3.2539 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=57 \[ \frac{4 \sqrt{5 x+3}}{77 \sqrt{1-2 x}}-\frac{6 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

[Out]

(4*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]) - (6*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(7*Sqrt[7])

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Rubi [A]  time = 0.0889073, antiderivative size = 57, 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 \[ \frac{4 \sqrt{5 x+3}}{77 \sqrt{1-2 x}}-\frac{6 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]) - (6*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(7*Sqrt[7])

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Rubi in Sympy [A]  time = 7.18883, size = 53, normalized size = 0.93 \[ - \frac{6 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{49} + \frac{4 \sqrt{5 x + 3}}{77 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/49 + 4*sqrt(5*x + 3)/(
77*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.111026, size = 60, normalized size = 1.05 \[ \frac{4 \sqrt{5 x+3}}{77 \sqrt{1-2 x}}-\frac{3 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]) - (3*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*
Sqrt[3 + 5*x])])/(7*Sqrt[7])

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Maple [B]  time = 0.022, size = 108, normalized size = 1.9 \[{\frac{1}{-539+1078\,x} \left ( 66\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-33\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -28\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/539*(66*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-33*7^(1/2
)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-28*(-10*x^2-x+3)^(1/2))*(3+
5*x)^(1/2)*(1-2*x)^(1/2)/(-1+2*x)/(-10*x^2-x+3)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22697, size = 86, normalized size = 1.51 \[ \frac{\sqrt{7}{\left (33 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 4 \, \sqrt{7} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{539 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/539*sqrt(7)*(33*(2*x - 1)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(
-2*x + 1))) - 4*sqrt(7)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.24892, size = 135, normalized size = 2.37 \[ \frac{3}{490} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{4 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{385 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/490*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))
)) - 4/385*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)