3.408 \(\int \frac{1}{(\sqrt{a+b x}+\sqrt{c+b x})^2} \, dx\)

Optimal. Leaf size=63 \[ \frac{(a-c)^2}{8 b \left (\sqrt{a+b x}+\sqrt{b x+c}\right )^4}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{2 b} \]

[Out]

(a - c)^2/(8*b*(Sqrt[a + b*x] + Sqrt[c + b*x])^4) + ArcTanh[Sqrt[a + b*x]/Sqrt[c + b*x]]/(2*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0985642, antiderivative size = 114, normalized size of antiderivative = 1.81, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {6689, 50, 63, 217, 206} \[ \frac{b x^2}{(a-c)^2}-\frac{(a+b x)^{3/2} \sqrt{b x+c}}{b (a-c)^2}+\frac{\sqrt{a+b x} \sqrt{b x+c}}{2 b (a-c)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{2 b}+\frac{x (a+c)}{(a-c)^2} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x] + Sqrt[c + b*x])^(-2),x]

[Out]

((a + c)*x)/(a - c)^2 + (b*x^2)/(a - c)^2 + (Sqrt[a + b*x]*Sqrt[c + b*x])/(2*b*(a - c)) - ((a + b*x)^(3/2)*Sqr
t[c + b*x])/(b*(a - c)^2) + ArcTanh[Sqrt[a + b*x]/Sqrt[c + b*x]]/(2*b)

Rule 6689

Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)^(n_.)])^(m_), x_Symbol] :> Dis
t[(a*e^2 - c*f^2)^m, Int[ExpandIntegrand[u/(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[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 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])

Rubi steps

\begin{align*} \int \frac{1}{\left (\sqrt{a+b x}+\sqrt{c+b x}\right )^2} \, dx &=\frac{\int \left (a \left (1+\frac{c}{a}\right )+2 b x-2 \sqrt{a+b x} \sqrt{c+b x}\right ) \, dx}{(a-c)^2}\\ &=\frac{(a+c) x}{(a-c)^2}+\frac{b x^2}{(a-c)^2}-\frac{2 \int \sqrt{a+b x} \sqrt{c+b x} \, dx}{(a-c)^2}\\ &=\frac{(a+c) x}{(a-c)^2}+\frac{b x^2}{(a-c)^2}-\frac{(a+b x)^{3/2} \sqrt{c+b x}}{b (a-c)^2}+\frac{\int \frac{\sqrt{a+b x}}{\sqrt{c+b x}} \, dx}{2 (a-c)}\\ &=\frac{(a+c) x}{(a-c)^2}+\frac{b x^2}{(a-c)^2}+\frac{\sqrt{a+b x} \sqrt{c+b x}}{2 b (a-c)}-\frac{(a+b x)^{3/2} \sqrt{c+b x}}{b (a-c)^2}+\frac{1}{4} \int \frac{1}{\sqrt{a+b x} \sqrt{c+b x}} \, dx\\ &=\frac{(a+c) x}{(a-c)^2}+\frac{b x^2}{(a-c)^2}+\frac{\sqrt{a+b x} \sqrt{c+b x}}{2 b (a-c)}-\frac{(a+b x)^{3/2} \sqrt{c+b x}}{b (a-c)^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+c+x^2}} \, dx,x,\sqrt{a+b x}\right )}{2 b}\\ &=\frac{(a+c) x}{(a-c)^2}+\frac{b x^2}{(a-c)^2}+\frac{\sqrt{a+b x} \sqrt{c+b x}}{2 b (a-c)}-\frac{(a+b x)^{3/2} \sqrt{c+b x}}{b (a-c)^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+b x}}\right )}{2 b}\\ &=\frac{(a+c) x}{(a-c)^2}+\frac{b x^2}{(a-c)^2}+\frac{\sqrt{a+b x} \sqrt{c+b x}}{2 b (a-c)}-\frac{(a+b x)^{3/2} \sqrt{c+b x}}{b (a-c)^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{c+b x}}\right )}{2 b}\\ \end{align*}

Mathematica [B]  time = 0.569051, size = 179, normalized size = 2.84 \[ \frac{2 b x \left (b x-\sqrt{a+b x} \sqrt{b x+c}\right )+a \left (2 b x-\sqrt{a+b x} \sqrt{b x+c}\right )+c \left (2 b x-\sqrt{a+b x} \sqrt{b x+c}\right )+\frac{\sqrt{b} (c-a)^3 \sqrt{\frac{b x+c}{c-a}} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{a+b x}}{\sqrt{b (c-a)}}\right )}{\sqrt{b (c-a)} \sqrt{b x+c}}+2 c^2}{2 b (a-c)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x] + Sqrt[c + b*x])^(-2),x]

[Out]

(2*c^2 + 2*b*x*(b*x - Sqrt[a + b*x]*Sqrt[c + b*x]) + a*(2*b*x - Sqrt[a + b*x]*Sqrt[c + b*x]) + c*(2*b*x - Sqrt
[a + b*x]*Sqrt[c + b*x]) + (Sqrt[b]*(-a + c)^3*Sqrt[(c + b*x)/(-a + c)]*ArcSinh[(Sqrt[b]*Sqrt[a + b*x])/Sqrt[b
*(-a + c)]])/(Sqrt[b*(-a + c)]*Sqrt[c + b*x]))/(2*b*(a - c)^2)

________________________________________________________________________________________

Maple [B]  time = 0.006, size = 377, normalized size = 6. \begin{align*}{\frac{ax}{ \left ( a-c \right ) ^{2}}}+{\frac{cx}{ \left ( a-c \right ) ^{2}}}+{\frac{b{x}^{2}}{ \left ( a-c \right ) ^{2}}}-{\frac{1}{ \left ( a-c \right ) ^{2}b}\sqrt{bx+a} \left ( bx+c \right ) ^{{\frac{3}{2}}}}-{\frac{a}{2\, \left ( a-c \right ) ^{2}b}\sqrt{bx+c}\sqrt{bx+a}}+{\frac{c}{2\, \left ( a-c \right ) ^{2}b}\sqrt{bx+c}\sqrt{bx+a}}+{\frac{{a}^{2}}{4\, \left ( a-c \right ) ^{2}}\sqrt{ \left ( bx+c \right ) \left ( bx+a \right ) }\ln \left ({ \left ({\frac{ab}{2}}+{\frac{bc}{2}}+{b}^{2}x \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+ \left ( ab+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+c}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{ac}{2\, \left ( a-c \right ) ^{2}}\sqrt{ \left ( bx+c \right ) \left ( bx+a \right ) }\ln \left ({ \left ({\frac{ab}{2}}+{\frac{bc}{2}}+{b}^{2}x \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+ \left ( ab+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+c}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{{b}^{2}}}}}+{\frac{{c}^{2}}{4\, \left ( a-c \right ) ^{2}}\sqrt{ \left ( bx+c \right ) \left ( bx+a \right ) }\ln \left ({ \left ({\frac{ab}{2}}+{\frac{bc}{2}}+{b}^{2}x \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+ \left ( ab+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+c}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(a-c)^2*a+x/(a-c)^2*c+b*x^2/(a-c)^2-1/(a-c)^2/b*(b*x+a)^(1/2)*(b*x+c)^(3/2)-1/2/(a-c)^2/b*(b*x+c)^(1/2)*(b*x
+a)^(1/2)*a+1/2/(a-c)^2/b*(b*x+c)^(1/2)*(b*x+a)^(1/2)*c+1/4/(a-c)^2*((b*x+c)*(b*x+a))^(1/2)/(b*x+c)^(1/2)/(b*x
+a)^(1/2)*ln((1/2*a*b+1/2*b*c+b^2*x)/(b^2)^(1/2)+(b^2*x^2+(a*b+b*c)*x+a*c)^(1/2))/(b^2)^(1/2)*a^2-1/2/(a-c)^2*
((b*x+c)*(b*x+a))^(1/2)/(b*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*b+1/2*b*c+b^2*x)/(b^2)^(1/2)+(b^2*x^2+(a*b+b*c)*
x+a*c)^(1/2))/(b^2)^(1/2)*a*c+1/4/(a-c)^2*((b*x+c)*(b*x+a))^(1/2)/(b*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*b+1/2*
b*c+b^2*x)/(b^2)^(1/2)+(b^2*x^2+(a*b+b*c)*x+a*c)^(1/2))/(b^2)^(1/2)*c^2

