33*(3 + 5*x)^(3/2)) + (6*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] - 6
*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]

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Rubi [A]  time = 0.121871, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{10 (1-2 x)^{3/2}}{33 (5 x+3)^{3/2}}+\frac{6 \sqrt{1-2 x}}{\sqrt{5 x+3}}-6 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-10*(1 - 2*x)^(3/2))/(33*(3 + 5*x)^(3/2)) + (6*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] - 6
*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]

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Rubi in Sympy [A]  time = 10.0923, size = 70, normalized size = 0.93 \[ - \frac{10 \left (- 2 x + 1\right )^{\frac{3}{2}}}{33 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{6 \sqrt{- 2 x + 1}}{\sqrt{5 x + 3}} - 6 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*(-2*x + 1)**(3/2)/(33*(5*x + 3)**(3/2)) + 6*sqrt(-2*x + 1)/sqrt(5*x + 3) - 6
*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))

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Mathematica [A]  time = 0.110884, size = 63, normalized size = 0.84 \[ \frac{2 \sqrt{1-2 x} (505 x+292)}{33 (5 x+3)^{3/2}}-3 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*(292 + 505*x))/(33*(3 + 5*x)^(3/2)) - 3*Sqrt[7]*ArcTan[(-20 - 3
7*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])]

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Maple [B]  time = 0.02, size = 147, normalized size = 2. \[{\frac{1}{33} \left ( 2475\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+2970\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+891\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1010\,x\sqrt{-10\,{x}^{2}-x+3}+584\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/33*(2475*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+2970*7
^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+891*7^(1/2)*arctan(1
/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1010*x*(-10*x^2-x+3)^(1/2)+584*(-10*x
^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.51638, size = 117, normalized size = 1.56 \[ 3 \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{404 \, x}{33 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1054}{165 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{44 \, x}{15 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{22}{15 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 404/33*x/sqrt(-10*
x^2 - x + 3) + 1054/165/sqrt(-10*x^2 - x + 3) + 44/15*x/(-10*x^2 - x + 3)^(3/2)
- 22/15/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.219398, size = 103, normalized size = 1.37 \[ \frac{99 \, \sqrt{7}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 2 \,{\left (505 \, x + 292\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{33 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/33*(99*sqrt(7)*(25*x^2 + 30*x + 9)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x +
3)*sqrt(-2*x + 1))) + 2*(505*x + 292)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 3
0*x + 9)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.254334, size = 267, normalized size = 3.56 \[ -\frac{1}{2640} \, \sqrt{5}{\left (\sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 792 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - 792 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2640*sqrt(5)*(sqrt(2)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4
*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 792*sqrt(70)*sqrt(2)*(p
i + 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)))) - 792*sqrt(2)*((sqrt(2)
*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))))