3.8.47 \(\int \frac {\sqrt {\frac {-1+5 x}{1+7 x}}}{x^2} \, dx\) [747]

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

[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 = {1978, 96, 95, 210} \begin {gather*} -12 \text {ArcTan}\left (\frac {\sqrt {7 x+1}}{\sqrt {5 x-1}}\right )-\frac {\sqrt {5 x-1} \sqrt {7 x+1}}{x} \end {gather*}

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 95

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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 210

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

Rule 1978

Int[(u_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[u*((a*e + b*e*
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 \text {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 = 0.17, size = 88, normalized size = 1.91 \begin {gather*} -\frac {\sqrt {\frac {-1+5 x}{1+7 x}} \left (\sqrt {-1+5 x} (1+7 x)+12 x \sqrt {1+7 x} \tan ^{-1}\left (\sqrt {35} x-\sqrt {-1+5 x} \sqrt {1+7 x}\right )\right )}{x \sqrt {-1+5 x}} \end {gather*}

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(105\) vs. \(2(38)=76\).
time = 0.13, size = 106, normalized size = 2.30

method	result	size
risch	\(-\frac {\left (1+7 x \right ) \sqrt {\frac {5 x -1}{1+7 x}}}{x}+\frac {6 \arctan \left (\frac {-2 x -2}{2 \sqrt {35 x^{2}-2 x -1}}\right ) \sqrt {\frac {5 x -1}{1+7 x}}\, \sqrt {\left (5 x -1\right ) \left (1+7 x \right )}}{5 x -1}\)	\(84\)
trager	\(-\frac {\left (1+7 x \right ) \sqrt {-\frac {1-5 x}{1+7 x}}}{x}+6 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {x \RootOf \left (\textit {\_Z}^{2}+1\right )+7 \sqrt {-\frac {1-5 x}{1+7 x}}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-\frac {1-5 x}{1+7 x}}}{x}\right )\)	\(91\)
default	\(-\frac {\sqrt {\frac {5 x -1}{1+7 x}}\, \left (1+7 x \right ) \left (-\left (35 x^{2}-2 x -1\right )^{\frac {3}{2}}+35 \sqrt {35 x^{2}-2 x -1}\, x^{2}+6 \arctan \left (\frac {1+x}{\sqrt {35 x^{2}-2 x -1}}\right ) x -2 \sqrt {35 x^{2}-2 x -1}\, x \right )}{\sqrt {\left (5 x -1\right ) \left (1+7 x \right )}\, x}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.47, size = 53, normalized size = 1.15 \begin {gather*} -\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 {gather*}

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 = 0.34, size = 46, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\frac {5 x - 1}{7 x + 1}}}{x^{2}}\, dx \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (38) = 76\).
time = 5.30, size = 114, normalized size = 2.48 \begin {gather*} {\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} \end {gather*}

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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Mupad [B]
time = 3.24, size = 74, normalized size = 1.61 \begin {gather*} 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 )} \end {gather*}

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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