3.430 \(\int \frac{1}{x (\sqrt{a+b x}+\sqrt{a+c x})} \, dx\)

Optimal. Leaf size=103 \[ -\frac{\sqrt{a+b x}}{x (b-c)}+\frac{\sqrt{a+c x}}{x (b-c)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)} \]

[Out]

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

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Rubi [A]  time = 0.0945461, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2103, 47, 63, 208} \[ -\frac{\sqrt{a+b x}}{x (b-c)}+\frac{\sqrt{a+c x}}{x (b-c)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(Sqrt[a + b*x] + Sqrt[a + c*x])),x]

[Out]

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

Rule 2103

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && 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}{x \left (\sqrt{a+b x}+\sqrt{a+c x}\right )} \, dx &=\frac{\int \frac{\sqrt{a+b x}}{x^2} \, dx}{b-c}-\frac{\int \frac{\sqrt{a+c x}}{x^2} \, dx}{b-c}\\ &=-\frac{\sqrt{a+b x}}{(b-c) x}+\frac{\sqrt{a+c x}}{(b-c) x}+\frac{b \int \frac{1}{x \sqrt{a+b x}} \, dx}{2 (b-c)}-\frac{c \int \frac{1}{x \sqrt{a+c x}} \, dx}{2 (b-c)}\\ &=-\frac{\sqrt{a+b x}}{(b-c) x}+\frac{\sqrt{a+c x}}{(b-c) x}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{b-c}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x}\right )}{b-c}\\ &=-\frac{\sqrt{a+b x}}{(b-c) x}+\frac{\sqrt{a+c x}}{(b-c) x}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)}\\ \end{align*}

Mathematica [A]  time = 0.198739, size = 135, normalized size = 1.31 \[ \frac{-\frac{a}{\sqrt{a+b x}}-\frac{b x}{\sqrt{a+b x}}-\frac{b x \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x}{a}+1}\right )}{\sqrt{a+b x}}+\frac{a}{\sqrt{a+c x}}+\frac{c x}{\sqrt{a+c x}}+\frac{c x \sqrt{\frac{c x}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{c x}{a}+1}\right )}{\sqrt{a+c x}}}{b x-c x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a/Sqrt[a + b*x]) - (b*x)/Sqrt[a + b*x] + a/Sqrt[a + c*x] + (c*x)/Sqrt[a + c*x] - (b*x*Sqrt[1 + (b*x)/a]*Arc
Tanh[Sqrt[1 + (b*x)/a]])/Sqrt[a + b*x] + (c*x*Sqrt[1 + (c*x)/a]*ArcTanh[Sqrt[1 + (c*x)/a]])/Sqrt[a + c*x])/(b*
x - c*x)

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Maple [A]  time = 0.004, size = 88, normalized size = 0.9 \begin{align*} 2\,{\frac{b}{b-c} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-2\,{\frac{c}{b-c} \left ( -1/2\,{\frac{\sqrt{cx+a}}{cx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.20451, size = 450, normalized size = 4.37 \begin{align*} \left [-\frac{\sqrt{a} b x \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + \sqrt{a} c x \log \left (\frac{c x - 2 \, \sqrt{c x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \, \sqrt{b x + a} a - 2 \, \sqrt{c x + a} a}{2 \,{\left (a b - a c\right )} x}, \frac{\sqrt{-a} b x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) - \sqrt{-a} c x \arctan \left (\frac{\sqrt{c x + a} \sqrt{-a}}{a}\right ) - \sqrt{b x + a} a + \sqrt{c x + a} a}{{\left (a b - a c\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (\sqrt{a + b x} + \sqrt{a + c x}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError