3.591 \(\int \frac{\sqrt{\frac{-1+5 x}{1+7 x}}}{x^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{\sqrt{5 x-1} \sqrt{7 x+1}}{x}-12 \tan ^{-1}\left (\frac{\sqrt{7 x+1}}{\sqrt{5 x-1}}\right ) \]

[Out]

-((Sqrt[-1 + 5*x]*Sqrt[1 + 7*x])/x) - 12*ArcTan[Sqrt[1 + 7*x]/Sqrt[-1 + 5*x]]

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Rubi [A]  time = 0.0789203, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{\sqrt{5 x-1} \sqrt{7 x+1}}{x}-12 \tan ^{-1}\left (\frac{\sqrt{7 x+1}}{\sqrt{5 x-1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(-1 + 5*x)/(1 + 7*x)]/x^2,x]

[Out]

-((Sqrt[-1 + 5*x]*Sqrt[1 + 7*x])/x) - 12*ArcTan[Sqrt[1 + 7*x]/Sqrt[-1 + 5*x]]

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Rubi in Sympy [A]  time = 3.25569, size = 39, normalized size = 0.85 \[ - 12 \operatorname{atan}{\left (\frac{\sqrt{7 x + 1}}{\sqrt{5 x - 1}} \right )} - \frac{\sqrt{5 x - 1} \sqrt{7 x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((-1+5*x)/(1+7*x))**(1/2)/x**2,x)

[Out]

-12*atan(sqrt(7*x + 1)/sqrt(5*x - 1)) - sqrt(5*x - 1)*sqrt(7*x + 1)/x

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Mathematica [A]  time = 0.0635688, size = 82, normalized size = 1.78 \[ -\frac{\sqrt{\frac{5 x-1}{7 x+1}} \left (\sqrt{5 x-1} (7 x+1)+6 x \sqrt{7 x+1} \tan ^{-1}\left (\frac{x+1}{\sqrt{5 x-1} \sqrt{7 x+1}}\right )\right )}{x \sqrt{5 x-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[(-1 + 5*x)/(1 + 7*x)]/x^2,x]

[Out]

-((Sqrt[(-1 + 5*x)/(1 + 7*x)]*(Sqrt[-1 + 5*x]*(1 + 7*x) + 6*x*Sqrt[1 + 7*x]*ArcT
an[(1 + x)/(Sqrt[-1 + 5*x]*Sqrt[1 + 7*x])]))/(x*Sqrt[-1 + 5*x]))

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Maple [B]  time = 0.031, size = 103, normalized size = 2.2 \[{\frac{1+7\,x}{x}\sqrt{{\frac{-1+5\,x}{1+7\,x}}} \left ( \left ( 35\,{x}^{2}-2\,x-1 \right ) ^{{\frac{3}{2}}}-35\,\sqrt{35\,{x}^{2}-2\,x-1}{x}^{2}+2\,\sqrt{35\,{x}^{2}-2\,x-1}x-6\,\arctan \left ({\frac{1+x}{\sqrt{35\,{x}^{2}-2\,x-1}}} \right ) x \right ){\frac{1}{\sqrt{ \left ( -1+5\,x \right ) \left ( 1+7\,x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((-1+5*x)/(1+7*x))^(1/2)/x^2,x)

[Out]

((-1+5*x)/(1+7*x))^(1/2)*(1+7*x)*((35*x^2-2*x-1)^(3/2)-35*(35*x^2-2*x-1)^(1/2)*x
^2+2*(35*x^2-2*x-1)^(1/2)*x-6*arctan((1+x)/(35*x^2-2*x-1)^(1/2))*x)/((-1+5*x)*(1
+7*x))^(1/2)/x

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Maxima [A]  time = 0.801891, size = 72, normalized size = 1.57 \[ -\frac{12 \, \sqrt{\frac{5 \, x - 1}{7 \, x + 1}}}{\frac{5 \, x - 1}{7 \, x + 1} + 1} + 12 \, \arctan \left (\sqrt{\frac{5 \, x - 1}{7 \, x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((5*x - 1)/(7*x + 1))/x^2,x, algorithm="maxima")

[Out]

-12*sqrt((5*x - 1)/(7*x + 1))/((5*x - 1)/(7*x + 1) + 1) + 12*arctan(sqrt((5*x -
1)/(7*x + 1)))

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Fricas [A]  time = 0.278061, size = 62, normalized size = 1.35 \[ \frac{12 \, x \arctan \left (\sqrt{\frac{5 \, x - 1}{7 \, x + 1}}\right ) -{\left (7 \, x + 1\right )} \sqrt{\frac{5 \, x - 1}{7 \, x + 1}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((5*x - 1)/(7*x + 1))/x^2,x, algorithm="fricas")

[Out]

(12*x*arctan(sqrt((5*x - 1)/(7*x + 1))) - (7*x + 1)*sqrt((5*x - 1)/(7*x + 1)))/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{5 x - 1}{7 x + 1}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((-1+5*x)/(1+7*x))**(1/2)/x**2,x)

[Out]

Integral(sqrt((5*x - 1)/(7*x + 1))/x**2, x)

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GIAC/XCAS [A]  time = 0.277704, size = 154, normalized size = 3.35 \[{\left (\sqrt{35} - 12 \, \arctan \left (\frac{1}{7} \, \sqrt{35}\right )\right )}{\rm sign}\left (7 \, x + 1\right ) + 12 \, \arctan \left (-\sqrt{35} x + \sqrt{35 \, x^{2} - 2 \, x - 1}\right ){\rm sign}\left (7 \, x + 1\right ) - \frac{2 \,{\left ({\left (\sqrt{35} x - \sqrt{35 \, x^{2} - 2 \, x - 1}\right )}{\rm sign}\left (7 \, x + 1\right ) + \sqrt{35}{\rm sign}\left (7 \, x + 1\right )\right )}}{{\left (\sqrt{35} x - \sqrt{35 \, x^{2} - 2 \, x - 1}\right )}^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((5*x - 1)/(7*x + 1))/x^2,x, algorithm="giac")

[Out]

(sqrt(35) - 12*arctan(1/7*sqrt(35)))*sign(7*x + 1) + 12*arctan(-sqrt(35)*x + sqr
t(35*x^2 - 2*x - 1))*sign(7*x + 1) - 2*((sqrt(35)*x - sqrt(35*x^2 - 2*x - 1))*si
gn(7*x + 1) + sqrt(35)*sign(7*x + 1))/((sqrt(35)*x - sqrt(35*x^2 - 2*x - 1))^2 +
1)