3.2478 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=64 \[ \frac{3 \sqrt{1-2 x} \sqrt{5 x+3}}{7 (3 x+2)}-\frac{37 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

[Out]

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

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Rubi [A]  time = 0.0887994, antiderivative size = 64, 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 \[ \frac{3 \sqrt{1-2 x} \sqrt{5 x+3}}{7 (3 x+2)}-\frac{37 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.39742, size = 56, normalized size = 0.88 \[ \frac{3 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{7 \left (3 x + 2\right )} - \frac{37 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{49} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(-2*x + 1)*sqrt(5*x + 3)/(7*(3*x + 2)) - 37*sqrt(7)*atan(sqrt(7)*sqrt(-2*x
+ 1)/(7*sqrt(5*x + 3)))/49

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Mathematica [A]  time = 0.0851971, size = 67, normalized size = 1.05 \[ \frac{3 \sqrt{1-2 x} \sqrt{5 x+3}}{7 (3 x+2)}-\frac{37 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{14 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 108, normalized size = 1.7 \[{\frac{1}{196+294\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 111\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+74\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +42\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/98*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(111*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10
*x^2-x+3)^(1/2))*x+74*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
+42*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 1.50693, size = 68, normalized size = 1.06 \[ \frac{37}{98} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{3 \, \sqrt{-10 \, x^{2} - x + 3}}{7 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

37/98*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 3/7*sqrt(-10*x
^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.222685, size = 86, normalized size = 1.34 \[ \frac{\sqrt{7}{\left (37 \,{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{7} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{98 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/98*sqrt(7)*(37*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-
2*x + 1))) + 6*sqrt(7)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.248944, size = 261, normalized size = 4.08 \[ \frac{37}{980} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{66 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{7 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

37/980*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)
))) + 66/7*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)))/(((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 +
280)