3.281 \(\int \frac{1}{\left (\sqrt{a+b x}+\sqrt{a+c x}\right )^3} \, dx\)

Optimal. Leaf size=164 \[ -\frac{2 a \sqrt{a+b x}}{x^2 (b-c)^3}+\frac{2 a \sqrt{a+c x}}{x^2 (b-c)^3}-\frac{(2 b+3 c) \sqrt{a+b x}}{x (b-c)^3}+\frac{(3 b+2 c) \sqrt{a+c x}}{x (b-c)^3}-\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}+\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3} \]

[Out]

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

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Rubi [A]  time = 0.410479, antiderivative size = 275, normalized size of antiderivative = 1.68, number of steps used = 16, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}-\frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}-\frac{2 a \sqrt{a+b x}}{x^2 (b-c)^3}+\frac{2 a \sqrt{a+c x}}{x^2 (b-c)^3}-\frac{b \sqrt{a+b x}}{x (b-c)^3}-\frac{(b+3 c) \sqrt{a+b x}}{x (b-c)^3}+\frac{c \sqrt{a+c x}}{x (b-c)^3}+\frac{(3 b+c) \sqrt{a+c x}}{x (b-c)^3}-\frac{b (b+3 c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3}+\frac{c (3 b+c) \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)^3} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x] + Sqrt[a + c*x])^(-3),x]

[Out]

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

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Rubi in Sympy [A]  time = 43.3717, size = 236, normalized size = 1.44 \[ - \frac{2 a \sqrt{a + b x}}{x^{2} \left (b - c\right )^{3}} + \frac{2 a \sqrt{a + c x}}{x^{2} \left (b - c\right )^{3}} - \frac{b \sqrt{a + b x}}{x \left (b - c\right )^{3}} + \frac{c \sqrt{a + c x}}{x \left (b - c\right )^{3}} - \frac{\sqrt{a + b x} \left (b + 3 c\right )}{x \left (b - c\right )^{3}} + \frac{\sqrt{a + c x} \left (3 b + c\right )}{x \left (b - c\right )^{3}} + \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a} \left (b - c\right )^{3}} - \frac{b \left (b + 3 c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a} \left (b - c\right )^{3}} - \frac{c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x}}{\sqrt{a}} \right )}}{\sqrt{a} \left (b - c\right )^{3}} + \frac{c \left (3 b + c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + c x}}{\sqrt{a}} \right )}}{\sqrt{a} \left (b - c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((b*x+a)**(1/2)+(c*x+a)**(1/2))**3,x)

[Out]

-2*a*sqrt(a + b*x)/(x**2*(b - c)**3) + 2*a*sqrt(a + c*x)/(x**2*(b - c)**3) - b*s
qrt(a + b*x)/(x*(b - c)**3) + c*sqrt(a + c*x)/(x*(b - c)**3) - sqrt(a + b*x)*(b
+ 3*c)/(x*(b - c)**3) + sqrt(a + c*x)*(3*b + c)/(x*(b - c)**3) + b**2*atanh(sqrt
(a + b*x)/sqrt(a))/(sqrt(a)*(b - c)**3) - b*(b + 3*c)*atanh(sqrt(a + b*x)/sqrt(a
))/(sqrt(a)*(b - c)**3) - c**2*atanh(sqrt(a + c*x)/sqrt(a))/(sqrt(a)*(b - c)**3)
+ c*(3*b + c)*atanh(sqrt(a + c*x)/sqrt(a))/(sqrt(a)*(b - c)**3)

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Mathematica [A]  time = 0.274927, size = 146, normalized size = 0.89 \[ \frac{-3 b c x^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+3 b c x^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )+\sqrt{a} \left (-3 c x \sqrt{a+b x}+3 b x \sqrt{a+c x}-2 a \sqrt{a+b x}-2 b x \sqrt{a+b x}+2 a \sqrt{a+c x}+2 c x \sqrt{a+c x}\right )}{\sqrt{a} x^2 (b-c)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x] + Sqrt[a + c*x])^(-3),x]

[Out]

(Sqrt[a]*(-2*a*Sqrt[a + b*x] - 2*b*x*Sqrt[a + b*x] - 3*c*x*Sqrt[a + b*x] + 2*a*S
qrt[a + c*x] + 3*b*x*Sqrt[a + c*x] + 2*c*x*Sqrt[a + c*x]) - 3*b*c*x^2*ArcTanh[Sq
rt[a + b*x]/Sqrt[a]] + 3*b*c*x^2*ArcTanh[Sqrt[a + c*x]/Sqrt[a]])/(Sqrt[a]*(b - c
)^3*x^2)

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Maple [B]  time = 0.004, size = 300, normalized size = 1.8 \[ 2\,{\frac{{b}^{2}}{ \left ( b-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }+8\,{\frac{a{b}^{2}}{ \left ( b-c \right ) ^{3}} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( -1/8\,{\frac{ \left ( bx+a \right ) ^{3/2}}{a}}-1/8\,\sqrt{bx+a} \right ) }+1/8\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-8\,{\frac{a{c}^{2}}{ \left ( b-c \right ) ^{3}} \left ({\frac{1}{{c}^{2}{x}^{2}} \left ( -1/8\,{\frac{ \left ( cx+a \right ) ^{3/2}}{a}}-1/8\,\sqrt{cx+a} \right ) }+1/8\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) } \right ) }+6\,{\frac{bc}{ \left ( b-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-6\,{\frac{bc}{ \left ( b-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{cx+a}}{cx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) } \right ) }-2\,{\frac{{c}^{2}}{ \left ( b-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{cx+a}}{cx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x)

[Out]

2/(b-c)^3*b^2*(-1/2*(b*x+a)^(1/2)/x/b-1/2*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2)
)+8/(b-c)^3*a*b^2*((-1/8/a*(b*x+a)^(3/2)-1/8*(b*x+a)^(1/2))/x^2/b^2+1/8/a^(3/2)*
arctanh((b*x+a)^(1/2)/a^(1/2)))-8/(b-c)^3*a*c^2*((-1/8/a*(c*x+a)^(3/2)-1/8*(c*x+
a)^(1/2))/c^2/x^2+1/8/a^(3/2)*arctanh((c*x+a)^(1/2)/a^(1/2)))+6/(b-c)^3*c*b*(-1/
2*(b*x+a)^(1/2)/x/b-1/2*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2))-6/(b-c)^3*b*c*(-
1/2*(c*x+a)^(1/2)/c/x-1/2/a^(1/2)*arctanh((c*x+a)^(1/2)/a^(1/2)))-2/(b-c)^3*c^2*
(-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. \[ \int \frac{1}{{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(b*x + a) + sqrt(c*x + a))^(-3),x, algorithm="maxima")

[Out]

integrate((sqrt(b*x + a) + sqrt(c*x + a))^(-3), x)

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Fricas [A]  time = 0.296504, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \, b c x^{2} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 3 \, b c x^{2} \log \left (\frac{{\left (c x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x + a} a}{x}\right ) + 2 \,{\left ({\left (2 \, b + 3 \, c\right )} x + 2 \, a\right )} \sqrt{b x + a} \sqrt{a} - 2 \,{\left ({\left (3 \, b + 2 \, c\right )} x + 2 \, a\right )} \sqrt{c x + a} \sqrt{a}}{2 \,{\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3}\right )} \sqrt{a} x^{2}}, \frac{3 \, b c x^{2} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) - 3 \, b c x^{2} \arctan \left (\frac{a}{\sqrt{c x + a} \sqrt{-a}}\right ) -{\left ({\left (2 \, b + 3 \, c\right )} x + 2 \, a\right )} \sqrt{b x + a} \sqrt{-a} +{\left ({\left (3 \, b + 2 \, c\right )} x + 2 \, a\right )} \sqrt{c x + a} \sqrt{-a}}{{\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3}\right )} \sqrt{-a} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(b*x + a) + sqrt(c*x + a))^(-3),x, algorithm="fricas")

[Out]

[-1/2*(3*b*c*x^2*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*a)/x) + 3*b*c*x^2*lo
g(((c*x + 2*a)*sqrt(a) - 2*sqrt(c*x + a)*a)/x) + 2*((2*b + 3*c)*x + 2*a)*sqrt(b*
x + a)*sqrt(a) - 2*((3*b + 2*c)*x + 2*a)*sqrt(c*x + a)*sqrt(a))/((b^3 - 3*b^2*c
+ 3*b*c^2 - c^3)*sqrt(a)*x^2), (3*b*c*x^2*arctan(a/(sqrt(b*x + a)*sqrt(-a))) - 3
*b*c*x^2*arctan(a/(sqrt(c*x + a)*sqrt(-a))) - ((2*b + 3*c)*x + 2*a)*sqrt(b*x + a
)*sqrt(-a) + ((3*b + 2*c)*x + 2*a)*sqrt(c*x + a)*sqrt(-a))/((b^3 - 3*b^2*c + 3*b
*c^2 - c^3)*sqrt(-a)*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (\sqrt{a + b x} + \sqrt{a + c x}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x+a)**(1/2)+(c*x+a)**(1/2))**3,x)

[Out]

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

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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(c*x + a))^(-3),x, algorithm="giac")

[Out]

Timed out