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]

-((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 = 0.0226765, 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, 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 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]

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 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 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])

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*}

Mathematica [A]  time = 0.0362176, 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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Maple [B]  time = 0.02, size = 106, normalized size = 2.3 \begin{align*} -{\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}+6\,\arctan \left ({\frac{1+x}{\sqrt{35\,{x}^{2}-2\,x-1}}} \right ) x-2\,\sqrt{35\,{x}^{2}-2\,x-1}x \right ){\frac{1}{\sqrt{ \left ( -1+5\,x \right ) \left ( 1+7\,x \right ) }}}} \end{align*}

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+6*arctan((1+x)/(35*x^2-2*
x-1)^(1/2))*x-2*(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 = 1.51165, size = 72, normalized size = 1.57 \begin{align*} -\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 ) \end{align*}

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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Fricas [A]  time = 1.87202, size = 111, normalized size = 2.41 \begin{align*} \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} \end{align*}

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

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError