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

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)

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Rubi [A]  time = 0.216258, 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 \[ \frac{b x^2}{(a-c)^2}-\frac{(a+b x)^{3/2} \sqrt{b x+c}}{b (a-c)^2}+\frac{\sqrt{a+b x} \sqrt{b x+c}}{2 b (a-c)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{2 b}+\frac{x (a+c)}{(a-c)^2} \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 b \int x\, dx}{\left (a - c\right )^{2}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{b x + c}}{\sqrt{a + b x}} \right )}}{2 b} - \frac{\sqrt{a + b x} \sqrt{b x + c}}{2 b \left (a - c\right )} - \frac{\sqrt{a + b x} \left (b x + c\right )^{\frac{3}{2}}}{b \left (a - c\right )^{2}} + \frac{\left (a + c\right ) \int a\, dx}{a \left (a - c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((b*x+a)**(1/2)+(b*x+c)**(1/2))**2,x)

[Out]

2*b*Integral(x, x)/(a - c)**2 + atanh(sqrt(b*x + c)/sqrt(a + b*x))/(2*b) - sqrt(
a + b*x)*sqrt(b*x + c)/(2*b*(a - c)) - sqrt(a + b*x)*(b*x + c)**(3/2)/(b*(a - c)
**2) + (a + c)*Integral(a, x)/(a*(a - c)**2)

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Mathematica [A]  time = 0.123261, size = 93, normalized size = 1.48 \[ \frac{4 b x (a+c)-2 \sqrt{a+b x} \sqrt{b x+c} (a+2 b x+c)+(a-c)^2 \log \left (2 \sqrt{a+b x} \sqrt{b x+c}+a+2 b x+c\right )+4 b^2 x^2}{4 b (a-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 377, normalized size = 6. \[{\frac{ax}{ \left ( a-c \right ) ^{2}}}+{\frac{cx}{ \left ( a-c \right ) ^{2}}}+{\frac{b{x}^{2}}{ \left ( a-c \right ) ^{2}}}-{\frac{1}{ \left ( a-c \right ) ^{2}b}\sqrt{bx+a} \left ( bx+c \right ) ^{{\frac{3}{2}}}}-{\frac{a}{2\, \left ( a-c \right ) ^{2}b}\sqrt{bx+c}\sqrt{bx+a}}+{\frac{c}{2\, \left ( a-c \right ) ^{2}b}\sqrt{bx+c}\sqrt{bx+a}}+{\frac{{a}^{2}}{4\, \left ( a-c \right ) ^{2}}\sqrt{ \left ( bx+c \right ) \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{ab}{2}}+{\frac{bc}{2}}+{b}^{2}x \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+ \left ( ab+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+c}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{ac}{2\, \left ( a-c \right ) ^{2}}\sqrt{ \left ( bx+c \right ) \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{ab}{2}}+{\frac{bc}{2}}+{b}^{2}x \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+ \left ( ab+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+c}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{{b}^{2}}}}}+{\frac{{c}^{2}}{4\, \left ( a-c \right ) ^{2}}\sqrt{ \left ( bx+c \right ) \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{ab}{2}}+{\frac{bc}{2}}+{b}^{2}x \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+ \left ( ab+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+c}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(b*x + a) + sqrt(b*x + c))^(-2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.275395, size = 711, normalized size = 11.29 \[ \frac{256 \, b^{4} x^{4} - a^{4} + 24 \, a^{3} c + 50 \, a^{2} c^{2} + 24 \, a c^{3} - c^{4} + 512 \,{\left (a b^{3} + b^{3} c\right )} x^{3} + 8 \,{\left (37 \, a^{2} b^{2} + 98 \, a b^{2} c + 37 \, b^{2} c^{2}\right )} x^{2} - 4 \,{\left (64 \, b^{3} x^{3} + a^{3} + 11 \, a^{2} c + 11 \, a c^{2} + c^{3} + 96 \,{\left (a b^{2} + b^{2} c\right )} x^{2} + 2 \,{\left (17 \, a^{2} b + 42 \, a b c + 17 \, b c^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b x + c} + 8 \,{\left (5 \, a^{3} b + 39 \, a^{2} b c + 39 \, a b c^{2} + 5 \, b c^{3}\right )} x - 4 \,{\left (a^{4} + 4 \, a^{3} c - 10 \, a^{2} c^{2} + 4 \, a c^{3} + c^{4} + 8 \,{\left (a^{2} b^{2} - 2 \, a b^{2} c + b^{2} c^{2}\right )} x^{2} - 4 \,{\left (a^{3} - a^{2} c - a c^{2} + c^{3} + 2 \,{\left (a^{2} b - 2 \, a b c + b c^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b x + c} + 8 \,{\left (a^{3} b - a^{2} b c - a b c^{2} + b c^{3}\right )} x\right )} \log \left (-2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b x + c} - a - c\right )}{16 \,{\left (a^{4} b + 4 \, a^{3} b c - 10 \, a^{2} b c^{2} + 4 \, a b c^{3} + b c^{4} + 8 \,{\left (a^{2} b^{3} - 2 \, a b^{3} c + b^{3} c^{2}\right )} x^{2} - 4 \,{\left (a^{3} b - a^{2} b c - a b c^{2} + b c^{3} + 2 \,{\left (a^{2} b^{2} - 2 \, a b^{2} c + b^{2} c^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b x + c} + 8 \,{\left (a^{3} b^{2} - a^{2} b^{2} c - a b^{2} c^{2} + b^{2} c^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(b*x + a) + sqrt(b*x + c))^(-2),x, algorithm="fricas")

[Out]

1/16*(256*b^4*x^4 - a^4 + 24*a^3*c + 50*a^2*c^2 + 24*a*c^3 - c^4 + 512*(a*b^3 +
b^3*c)*x^3 + 8*(37*a^2*b^2 + 98*a*b^2*c + 37*b^2*c^2)*x^2 - 4*(64*b^3*x^3 + a^3
+ 11*a^2*c + 11*a*c^2 + c^3 + 96*(a*b^2 + b^2*c)*x^2 + 2*(17*a^2*b + 42*a*b*c +
17*b*c^2)*x)*sqrt(b*x + a)*sqrt(b*x + c) + 8*(5*a^3*b + 39*a^2*b*c + 39*a*b*c^2
+ 5*b*c^3)*x - 4*(a^4 + 4*a^3*c - 10*a^2*c^2 + 4*a*c^3 + c^4 + 8*(a^2*b^2 - 2*a*
b^2*c + b^2*c^2)*x^2 - 4*(a^3 - a^2*c - a*c^2 + c^3 + 2*(a^2*b - 2*a*b*c + b*c^2
)*x)*sqrt(b*x + a)*sqrt(b*x + c) + 8*(a^3*b - a^2*b*c - a*b*c^2 + b*c^3)*x)*log(
-2*b*x + 2*sqrt(b*x + a)*sqrt(b*x + c) - a - c))/(a^4*b + 4*a^3*b*c - 10*a^2*b*c
^2 + 4*a*b*c^3 + b*c^4 + 8*(a^2*b^3 - 2*a*b^3*c + b^3*c^2)*x^2 - 4*(a^3*b - a^2*
b*c - a*b*c^2 + b*c^3 + 2*(a^2*b^2 - 2*a*b^2*c + b^2*c^2)*x)*sqrt(b*x + a)*sqrt(
b*x + c) + 8*(a^3*b^2 - a^2*b^2*c - a*b^2*c^2 + b^2*c^3)*x)

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Sympy [A]  time = 3.59794, size = 388, normalized size = 6.16 \[ \begin{cases} \frac{2 a \log{\left (\sqrt{a + b x} + \sqrt{b x + c} \right )}}{4 a b + 8 b^{2} x + 4 b c + 8 b \sqrt{a + b x} \sqrt{b x + c}} + \frac{a}{4 a b + 8 b^{2} x + 4 b c + 8 b \sqrt{a + b x} \sqrt{b x + c}} + \frac{4 b x \log{\left (\sqrt{a + b x} + \sqrt{b x + c} \right )}}{4 a b + 8 b^{2} x + 4 b c + 8 b \sqrt{a + b x} \sqrt{b x + c}} + \frac{2 b x}{4 a b + 8 b^{2} x + 4 b c + 8 b \sqrt{a + b x} \sqrt{b x + c}} + \frac{2 c \log{\left (\sqrt{a + b x} + \sqrt{b x + c} \right )}}{4 a b + 8 b^{2} x + 4 b c + 8 b \sqrt{a + b x} \sqrt{b x + c}} + \frac{c}{4 a b + 8 b^{2} x + 4 b c + 8 b \sqrt{a + b x} \sqrt{b x + c}} + \frac{4 \sqrt{a + b x} \sqrt{b x + c} \log{\left (\sqrt{a + b x} + \sqrt{b x + c} \right )}}{4 a b + 8 b^{2} x + 4 b c + 8 b \sqrt{a + b x} \sqrt{b x + c}} & \text{for}\: b \neq 0 \\\frac{x}{\left (\sqrt{a} + \sqrt{c}\right )^{2}} & \text{otherwise} \end{cases} \]

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((2*a*log(sqrt(a + b*x) + sqrt(b*x + c))/(4*a*b + 8*b**2*x + 4*b*c + 8*
b*sqrt(a + b*x)*sqrt(b*x + c)) + a/(4*a*b + 8*b**2*x + 4*b*c + 8*b*sqrt(a + b*x)
*sqrt(b*x + c)) + 4*b*x*log(sqrt(a + b*x) + sqrt(b*x + c))/(4*a*b + 8*b**2*x + 4
*b*c + 8*b*sqrt(a + b*x)*sqrt(b*x + c)) + 2*b*x/(4*a*b + 8*b**2*x + 4*b*c + 8*b*
sqrt(a + b*x)*sqrt(b*x + c)) + 2*c*log(sqrt(a + b*x) + sqrt(b*x + c))/(4*a*b + 8
*b**2*x + 4*b*c + 8*b*sqrt(a + b*x)*sqrt(b*x + c)) + c/(4*a*b + 8*b**2*x + 4*b*c
+ 8*b*sqrt(a + b*x)*sqrt(b*x + c)) + 4*sqrt(a + b*x)*sqrt(b*x + c)*log(sqrt(a +
b*x) + sqrt(b*x + c))/(4*a*b + 8*b**2*x + 4*b*c + 8*b*sqrt(a + b*x)*sqrt(b*x +
c)), Ne(b, 0)), (x/(sqrt(a) + sqrt(c))**2, True))

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(b*x + a) + sqrt(b*x + c))^(-2),x, algorithm="giac")

[Out]

Timed out