3.3055 \(\int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} \, dx\)

Optimal. Leaf size=113 \[ \frac{\left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{4 a^{3/2}}+\frac{x \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{2 a} \]

[Out]

((2*a + b*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x]*x)/(2*a) + ((4*a*c - b^2*d)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sq
rt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(4*a^(3/2))

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Rubi [A]  time = 0.117751, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1969, 1357, 720, 724, 206} \[ \frac{\left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{4 a^{3/2}}+\frac{x \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[d/x] + c/x],x]

[Out]

((2*a + b*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x]*x)/(2*a) + ((4*a*c - b^2*d)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sq
rt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(4*a^(3/2))

Rule 1969

Int[((a_.) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> -Dist[d, Subst[Int[(a + b*x^n +
(c*x^(2*n))/d^(2*n))^p/x^2, x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, -2*n] && IntegerQ[2*n]

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

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

Rubi steps

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

Mathematica [F]  time = 0.149096, size = 0, normalized size = 0. \[ \int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [B]  time = 0.129, size = 213, normalized size = 1.9 \begin{align*}{\frac{1}{4}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }}\sqrt{x} \left ( 2\,{a}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}\sqrt{x}b-\ln \left ({\frac{1}{2} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ){\frac{1}{\sqrt{a}}}} \right ) da{b}^{2}+4\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{x}+4\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{2}c \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{a}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)*(2*a^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^(1/2)*b-l
n(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*d*a*b^2+4*a^(5/2)*(
b*(d/x)^(1/2)*x+a*x+c)^(1/2)*x^(1/2)+4*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2
*a*x^(1/2))/a^(1/2))*a^2*c)/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/a^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.75088, size = 221, normalized size = 1.96 \begin{align*} \frac{{\left (2 \, \sqrt{a d x + \sqrt{d x} b d + c d}{\left (\frac{b d}{a} + 2 \, \sqrt{d x}\right )} + \frac{{\left (b^{2} d^{2} - 4 \, a c d\right )} \log \left ({\left | -b d - 2 \,{\left (\sqrt{d x} \sqrt{a} - \sqrt{a d x + \sqrt{d x} b d + c d}\right )} \sqrt{a} \right |}\right )}{a^{\frac{3}{2}}}\right )} \mathrm{sgn}\left (x\right )}{4 \, d} - \frac{{\left (b^{2} d \log \left ({\left | -b d + 2 \, \sqrt{c d} \sqrt{a} \right |}\right ) - 4 \, a c \log \left ({\left | -b d + 2 \, \sqrt{c d} \sqrt{a} \right |}\right ) + 2 \, \sqrt{c d} \sqrt{a} b\right )} \mathrm{sgn}\left (x\right )}{4 \, a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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