3.2486 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=55 \[ \frac{6 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}}-\frac{10 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]

[Out]

(-10*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x]) + (6*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3
+ 5*x])])/Sqrt[7]

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Rubi [A]  time = 0.0879988, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{6 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}}-\frac{10 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x]) + (6*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3
+ 5*x])])/Sqrt[7]

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Rubi in Sympy [A]  time = 7.03536, size = 53, normalized size = 0.96 \[ - \frac{10 \sqrt{- 2 x + 1}}{11 \sqrt{5 x + 3}} + \frac{6 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*sqrt(-2*x + 1)/(11*sqrt(5*x + 3)) + 6*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7
*sqrt(5*x + 3)))/7

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Mathematica [A]  time = 0.0695873, size = 58, normalized size = 1.05 \[ \frac{3 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{\sqrt{7}}-\frac{10 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x]) + (3*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x
]*Sqrt[3 + 5*x])])/Sqrt[7]

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Maple [B]  time = 0.02, size = 101, normalized size = 1.8 \[ -{\frac{1}{77} \left ( 165\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+99\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +70\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/77*(165*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+99*7^(1/
2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+70*(-10*x^2-x+3)^(1/2))*(1
-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23723, size = 86, normalized size = 1.56 \[ -\frac{\sqrt{7}{\left (33 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 10 \, \sqrt{7} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{77 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/77*sqrt(7)*(33*(5*x + 3)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(
-2*x + 1))) + 10*sqrt(7)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.236446, size = 180, normalized size = 3.27 \[ -\frac{3}{70} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1}{22} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/70*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
)) - 1/22*sqrt(10)*((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)))