3.2237 \(\int \frac {\sqrt {a+b \sqrt {x}}}{x^2} \, dx\)

Optimal. Leaf size=77 \[ \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {b \sqrt {a+b \sqrt {x}}}{2 a \sqrt {x}}-\frac {\sqrt {a+b \sqrt {x}}}{x} \]

[Out]

1/2*b^2*arctanh((a+b*x^(1/2))^(1/2)/a^(1/2))/a^(3/2)-(a+b*x^(1/2))^(1/2)/x-1/2*b*(a+b*x^(1/2))^(1/2)/a/x^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {266, 47, 51, 63, 208} \[ \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {b \sqrt {a+b \sqrt {x}}}{2 a \sqrt {x}}-\frac {\sqrt {a+b \sqrt {x}}}{x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[x]]/x^2,x]

[Out]

-(Sqrt[a + b*Sqrt[x]]/x) - (b*Sqrt[a + b*Sqrt[x]])/(2*a*Sqrt[x]) + (b^2*ArcTanh[Sqrt[a + b*Sqrt[x]]/Sqrt[a]])/
(2*a^(3/2))

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b \sqrt {x}}}{x^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{x}+\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{x}-\frac {b \sqrt {a+b \sqrt {x}}}{2 a \sqrt {x}}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sqrt {x}\right )}{4 a}\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{x}-\frac {b \sqrt {a+b \sqrt {x}}}{2 a \sqrt {x}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sqrt {x}}\right )}{2 a}\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{x}-\frac {b \sqrt {a+b \sqrt {x}}}{2 a \sqrt {x}}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{2 a^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 43, normalized size = 0.56 \[ -\frac {4 b^2 \left (a+b \sqrt {x}\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {\sqrt {x} b}{a}+1\right )}{3 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Sqrt[x]]/x^2,x]

[Out]

(-4*b^2*(a + b*Sqrt[x])^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (b*Sqrt[x])/a])/(3*a^3)

________________________________________________________________________________________

fricas [A]  time = 1.42, size = 133, normalized size = 1.73 \[ \left [\frac {\sqrt {a} b^{2} x \log \left (\frac {b x + 2 \, \sqrt {b \sqrt {x} + a} \sqrt {a} \sqrt {x} + 2 \, a \sqrt {x}}{x}\right ) - 2 \, {\left (a b \sqrt {x} + 2 \, a^{2}\right )} \sqrt {b \sqrt {x} + a}}{4 \, a^{2} x}, -\frac {\sqrt {-a} b^{2} x \arctan \left (\frac {\sqrt {b \sqrt {x} + a} \sqrt {-a}}{a}\right ) + {\left (a b \sqrt {x} + 2 \, a^{2}\right )} \sqrt {b \sqrt {x} + a}}{2 \, a^{2} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(sqrt(a)*b^2*x*log((b*x + 2*sqrt(b*sqrt(x) + a)*sqrt(a)*sqrt(x) + 2*a*sqrt(x))/x) - 2*(a*b*sqrt(x) + 2*a^
2)*sqrt(b*sqrt(x) + a))/(a^2*x), -1/2*(sqrt(-a)*b^2*x*arctan(sqrt(b*sqrt(x) + a)*sqrt(-a)/a) + (a*b*sqrt(x) +
2*a^2)*sqrt(b*sqrt(x) + a))/(a^2*x)]

________________________________________________________________________________________

giac [A]  time = 0.16, size = 72, normalized size = 0.94 \[ -\frac {\frac {b^{3} \arctan \left (\frac {\sqrt {b \sqrt {x} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {{\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} b^{3} + \sqrt {b \sqrt {x} + a} a b^{3}}{a b^{2} x}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^3*arctan(sqrt(b*sqrt(x) + a)/sqrt(-a))/(sqrt(-a)*a) + ((b*sqrt(x) + a)^(3/2)*b^3 + sqrt(b*sqrt(x) + a)
*a*b^3)/(a*b^2*x))/b

________________________________________________________________________________________

maple [A]  time = 0.01, size = 59, normalized size = 0.77 \[ 4 \left (\frac {\arctanh \left (\frac {\sqrt {b \sqrt {x}+a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}+\frac {-\frac {\left (b \sqrt {x}+a \right )^{\frac {3}{2}}}{8 a}-\frac {\sqrt {b \sqrt {x}+a}}{8}}{b^{2} x}\right ) b^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*b^2*((-1/8/a*(b*x^(1/2)+a)^(3/2)-1/8*(b*x^(1/2)+a)^(1/2))/x/b^2+1/8/a^(3/2)*arctanh((b*x^(1/2)+a)^(1/2)/a^(1
/2)))

________________________________________________________________________________________

maxima [A]  time = 2.04, size = 100, normalized size = 1.30 \[ -\frac {b^{2} \log \left (\frac {\sqrt {b \sqrt {x} + a} - \sqrt {a}}{\sqrt {b \sqrt {x} + a} + \sqrt {a}}\right )}{4 \, a^{\frac {3}{2}}} - \frac {{\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} b^{2} + \sqrt {b \sqrt {x} + a} a b^{2}}{2 \, {\left ({\left (b \sqrt {x} + a\right )}^{2} a - 2 \, {\left (b \sqrt {x} + a\right )} a^{2} + a^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b^2*log((sqrt(b*sqrt(x) + a) - sqrt(a))/(sqrt(b*sqrt(x) + a) + sqrt(a)))/a^(3/2) - 1/2*((b*sqrt(x) + a)^(
3/2)*b^2 + sqrt(b*sqrt(x) + a)*a*b^2)/((b*sqrt(x) + a)^2*a - 2*(b*sqrt(x) + a)*a^2 + a^3)

________________________________________________________________________________________

mupad [B]  time = 1.57, size = 54, normalized size = 0.70 \[ \frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,\sqrt {x}}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {\sqrt {a+b\,\sqrt {x}}}{2\,x}-\frac {{\left (a+b\,\sqrt {x}\right )}^{3/2}}{2\,a\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^(1/2)/x^2,x)

[Out]

(b^2*atanh((a + b*x^(1/2))^(1/2)/a^(1/2)))/(2*a^(3/2)) - (a + b*x^(1/2))^(1/2)/(2*x) - (a + b*x^(1/2))^(3/2)/(
2*a*x)

________________________________________________________________________________________

sympy [A]  time = 4.24, size = 105, normalized size = 1.36 \[ - \frac {a}{\sqrt {b} x^{\frac {5}{4}} \sqrt {\frac {a}{b \sqrt {x}} + 1}} - \frac {3 \sqrt {b}}{2 x^{\frac {3}{4}} \sqrt {\frac {a}{b \sqrt {x}} + 1}} - \frac {b^{\frac {3}{2}}}{2 a \sqrt [4]{x} \sqrt {\frac {a}{b \sqrt {x}} + 1}} + \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt [4]{x}} \right )}}{2 a^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(sqrt(b)*x**(5/4)*sqrt(a/(b*sqrt(x)) + 1)) - 3*sqrt(b)/(2*x**(3/4)*sqrt(a/(b*sqrt(x)) + 1)) - b**(3/2)/(2*a
*x**(1/4)*sqrt(a/(b*sqrt(x)) + 1)) + b**2*asinh(sqrt(a)/(sqrt(b)*x**(1/4)))/(2*a**(3/2))

________________________________________________________________________________________