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

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]

-12*arctan((1+7*x)^(1/2)/(-1+5*x)^(1/2))-(-1+5*x)^(1/2)*(1+7*x)^(1/2)/x

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Rubi [A]  time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1958, 94, 93, 204} \[ -\frac {\sqrt {5 x-1} \sqrt {7 x+1}}{x}-12 \tan ^{-1}\left (\frac {\sqrt {7 x+1}}{\sqrt {5 x-1}}\right ) \]

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]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1958

Int[(u_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[(u*(e*(a + b*x
^n))^p)/(c + d*x^n)^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[b*d*e, 0] && GtQ[c - (a*d)/b, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {\frac {-1+5 x}{1+7 x}}}{x^2} \, dx &=\int \frac {\sqrt {-1+5 x}}{x^2 \sqrt {1+7 x}} \, dx\\ &=-\frac {\sqrt {-1+5 x} \sqrt {1+7 x}}{x}+6 \int \frac {1}{x \sqrt {-1+5 x} \sqrt {1+7 x}} \, dx\\ &=-\frac {\sqrt {-1+5 x} \sqrt {1+7 x}}{x}+12 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\frac {\sqrt {1+7 x}}{\sqrt {-1+5 x}}\right )\\ &=-\frac {\sqrt {-1+5 x} \sqrt {1+7 x}}{x}-12 \tan ^{-1}\left (\frac {\sqrt {1+7 x}}{\sqrt {-1+5 x}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 79, normalized size = 1.72 \[ \frac {\sqrt {\frac {5 x-1}{7 x+1}} \left (12 x \sqrt {7 x+1} \tan ^{-1}\left (\frac {\sqrt {5 x-1}}{\sqrt {7 x+1}}\right )-\sqrt {5 x-1} (7 x+1)\right )}{x \sqrt {5 x-1}} \]

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

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fricas [A]  time = 0.40, size = 46, normalized size = 1.00 \[ \frac {12 \, x \arctan \left (\sqrt {\frac {5 \, x - 1}{7 \, x + 1}}\right ) - {\left (7 \, x + 1\right )} \sqrt {\frac {5 \, x - 1}{7 \, x + 1}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1+5*x)/(1+7*x))^(1/2)/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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giac [B]  time = 0.40, size = 114, normalized size = 2.48 \[ {\left (\sqrt {35} - 12 \, \arctan \left (\frac {1}{7} \, \sqrt {35}\right )\right )} \mathrm {sgn}\left (7 \, x + 1\right ) + 12 \, \arctan \left (-\sqrt {35} x + \sqrt {35 \, x^{2} - 2 \, x - 1}\right ) \mathrm {sgn}\left (7 \, x + 1\right ) - \frac {2 \, {\left ({\left (\sqrt {35} x - \sqrt {35 \, x^{2} - 2 \, x - 1}\right )} \mathrm {sgn}\left (7 \, x + 1\right ) + \sqrt {35} \mathrm {sgn}\left (7 \, x + 1\right )\right )}}{{\left (\sqrt {35} x - \sqrt {35 \, x^{2} - 2 \, x - 1}\right )}^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(35) - 12*arctan(1/7*sqrt(35)))*sgn(7*x + 1) + 12*arctan(-sqrt(35)*x + sqrt(35*x^2 - 2*x - 1))*sgn(7*x +
1) - 2*((sqrt(35)*x - sqrt(35*x^2 - 2*x - 1))*sgn(7*x + 1) + sqrt(35)*sgn(7*x + 1))/((sqrt(35)*x - sqrt(35*x^2
- 2*x - 1))^2 + 1)

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maple [B]  time = 0.02, size = 103, normalized size = 2.24 \[ \frac {\sqrt {\frac {5 x -1}{7 x +1}}\, \left (7 x +1\right ) \left (-35 \sqrt {35 x^{2}-2 x -1}\, x^{2}-6 x \arctan \left (\frac {x +1}{\sqrt {35 x^{2}-2 x -1}}\right )+2 \sqrt {35 x^{2}-2 x -1}\, x +\left (35 x^{2}-2 x -1\right )^{\frac {3}{2}}\right )}{\sqrt {\left (5 x -1\right ) \left (7 x +1\right )}\, x} \]

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((x+1)/(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 = 1.43, size = 53, normalized size = 1.15 \[ -\frac {12 \, \sqrt {\frac {5 \, x - 1}{7 \, x + 1}}}{\frac {5 \, x - 1}{7 \, x + 1} + 1} + 12 \, \arctan \left (\sqrt {\frac {5 \, x - 1}{7 \, x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1+5*x)/(1+7*x))^(1/2)/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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mupad [B]  time = 3.24, size = 74, normalized size = 1.61 \[ 12\,\mathrm {atan}\left (\frac {\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,\sqrt {\frac {5\,x-1}{7\,x+1}}}{35}\right )-\frac {12\,\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,\sqrt {\frac {5\,x-1}{7\,x+1}}}{25\,\left (\frac {7\,x-\frac {7}{5}}{7\,x+1}+\frac {7}{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12*atan((5^(1/2)*7^(1/2)*35^(1/2)*((5*x - 1)/(7*x + 1))^(1/2))/35) - (12*5^(1/2)*7^(1/2)*35^(1/2)*((5*x - 1)/(
7*x + 1))^(1/2))/(25*((7*x - 7/5)/(7*x + 1) + 7/5))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {5 x - 1}{7 x + 1}}}{x^{2}}\, dx \]

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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