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

Optimal. Leaf size=51 \[ \frac{c x \sqrt{a+\frac{b}{x}}}{a}-\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0331384, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {375, 78, 63, 208} \[ \frac{c x \sqrt{a+\frac{b}{x}}}{a}-\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d/x)/Sqrt[a + b/x],x]

[Out]

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

Rule 375

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

Rule 78

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

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

Mathematica [A]  time = 0.0339721, size = 53, normalized size = 1.04 \[ \frac{2 \left (a d-\frac{b c}{2}\right ) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{c x \sqrt{a+\frac{b}{x}}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d/x)/Sqrt[a + b/x],x]

[Out]

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

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Maple [B]  time = 0.01, size = 173, normalized size = 3.4 \begin{align*}{\frac{x}{2\,b}\sqrt{{\frac{ax+b}{x}}} \left ( 2\,\sqrt{a{x}^{2}+bx}{a}^{3/2}d+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) abd-2\,{a}^{3/2}\sqrt{ \left ( ax+b \right ) x}d+2\,\sqrt{a}\sqrt{ \left ( ax+b \right ) x}bc+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) abd-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){b}^{2}c \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{a}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29481, size = 277, normalized size = 5.43 \begin{align*} \left [\frac{2 \, a c x \sqrt{\frac{a x + b}{x}} -{\left (b c - 2 \, a d\right )} \sqrt{a} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right )}{2 \, a^{2}}, \frac{a c x \sqrt{\frac{a x + b}{x}} +{\left (b c - 2 \, a d\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right )}{a^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 24.4353, size = 82, normalized size = 1.61 \begin{align*} \frac{\sqrt{b} c \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{a} - \frac{2 d \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{a}} \sqrt{a + \frac{b}{x}}} \right )}}{a \sqrt{- \frac{1}{a}}} - \frac{b c \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*c*sqrt(x)*sqrt(a*x/b + 1)/a - 2*d*atan(1/(sqrt(-1/a)*sqrt(a + b/x)))/(a*sqrt(-1/a)) - b*c*asinh(sqrt(a
)*sqrt(x)/sqrt(b))/a**(3/2)

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Giac [A]  time = 1.18868, size = 99, normalized size = 1.94 \begin{align*} -b{\left (\frac{c \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a} - \frac{{\left (b c - 2 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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