3.2243 \(\int \frac{1}{\sqrt{a+b \sqrt{x}} x^2} \, dx\)

Optimal. Leaf size=80 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{3 b \sqrt{a+b \sqrt{x}}}{2 a^2 \sqrt{x}}-\frac{\sqrt{a+b \sqrt{x}}}{a x} \]

[Out]

-(Sqrt[a + b*Sqrt[x]]/(a*x)) + (3*b*Sqrt[a + b*Sqrt[x]])/(2*a^2*Sqrt[x]) - (3*b^2*ArcTanh[Sqrt[a + b*Sqrt[x]]/
Sqrt[a]])/(2*a^(5/2))

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Rubi [A]  time = 0.0351734, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {266, 51, 63, 208} \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{3 b \sqrt{a+b \sqrt{x}}}{2 a^2 \sqrt{x}}-\frac{\sqrt{a+b \sqrt{x}}}{a x} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*Sqrt[x]]*x^2),x]

[Out]

-(Sqrt[a + b*Sqrt[x]]/(a*x)) + (3*b*Sqrt[a + b*Sqrt[x]])/(2*a^2*Sqrt[x]) - (3*b^2*ArcTanh[Sqrt[a + b*Sqrt[x]]/
Sqrt[a]])/(2*a^(5/2))

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]]

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]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+b \sqrt{x}} x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{a x}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )}{2 a}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{a x}+\frac{3 b \sqrt{a+b \sqrt{x}}}{2 a^2 \sqrt{x}}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )}{4 a^2}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{a x}+\frac{3 b \sqrt{a+b \sqrt{x}}}{2 a^2 \sqrt{x}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sqrt{x}}\right )}{2 a^2}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{a x}+\frac{3 b \sqrt{a+b \sqrt{x}}}{2 a^2 \sqrt{x}}-\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.0072077, size = 41, normalized size = 0.51 \[ -\frac{4 b^2 \sqrt{a+b \sqrt{x}} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{\sqrt{x} b}{a}+1\right )}{a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*Sqrt[x]]*x^2),x]

[Out]

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

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Maple [A]  time = 0.006, size = 72, normalized size = 0.9 \begin{align*} 4\,{b}^{2} \left ( -1/4\,{\frac{\sqrt{a+b\sqrt{x}}}{xa{b}^{2}}}-3/4\,{\frac{1}{a} \left ( -1/2\,{\frac{\sqrt{a+b\sqrt{x}}}{ab\sqrt{x}}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) } \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.2653, size = 360, normalized size = 4.5 \begin{align*} \left [\frac{3 \, \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 (3 \, a b \sqrt{x} - 2 \, a^{2}\right )} \sqrt{b \sqrt{x} + a}}{4 \, a^{3} x}, \frac{3 \, \sqrt{-a} b^{2} x \arctan \left (\frac{\sqrt{b \sqrt{x} + a} \sqrt{-a}}{a}\right ) +{\left (3 \, a b \sqrt{x} - 2 \, a^{2}\right )} \sqrt{b \sqrt{x} + a}}{2 \, a^{3} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(3*sqrt(a)*b^2*x*log((b*x - 2*sqrt(b*sqrt(x) + a)*sqrt(a)*sqrt(x) + 2*a*sqrt(x))/x) + 2*(3*a*b*sqrt(x) -
2*a^2)*sqrt(b*sqrt(x) + a))/(a^3*x), 1/2*(3*sqrt(-a)*b^2*x*arctan(sqrt(b*sqrt(x) + a)*sqrt(-a)/a) + (3*a*b*sqr
t(x) - 2*a^2)*sqrt(b*sqrt(x) + a))/(a^3*x)]

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Sympy [A]  time = 4.45126, size = 110, normalized size = 1.38 \begin{align*} - \frac{1}{\sqrt{b} x^{\frac{5}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{\sqrt{b}}{2 a x^{\frac{3}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{3 b^{\frac{3}{2}}}{2 a^{2} \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )}}{2 a^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11085, size = 89, normalized size = 1.11 \begin{align*} \frac{1}{2} \, b^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} - 5 \, \sqrt{b \sqrt{x} + a} a}{a^{2} b^{2} x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*(3*arctan(sqrt(b*sqrt(x) + a)/sqrt(-a))/(sqrt(-a)*a^2) + (3*(b*sqrt(x) + a)^(3/2) - 5*sqrt(b*sqrt(x) +
a)*a)/(a^2*b^2*x))