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

Optimal. Leaf size=97 \[ \frac{2 \sqrt{a+b x}}{a-c}-\frac{2 \sqrt{b x+c}}{a-c}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a-c}+\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{a-c} \]

[Out]

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

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Rubi [A]  time = 0.103954, antiderivative size = 97, 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 = {2104, 50, 63, 208} \[ \frac{2 \sqrt{a+b x}}{a-c}-\frac{2 \sqrt{b x+c}}{a-c}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a-c}+\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{a-c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2104

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

Rule 50

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

Mathematica [A]  time = 0.0721256, size = 75, normalized size = 0.77 \[ \frac{2 \left (\sqrt{a+b x}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\sqrt{b x+c}+\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )\right )}{a-c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 73, normalized size = 0.8 \begin{align*}{\frac{1}{a-c} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }-{\frac{1}{a-c} \left ( 2\,\sqrt{bx+c}-2\,\sqrt{c}{\it Artanh} \left ({\frac{\sqrt{bx+c}}{\sqrt{c}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a-c)*(2*(b*x+a)^(1/2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))-1/(a-c)*(2*(b*x+c)^(1/2)-2*c^(1/2)*arctanh(
(b*x+c)^(1/2)/c^(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{b x + c}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.02143, size = 794, normalized size = 8.19 \begin{align*} \left [-\frac{\sqrt{a} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + \sqrt{c} \log \left (\frac{b x - 2 \, \sqrt{b x + c} \sqrt{c} + 2 \, c}{x}\right ) - 2 \, \sqrt{b x + a} + 2 \, \sqrt{b x + c}}{a - c}, -\frac{2 \, \sqrt{-c} \arctan \left (\frac{\sqrt{b x + c} \sqrt{-c}}{c}\right ) + \sqrt{a} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \, \sqrt{b x + a} + 2 \, \sqrt{b x + c}}{a - c}, \frac{2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) - \sqrt{c} \log \left (\frac{b x - 2 \, \sqrt{b x + c} \sqrt{c} + 2 \, c}{x}\right ) + 2 \, \sqrt{b x + a} - 2 \, \sqrt{b x + c}}{a - c}, \frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) - \sqrt{-c} \arctan \left (\frac{\sqrt{b x + c} \sqrt{-c}}{c}\right ) + \sqrt{b x + a} - \sqrt{b x + c}\right )}}{a - c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{b x + c}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError