3.2939 \(\int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^2} \, dx\)

Optimal. Leaf size=67 \[ -\frac {\sqrt {a+b \sqrt {c x^2}}}{x}-\frac {b \sqrt {c x^2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{\sqrt {a} x} \]

[Out]

-b*arctanh((a+b*(c*x^2)^(1/2))^(1/2)/a^(1/2))*(c*x^2)^(1/2)/x/a^(1/2)-(a+b*(c*x^2)^(1/2))^(1/2)/x

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Rubi [A]  time = 0.03, antiderivative size = 67, 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 = {368, 47, 63, 208} \[ -\frac {\sqrt {a+b \sqrt {c x^2}}}{x}-\frac {b \sqrt {c x^2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{\sqrt {a} x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[c*x^2]]/x^2,x]

[Out]

-(Sqrt[a + b*Sqrt[c*x^2]]/x) - (b*Sqrt[c*x^2]*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]])/(Sqrt[a]*x)

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 368

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*((c*x^q
)^(1/q))^(m + 1)), Subst[Int[x^m*(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q
}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^2} \, dx &=\frac {\sqrt {c x^2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\sqrt {c x^2}\right )}{x}\\ &=-\frac {\sqrt {a+b \sqrt {c x^2}}}{x}+\frac {\left (b \sqrt {c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{2 x}\\ &=-\frac {\sqrt {a+b \sqrt {c x^2}}}{x}+\frac {\sqrt {c x^2} \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sqrt {c x^2}}\right )}{x}\\ &=-\frac {\sqrt {a+b \sqrt {c x^2}}}{x}-\frac {b \sqrt {c x^2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{\sqrt {a} x}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 87, normalized size = 1.30 \[ -\frac {b \sqrt {c x^2} \sqrt {\frac {b \sqrt {c x^2}}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b \sqrt {c x^2}}{a}+1}\right )+a+b \sqrt {c x^2}}{x \sqrt {a+b \sqrt {c x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^2,x]

[Out]

-((a + b*Sqrt[c*x^2] + b*Sqrt[c*x^2]*Sqrt[1 + (b*Sqrt[c*x^2])/a]*ArcTanh[Sqrt[1 + (b*Sqrt[c*x^2])/a]])/(x*Sqrt
[a + b*Sqrt[c*x^2]]))

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fricas [A]  time = 1.11, size = 180, normalized size = 2.69 \[ \left [\frac {b x \sqrt {\frac {c}{a}} \log \left (\frac {b c x^{2} - 2 \, \sqrt {\sqrt {c x^{2}} b + a} a x \sqrt {\frac {c}{a}} + 2 \, \sqrt {c x^{2}} a}{x^{2}}\right ) - 2 \, \sqrt {\sqrt {c x^{2}} b + a}}{2 \, x}, -\frac {b x \sqrt {-\frac {c}{a}} \arctan \left (-\frac {{\left (a b c x^{2} \sqrt {-\frac {c}{a}} - \sqrt {c x^{2}} a^{2} \sqrt {-\frac {c}{a}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{b^{2} c^{2} x^{3} - a^{2} c x}\right ) + \sqrt {\sqrt {c x^{2}} b + a}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^2,x, algorithm="fricas")

[Out]

[1/2*(b*x*sqrt(c/a)*log((b*c*x^2 - 2*sqrt(sqrt(c*x^2)*b + a)*a*x*sqrt(c/a) + 2*sqrt(c*x^2)*a)/x^2) - 2*sqrt(sq
rt(c*x^2)*b + a))/x, -(b*x*sqrt(-c/a)*arctan(-(a*b*c*x^2*sqrt(-c/a) - sqrt(c*x^2)*a^2*sqrt(-c/a))*sqrt(sqrt(c*
x^2)*b + a)/(b^2*c^2*x^3 - a^2*c*x)) + sqrt(sqrt(c*x^2)*b + a))/x]

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giac [A]  time = 0.16, size = 54, normalized size = 0.81 \[ \frac {\frac {b^{2} c \arctan \left (\frac {\sqrt {b \sqrt {c} x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\sqrt {b \sqrt {c} x + a} b \sqrt {c}}{x}}{b \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^2,x, algorithm="giac")

[Out]

(b^2*c*arctan(sqrt(b*sqrt(c)*x + a)/sqrt(-a))/sqrt(-a) - sqrt(b*sqrt(c)*x + a)*b*sqrt(c)/x)/(b*sqrt(c))

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maple [A]  time = 0.01, size = 54, normalized size = 0.81 \[ -\frac {\sqrt {c \,x^{2}}\, b \arctanh \left (\frac {\sqrt {a +\sqrt {c \,x^{2}}\, b}}{\sqrt {a}}\right )+\sqrt {a +\sqrt {c \,x^{2}}\, b}\, \sqrt {a}}{\sqrt {a}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+(c*x^2)^(1/2)*b)^(1/2)/x^2,x)

[Out]

-(arctanh((a+(c*x^2)^(1/2)*b)^(1/2)/a^(1/2))*b*(c*x^2)^(1/2)+(a+(c*x^2)^(1/2)*b)^(1/2)*a^(1/2))/x/a^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\sqrt {c x^{2}} b + a}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^2)^(1/2))^(1/2)/x^2,x, algorithm="maxima")

[Out]

integrate(sqrt(sqrt(c*x^2)*b + a)/x^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+b\,\sqrt {c\,x^2}}}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*(c*x^2)^(1/2))^(1/2)/x^2,x)

[Out]

int((a + b*(c*x^2)^(1/2))^(1/2)/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b \sqrt {c x^{2}}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**2,x)

[Out]

Integral(sqrt(a + b*sqrt(c*x**2))/x**2, x)

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