3.1065 \(\int \frac{\sqrt{c+\frac{d}{x}}}{\sqrt{a+\frac{b}{x}} x} \, dx\)

Optimal. Leaf size=93 \[ \frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a}}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0863921, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {446, 105, 63, 217, 206, 93, 208} \[ \frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a}}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d/x]/(Sqrt[a + b/x]*x),x]

[Out]

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

Rule 446

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

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*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{\sqrt{c+\frac{d}{x}}}{\sqrt{a+\frac{b}{x}} x} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=-\left (c \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\frac{1}{x}\right )\right )-d \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\frac{1}{x}\right )\\ &=-\left ((2 c) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{c+\frac{d}{x}}}\right )\right )-\frac{(2 d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{b}\\ &=\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a}}-\frac{(2 d) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{c+\frac{d}{x}}}\right )}{b}\\ &=\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a}}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.460271, size = 142, normalized size = 1.53 \[ \frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a}}-\frac{2 \sqrt{d} x \sqrt{c+\frac{d}{x}} \sqrt{b c-a d} \sqrt{\frac{b (c x+d)}{x (b c-a d)}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{b c x+b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.043, size = 143, normalized size = 1.5 \begin{align*} -{x\sqrt{{\frac{ax+b}{x}}}\sqrt{{\frac{cx+d}{x}}} \left ( \ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{bd}\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }+2\,bd \right ) } \right ) d\sqrt{ac}-\ln \left ({\frac{1}{2} \left ( 2\,xac+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) c\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }}}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.10637, size = 1628, normalized size = 17.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(c/a)*log(-8*a^2*c^2*x^2 - b^2*c^2 - 6*a*b*c*d - a^2*d^2 - 4*(2*a^2*c*x^2 + (a*b*c + a^2*d)*x)*sqrt(c
/a)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) - 8*(a*b*c^2 + a^2*c*d)*x) + 1/2*sqrt(d/b)*log(-(8*b^2*d^2 + (b^2*c^2
+ 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*b^2*d*x + (b^2*c + a*b*d)*x^2)*sqrt(d/b)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x)
+ 8*(b^2*c*d + a*b*d^2)*x)/x^2), -sqrt(-c/a)*arctan(2*a*x*sqrt(-c/a)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x)/(2*a
*c*x + b*c + a*d)) + 1/2*sqrt(d/b)*log(-(8*b^2*d^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*b^2*d*x + (b^2
*c + a*b*d)*x^2)*sqrt(d/b)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) + 8*(b^2*c*d + a*b*d^2)*x)/x^2), sqrt(-d/b)*arc
tan(1/2*(2*b*d*x + (b*c + a*d)*x^2)*sqrt(-d/b)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x)/(a*c*d*x^2 + b*d^2 + (b*c*d
+ a*d^2)*x)) + 1/2*sqrt(c/a)*log(-8*a^2*c^2*x^2 - b^2*c^2 - 6*a*b*c*d - a^2*d^2 - 4*(2*a^2*c*x^2 + (a*b*c + a
^2*d)*x)*sqrt(c/a)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) - 8*(a*b*c^2 + a^2*c*d)*x), -sqrt(-c/a)*arctan(2*a*x*sq
rt(-c/a)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x)/(2*a*c*x + b*c + a*d)) + sqrt(-d/b)*arctan(1/2*(2*b*d*x + (b*c +
a*d)*x^2)*sqrt(-d/b)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x)/(a*c*d*x^2 + b*d^2 + (b*c*d + a*d^2)*x))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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