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

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

[Out]

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

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Rubi [A]  time = 0.0708231, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {6690, 50, 63, 208} \[ \frac{2 \sqrt{a+b x}}{b-c}-\frac{2 \sqrt{a+c x}}{b-c}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{b-c}+\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{b-c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6690

Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)^(n_.)])^(m_), x_Symbol] :> Dis
t[(b*e^2 - d*f^2)^m, Int[ExpandIntegrand[(u*x^(m*n))/(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^m, x], x], x] /;
FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m, 0] && EqQ[a*e^2 - c*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}{\sqrt{a+b x}+\sqrt{a+c x}} \, dx &=\frac{\int \left (\frac{\sqrt{a+b x}}{x}-\frac{\sqrt{a+c x}}{x}\right ) \, dx}{b-c}\\ &=\frac{\int \frac{\sqrt{a+b x}}{x} \, dx}{b-c}-\frac{\int \frac{\sqrt{a+c x}}{x} \, dx}{b-c}\\ &=\frac{2 \sqrt{a+b x}}{b-c}-\frac{2 \sqrt{a+c x}}{b-c}+\frac{a \int \frac{1}{x \sqrt{a+b x}} \, dx}{b-c}-\frac{a \int \frac{1}{x \sqrt{a+c x}} \, dx}{b-c}\\ &=\frac{2 \sqrt{a+b x}}{b-c}-\frac{2 \sqrt{a+c x}}{b-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 (b-c)}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x}\right )}{(b-c) c}\\ &=\frac{2 \sqrt{a+b x}}{b-c}-\frac{2 \sqrt{a+c x}}{b-c}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{b-c}+\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{b-c}\\ \end{align*}

Mathematica [A]  time = 0.0483586, size = 75, normalized size = 0.77 \[ \frac{2 \left (\sqrt{a+b x}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\sqrt{a+c x}+\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )\right )}{b-c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 73, normalized size = 0.8 \begin{align*}{\frac{1}{b-c} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }-{\frac{1}{b-c} \left ( 2\,\sqrt{cx+a}-2\,\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/((b*x+a)^(1/2)+(c*x+a)^(1/2)),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.29619, size = 393, normalized size = 4.05 \begin{align*} \left [-\frac{\sqrt{a} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + \sqrt{a} \log \left (\frac{c x - 2 \, \sqrt{c x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \, \sqrt{b x + a} + 2 \, \sqrt{c x + a}}{b - c}, \frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) - \sqrt{-a} \arctan \left (\frac{\sqrt{c x + a} \sqrt{-a}}{a}\right ) + \sqrt{b x + a} - \sqrt{c x + a}\right )}}{b - c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError