3.254 \(\int \frac{1}{x^2 \left (\sqrt{a+b x}+\sqrt{c+b x}\right )^3} \, dx\)

Optimal. Leaf size=162 \[ \frac{8 b \sqrt{a+b x}}{(a-c)^3}-\frac{8 b \sqrt{b x+c}}{(a-c)^3}-\frac{(a+3 c) \sqrt{a+b x}}{x (a-c)^3}+\frac{(3 a+c) \sqrt{b x+c}}{x (a-c)^3}-\frac{3 b (3 a+c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (a-c)^3}-\frac{3 b (a+3 c) \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{\sqrt{c} (c-a)^3} \]

[Out]

(8*b*Sqrt[a + b*x])/(a - c)^3 - ((a + 3*c)*Sqrt[a + b*x])/((a - c)^3*x) - (8*b*S
qrt[c + b*x])/(a - c)^3 + ((3*a + c)*Sqrt[c + b*x])/((a - c)^3*x) - (3*b*(3*a +
c)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(Sqrt[a]*(a - c)^3) - (3*b*(a + 3*c)*ArcTanh[
Sqrt[c + b*x]/Sqrt[c]])/(Sqrt[c]*(-a + c)^3)

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Rubi [A]  time = 0.564657, antiderivative size = 223, normalized size of antiderivative = 1.38, number of steps used = 14, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{8 b \sqrt{a+b x}}{(a-c)^3}-\frac{8 b \sqrt{b x+c}}{(a-c)^3}-\frac{(a+3 c) \sqrt{a+b x}}{x (a-c)^3}+\frac{(3 a+c) \sqrt{b x+c}}{x (a-c)^3}-\frac{b (a+3 c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (a-c)^3}-\frac{8 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(a-c)^3}+\frac{b (3 a+c) \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{\sqrt{c} (a-c)^3}+\frac{8 b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{(a-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

(8*b*Sqrt[a + b*x])/(a - c)^3 - ((a + 3*c)*Sqrt[a + b*x])/((a - c)^3*x) - (8*b*S
qrt[c + b*x])/(a - c)^3 + ((3*a + c)*Sqrt[c + b*x])/((a - c)^3*x) - (8*Sqrt[a]*b
*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(a - c)^3 - (b*(a + 3*c)*ArcTanh[Sqrt[a + b*x]/
Sqrt[a]])/(Sqrt[a]*(a - c)^3) + (8*b*Sqrt[c]*ArcTanh[Sqrt[c + b*x]/Sqrt[c]])/(a
- c)^3 + (b*(3*a + c)*ArcTanh[Sqrt[c + b*x]/Sqrt[c]])/((a - c)^3*Sqrt[c])

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Rubi in Sympy [A]  time = 41.0077, size = 196, normalized size = 1.21 \[ - \frac{8 \sqrt{a} b \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\left (a - c\right )^{3}} + \frac{8 b \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{b x + c}}{\sqrt{c}} \right )}}{\left (a - c\right )^{3}} + \frac{8 b \sqrt{a + b x}}{\left (a - c\right )^{3}} - \frac{8 b \sqrt{b x + c}}{\left (a - c\right )^{3}} + \frac{b \left (3 a + c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c}}{\sqrt{c}} \right )}}{\sqrt{c} \left (a - c\right )^{3}} - \frac{\left (a + 3 c\right ) \sqrt{a + b x}}{x \left (a - c\right )^{3}} + \frac{\left (3 a + c\right ) \sqrt{b x + c}}{x \left (a - c\right )^{3}} - \frac{b \left (a + 3 c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a} \left (a - c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*sqrt(a)*b*atanh(sqrt(a + b*x)/sqrt(a))/(a - c)**3 + 8*b*sqrt(c)*atanh(sqrt(b*
x + c)/sqrt(c))/(a - c)**3 + 8*b*sqrt(a + b*x)/(a - c)**3 - 8*b*sqrt(b*x + c)/(a
- c)**3 + b*(3*a + c)*atanh(sqrt(b*x + c)/sqrt(c))/(sqrt(c)*(a - c)**3) - (a +
3*c)*sqrt(a + b*x)/(x*(a - c)**3) + (3*a + c)*sqrt(b*x + c)/(x*(a - c)**3) - b*(
a + 3*c)*atanh(sqrt(a + b*x)/sqrt(a))/(sqrt(a)*(a - c)**3)

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Mathematica [A]  time = 0.511731, size = 112, normalized size = 0.69 \[ \frac{\frac{\sqrt{b x+c} (3 a-8 b x+c)}{x}-\frac{\sqrt{a+b x} (a-8 b x+3 c)}{x}-\frac{3 b (3 a+c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}+\frac{3 b (a+3 c) \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{\sqrt{c}}}{(a-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-(((a + 3*c - 8*b*x)*Sqrt[a + b*x])/x) + ((3*a + c - 8*b*x)*Sqrt[c + b*x])/x -
(3*b*(3*a + c)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/Sqrt[a] + (3*b*(a + 3*c)*ArcTanh[
Sqrt[c + b*x]/Sqrt[c]])/Sqrt[c])/(a - c)^3

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Maple [A]  time = 0.004, size = 252, normalized size = 1.6 \[ 2\,{\frac{ab}{ \left ( a-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }+6\,{\frac{bc}{ \left ( a-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-6\,{\frac{ab}{ \left ( a-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{bx+c}}{bx}}-1/2\,{\frac{1}{\sqrt{c}}{\it Artanh} \left ({\frac{\sqrt{bx+c}}{\sqrt{c}}} \right ) } \right ) }-2\,{\frac{bc}{ \left ( a-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{bx+c}}{bx}}-1/2\,{\frac{1}{\sqrt{c}}{\it Artanh} \left ({\frac{\sqrt{bx+c}}{\sqrt{c}}} \right ) } \right ) }+4\,{\frac{b}{ \left ( a-c \right ) ^{3}} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }-4\,{\frac{b}{ \left ( a-c \right ) ^{3}} \left ( 2\,\sqrt{bx+c}-2\,\sqrt{c}{\it Artanh} \left ({\frac{\sqrt{bx+c}}{\sqrt{c}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(a-c)^3*a*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/(a-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/(a-c)^3*a*b*(-1/2*(b*x+c)^(1/2)/x/b-1/2/c^(1/2)*arctanh((b*x+c)^(1/2)/c^(1
/2)))-2/(a-c)^3*c*b*(-1/2*(b*x+c)^(1/2)/x/b-1/2/c^(1/2)*arctanh((b*x+c)^(1/2)/c^
(1/2)))+4/(a-c)^3*b*(2*(b*x+a)^(1/2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))-4
/(a-c)^3*b*(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. \[ \int \frac{1}{x^{2}{\left (\sqrt{b x + a} + \sqrt{b x + c}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^2*(sqrt(b*x + a) + sqrt(b*x + c))^3),x, algorithm="maxima")

[Out]

integrate(1/(x^2*(sqrt(b*x + a) + sqrt(b*x + c))^3), x)

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Fricas [A]  time = 0.317618, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (3 \, a b + b c\right )} \sqrt{c} x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 3 \,{\left (a b + 3 \, b c\right )} \sqrt{a} x \log \left (\frac{{\left (b x + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{b x + c} c}{x}\right ) - 2 \,{\left (8 \, b x - a - 3 \, c\right )} \sqrt{b x + a} \sqrt{a} \sqrt{c} + 2 \,{\left (8 \, b x - 3 \, a - c\right )} \sqrt{b x + c} \sqrt{a} \sqrt{c}}{2 \,{\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )} \sqrt{a} \sqrt{c} x}, -\frac{6 \,{\left (a b + 3 \, b c\right )} \sqrt{a} x \arctan \left (\frac{c}{\sqrt{b x + c} \sqrt{-c}}\right ) + 3 \,{\left (3 \, a b + b c\right )} \sqrt{-c} x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) - 2 \,{\left (8 \, b x - a - 3 \, c\right )} \sqrt{b x + a} \sqrt{a} \sqrt{-c} + 2 \,{\left (8 \, b x - 3 \, a - c\right )} \sqrt{b x + c} \sqrt{a} \sqrt{-c}}{2 \,{\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )} \sqrt{a} \sqrt{-c} x}, \frac{6 \,{\left (3 \, a b + b c\right )} \sqrt{c} x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) - 3 \,{\left (a b + 3 \, b c\right )} \sqrt{-a} x \log \left (\frac{{\left (b x + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{b x + c} c}{x}\right ) + 2 \,{\left (8 \, b x - a - 3 \, c\right )} \sqrt{b x + a} \sqrt{-a} \sqrt{c} - 2 \,{\left (8 \, b x - 3 \, a - c\right )} \sqrt{b x + c} \sqrt{-a} \sqrt{c}}{2 \,{\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )} \sqrt{-a} \sqrt{c} x}, \frac{3 \,{\left (3 \, a b + b c\right )} \sqrt{-c} x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) - 3 \,{\left (a b + 3 \, b c\right )} \sqrt{-a} x \arctan \left (\frac{c}{\sqrt{b x + c} \sqrt{-c}}\right ) +{\left (8 \, b x - a - 3 \, c\right )} \sqrt{b x + a} \sqrt{-a} \sqrt{-c} -{\left (8 \, b x - 3 \, a - c\right )} \sqrt{b x + c} \sqrt{-a} \sqrt{-c}}{{\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )} \sqrt{-a} \sqrt{-c} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^2*(sqrt(b*x + a) + sqrt(b*x + c))^3),x, algorithm="fricas")

[Out]

[-1/2*(3*(3*a*b + b*c)*sqrt(c)*x*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*a)/x
) + 3*(a*b + 3*b*c)*sqrt(a)*x*log(((b*x + 2*c)*sqrt(c) - 2*sqrt(b*x + c)*c)/x) -
2*(8*b*x - a - 3*c)*sqrt(b*x + a)*sqrt(a)*sqrt(c) + 2*(8*b*x - 3*a - c)*sqrt(b*
x + c)*sqrt(a)*sqrt(c))/((a^3 - 3*a^2*c + 3*a*c^2 - c^3)*sqrt(a)*sqrt(c)*x), -1/
2*(6*(a*b + 3*b*c)*sqrt(a)*x*arctan(c/(sqrt(b*x + c)*sqrt(-c))) + 3*(3*a*b + b*c
)*sqrt(-c)*x*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*a)/x) - 2*(8*b*x - a - 3
*c)*sqrt(b*x + a)*sqrt(a)*sqrt(-c) + 2*(8*b*x - 3*a - c)*sqrt(b*x + c)*sqrt(a)*s
qrt(-c))/((a^3 - 3*a^2*c + 3*a*c^2 - c^3)*sqrt(a)*sqrt(-c)*x), 1/2*(6*(3*a*b + b
*c)*sqrt(c)*x*arctan(a/(sqrt(b*x + a)*sqrt(-a))) - 3*(a*b + 3*b*c)*sqrt(-a)*x*lo
g(((b*x + 2*c)*sqrt(c) - 2*sqrt(b*x + c)*c)/x) + 2*(8*b*x - a - 3*c)*sqrt(b*x +
a)*sqrt(-a)*sqrt(c) - 2*(8*b*x - 3*a - c)*sqrt(b*x + c)*sqrt(-a)*sqrt(c))/((a^3
- 3*a^2*c + 3*a*c^2 - c^3)*sqrt(-a)*sqrt(c)*x), (3*(3*a*b + b*c)*sqrt(-c)*x*arct
an(a/(sqrt(b*x + a)*sqrt(-a))) - 3*(a*b + 3*b*c)*sqrt(-a)*x*arctan(c/(sqrt(b*x +
c)*sqrt(-c))) + (8*b*x - a - 3*c)*sqrt(b*x + a)*sqrt(-a)*sqrt(-c) - (8*b*x - 3*
a - c)*sqrt(b*x + c)*sqrt(-a)*sqrt(-c))/((a^3 - 3*a^2*c + 3*a*c^2 - c^3)*sqrt(-a
)*sqrt(-c)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (\sqrt{a + b x} + \sqrt{b x + c}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*(sqrt(a + b*x) + sqrt(b*x + c))**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(1/(x^2*(sqrt(b*x + a) + sqrt(b*x + c))^3),x, algorithm="giac")

[Out]

Timed out