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\sqrt{b x + a} + \sqrt{b x + c}\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((sqrt(b*x + a) + sqrt(b*x + c))^(-2), x)

________________________________________________________________________________________

Fricas [B]  time = 0.993924, size = 247, normalized size = 3.92 \begin{align*} \frac{4 \, b^{2} x^{2} - 2 \,{\left (2 \, b x + a + c\right )} \sqrt{b x + a} \sqrt{b x + c} + 4 \,{\left (a b + b c\right )} x -{\left (a^{2} - 2 \, a c + c^{2}\right )} \log \left (-2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b x + c} - a - c\right )}{4 \,{\left (a^{2} b - 2 \, a b c + b c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*b^2*x^2 - 2*(2*b*x + a + c)*sqrt(b*x + a)*sqrt(b*x + c) + 4*(a*b + b*c)*x - (a^2 - 2*a*c + c^2)*log(-2*
b*x + 2*sqrt(b*x + a)*sqrt(b*x + c) - a - c))/(a^2*b - 2*a*b*c + b*c^2)

________________________________________________________________________________________

Sympy [A]  time = 0.992041, size = 318, normalized size = 5.05 \begin{align*} \begin{cases} \frac{a \log{\left (\sqrt{a + b x} + \sqrt{b x + c} \right )}}{2 a b + 4 b^{2} x + 2 b c + 4 b \sqrt{a + b x} \sqrt{b x + c}} + \frac{2 b x \log{\left (\sqrt{a + b x} + \sqrt{b x + c} \right )}}{2 a b + 4 b^{2} x + 2 b c + 4 b \sqrt{a + b x} \sqrt{b x + c}} + \frac{c \log{\left (\sqrt{a + b x} + \sqrt{b x + c} \right )}}{2 a b + 4 b^{2} x + 2 b c + 4 b \sqrt{a + b x} \sqrt{b x + c}} + \frac{2 \sqrt{a + b x} \sqrt{b x + c} \log{\left (\sqrt{a + b x} + \sqrt{b x + c} \right )}}{2 a b + 4 b^{2} x + 2 b c + 4 b \sqrt{a + b x} \sqrt{b x + c}} - \frac{\sqrt{a + b x} \sqrt{b x + c}}{2 a b + 4 b^{2} x + 2 b c + 4 b \sqrt{a + b x} \sqrt{b x + c}} & \text{for}\: b \neq 0 \\\frac{x}{\left (\sqrt{a} + \sqrt{c}\right )^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*log(sqrt(a + b*x) + sqrt(b*x + c))/(2*a*b + 4*b**2*x + 2*b*c + 4*b*sqrt(a + b*x)*sqrt(b*x + c)) +
2*b*x*log(sqrt(a + b*x) + sqrt(b*x + c))/(2*a*b + 4*b**2*x + 2*b*c + 4*b*sqrt(a + b*x)*sqrt(b*x + c)) + c*log
(sqrt(a + b*x) + sqrt(b*x + c))/(2*a*b + 4*b**2*x + 2*b*c + 4*b*sqrt(a + b*x)*sqrt(b*x + c)) + 2*sqrt(a + b*x)
*sqrt(b*x + c)*log(sqrt(a + b*x) + sqrt(b*x + c))/(2*a*b + 4*b**2*x + 2*b*c + 4*b*sqrt(a + b*x)*sqrt(b*x + c))
- sqrt(a + b*x)*sqrt(b*x + c)/(2*a*b + 4*b**2*x + 2*b*c + 4*b*sqrt(a + b*x)*sqrt(b*x + c)), Ne(b, 0)), (x/(sq
rt(a) + sqrt(c))**2, True))

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out