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

Optimal. Leaf size=104 \[ \frac{2 \sqrt{d} \sqrt{b c-a d} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2}+\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a} c^2}+\frac{x \sqrt{a+\frac{b}{x}}}{c} \]

[Out]

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

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Rubi [A]  time = 0.110625, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {375, 99, 156, 63, 208, 205} \[ \frac{2 \sqrt{d} \sqrt{b c-a d} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2}+\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a} c^2}+\frac{x \sqrt{a+\frac{b}{x}}}{c} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]/(c + d/x),x]

[Out]

(Sqrt[a + b/x]*x)/c + (2*Sqrt[d]*Sqrt[b*c - a*d]*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/c^2 + ((b*c
- 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(Sqrt[a]*c^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 99

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

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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 205

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

Rubi steps

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

Mathematica [A]  time = 0.191681, size = 100, normalized size = 0.96 \[ \frac{2 \sqrt{d} \sqrt{b c-a d} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )+\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a}}+c x \sqrt{a+\frac{b}{x}}}{c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x]/(c + d/x),x]

[Out]

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

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Maple [B]  time = 0.062, size = 287, normalized size = 2.8 \begin{align*} -{\frac{x}{2\,{c}^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( 2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) \sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}acd-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) \sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}b{c}^{2}+2\,\ln \left ({\frac{1}{cx+d} \left ( 2\,\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}\sqrt{ \left ( ax+b \right ) x}c-2\,adx+bcx-bd \right ) } \right ){a}^{3/2}{d}^{2}-2\,\ln \left ({\frac{1}{cx+d} \left ( 2\,\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}\sqrt{ \left ( ax+b \right ) x}c-2\,adx+bcx-bd \right ) } \right ) \sqrt{a}bcd-2\,\sqrt{ \left ( ax+b \right ) x}{c}^{2}\sqrt{a}\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}} \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{\frac{ \left ( ad-bc \right ) d}{{c}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a + b/x)/(c + d/x), x)

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Fricas [A]  time = 1.49723, size = 1110, normalized size = 10.67 \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 \, \sqrt{-b c d + a d^{2}} a \log \left (\frac{b d -{\left (b c - 2 \, a d\right )} x + 2 \, \sqrt{-b c d + a d^{2}} x \sqrt{\frac{a x + b}{x}}}{c x + d}\right )}{2 \, a c^{2}}, \frac{2 \, a c x \sqrt{\frac{a x + b}{x}} - 4 \, \sqrt{b c d - a d^{2}} a \arctan \left (\frac{\sqrt{b c d - a d^{2}} x \sqrt{\frac{a x + b}{x}}}{a d x + b d}\right ) -{\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 c^{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 ) + \sqrt{-b c d + a d^{2}} a \log \left (\frac{b d -{\left (b c - 2 \, a d\right )} x + 2 \, \sqrt{-b c d + a d^{2}} x \sqrt{\frac{a x + b}{x}}}{c x + d}\right )}{a c^{2}}, \frac{a c x \sqrt{\frac{a x + b}{x}} - 2 \, \sqrt{b c d - a d^{2}} a \arctan \left (\frac{\sqrt{b c d - a d^{2}} x \sqrt{\frac{a x + b}{x}}}{a d x + b d}\right ) -{\left (b c - 2 \, a d\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right )}{a c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(1/2)/(c+d/x),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) + 2*sqr
t(-b*c*d + a*d^2)*a*log((b*d - (b*c - 2*a*d)*x + 2*sqrt(-b*c*d + a*d^2)*x*sqrt((a*x + b)/x))/(c*x + d)))/(a*c^
2), 1/2*(2*a*c*x*sqrt((a*x + b)/x) - 4*sqrt(b*c*d - a*d^2)*a*arctan(sqrt(b*c*d - a*d^2)*x*sqrt((a*x + b)/x)/(a
*d*x + b*d)) - (b*c - 2*a*d)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b))/(a*c^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) + sqrt(-b*c*d + a*d^2)*a*log((b*d - (b*
c - 2*a*d)*x + 2*sqrt(-b*c*d + a*d^2)*x*sqrt((a*x + b)/x))/(c*x + d)))/(a*c^2), (a*c*x*sqrt((a*x + b)/x) - 2*s
qrt(b*c*d - a*d^2)*a*arctan(sqrt(b*c*d - a*d^2)*x*sqrt((a*x + b)/x)/(a*d*x + b*d)) - (b*c - 2*a*d)*sqrt(-a)*ar
ctan(sqrt(-a)*sqrt((a*x + b)/x)/a))/(a*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError