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

Optimal. Leaf size=157 \[ \frac{8 (a+b x)^{3/2}}{3 (a-c)^3}+\frac{2 (a+3 c) \sqrt{a+b x}}{(a-c)^3}-\frac{8 (b x+c)^{3/2}}{3 (a-c)^3}-\frac{2 (3 a+c) \sqrt{b x+c}}{(a-c)^3}-\frac{2 \sqrt{a} (a+3 c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(a-c)^3}+\frac{2 \sqrt{c} (3 a+c) \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{(a-c)^3} \]

[Out]

(2*(a + 3*c)*Sqrt[a + b*x])/(a - c)^3 + (8*(a + b*x)^(3/2))/(3*(a - c)^3) - (2*(
3*a + c)*Sqrt[c + b*x])/(a - c)^3 - (8*(c + b*x)^(3/2))/(3*(a - c)^3) - (2*Sqrt[
a]*(a + 3*c)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(a - c)^3 + (2*Sqrt[c]*(3*a + c)*Ar
cTanh[Sqrt[c + b*x]/Sqrt[c]])/(a - c)^3

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Rubi [A]  time = 0.46371, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{8 (a+b x)^{3/2}}{3 (a-c)^3}+\frac{2 (a+3 c) \sqrt{a+b x}}{(a-c)^3}-\frac{8 (b x+c)^{3/2}}{3 (a-c)^3}-\frac{2 (3 a+c) \sqrt{b x+c}}{(a-c)^3}-\frac{2 \sqrt{a} (a+3 c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(a-c)^3}+\frac{2 \sqrt{c} (3 a+c) \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{(a-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + 3*c)*Sqrt[a + b*x])/(a - c)^3 + (8*(a + b*x)^(3/2))/(3*(a - c)^3) - (2*(
3*a + c)*Sqrt[c + b*x])/(a - c)^3 - (8*(c + b*x)^(3/2))/(3*(a - c)^3) - (2*Sqrt[
a]*(a + 3*c)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(a - c)^3 + (2*Sqrt[c]*(3*a + c)*Ar
cTanh[Sqrt[c + b*x]/Sqrt[c]])/(a - c)^3

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Rubi in Sympy [A]  time = 34.0811, size = 138, normalized size = 0.88 \[ - \frac{2 \sqrt{a} \left (a + 3 c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\left (a - c\right )^{3}} + \frac{2 \sqrt{c} \left (3 a + c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c}}{\sqrt{c}} \right )}}{\left (a - c\right )^{3}} + \frac{2 \left (a + 3 c\right ) \sqrt{a + b x}}{\left (a - c\right )^{3}} + \frac{8 \left (a + b x\right )^{\frac{3}{2}}}{3 \left (a - c\right )^{3}} - \frac{2 \left (3 a + c\right ) \sqrt{b x + c}}{\left (a - c\right )^{3}} - \frac{8 \left (b x + c\right )^{\frac{3}{2}}}{3 \left (a - c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a)*(a + 3*c)*atanh(sqrt(a + b*x)/sqrt(a))/(a - c)**3 + 2*sqrt(c)*(3*a +
c)*atanh(sqrt(b*x + c)/sqrt(c))/(a - c)**3 + 2*(a + 3*c)*sqrt(a + b*x)/(a - c)**
3 + 8*(a + b*x)**(3/2)/(3*(a - c)**3) - 2*(3*a + c)*sqrt(b*x + c)/(a - c)**3 - 8
*(b*x + c)**(3/2)/(3*(a - c)**3)

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Mathematica [A]  time = 0.242034, size = 142, normalized size = 0.9 \[ \frac{2 \left (-9 a \sqrt{b x+c}+9 c \sqrt{a+b x}-3 \sqrt{a} (a+3 c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+3 \sqrt{c} (3 a+c) \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )+7 a \sqrt{a+b x}+4 b x \sqrt{a+b x}-7 c \sqrt{b x+c}-4 b x \sqrt{b x+c}\right )}{3 (a-c)^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(7*a*Sqrt[a + b*x] + 9*c*Sqrt[a + b*x] + 4*b*x*Sqrt[a + b*x] - 9*a*Sqrt[c + b
*x] - 7*c*Sqrt[c + b*x] - 4*b*x*Sqrt[c + b*x] - 3*Sqrt[a]*(a + 3*c)*ArcTanh[Sqrt
[a + b*x]/Sqrt[a]] + 3*Sqrt[c]*(3*a + c)*ArcTanh[Sqrt[c + b*x]/Sqrt[c]]))/(3*(a
- c)^3)

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Maple [A]  time = 0.005, size = 181, normalized size = 1.2 \[{\frac{a}{ \left ( a-c \right ) ^{3}} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }+{\frac{8}{3\, \left ( a-c \right ) ^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{8}{3\, \left ( a-c \right ) ^{3}} \left ( bx+c \right ) ^{{\frac{3}{2}}}}+3\,{\frac{c}{ \left ( a-c \right ) ^{3}} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }-3\,{\frac{a}{ \left ( a-c \right ) ^{3}} \left ( 2\,\sqrt{bx+c}-2\,\sqrt{c}{\it Artanh} \left ({\frac{\sqrt{bx+c}}{\sqrt{c}}} \right ) \right ) }-{\frac{c}{ \left ( a-c \right ) ^{3}} \left ( 2\,\sqrt{bx+c}-2\,\sqrt{c}{\it Artanh} \left ({\frac{\sqrt{bx+c}}{\sqrt{c}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.31121, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (a + 3 \, c\right )} \sqrt{a} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 3 \,{\left (3 \, a + c\right )} \sqrt{c} \log \left (\frac{b x - 2 \, \sqrt{b x + c} \sqrt{c} + 2 \, c}{x}\right ) - 2 \,{\left (4 \, b x + 7 \, a + 9 \, c\right )} \sqrt{b x + a} + 2 \,{\left (4 \, b x + 9 \, a + 7 \, c\right )} \sqrt{b x + c}}{3 \,{\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )}}, \frac{6 \,{\left (3 \, a + c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{b x + c}}{\sqrt{-c}}\right ) - 3 \,{\left (a + 3 \, c\right )} \sqrt{a} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (4 \, b x + 7 \, a + 9 \, c\right )} \sqrt{b x + a} - 2 \,{\left (4 \, b x + 9 \, a + 7 \, c\right )} \sqrt{b x + c}}{3 \,{\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )}}, -\frac{6 \, \sqrt{-a}{\left (a + 3 \, c\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) + 3 \,{\left (3 \, a + c\right )} \sqrt{c} \log \left (\frac{b x - 2 \, \sqrt{b x + c} \sqrt{c} + 2 \, c}{x}\right ) - 2 \,{\left (4 \, b x + 7 \, a + 9 \, c\right )} \sqrt{b x + a} + 2 \,{\left (4 \, b x + 9 \, a + 7 \, c\right )} \sqrt{b x + c}}{3 \,{\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )}}, -\frac{2 \,{\left (3 \, \sqrt{-a}{\left (a + 3 \, c\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) - 3 \,{\left (3 \, a + c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{b x + c}}{\sqrt{-c}}\right ) -{\left (4 \, b x + 7 \, a + 9 \, c\right )} \sqrt{b x + a} +{\left (4 \, b x + 9 \, a + 7 \, c\right )} \sqrt{b x + c}\right )}}{3 \,{\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/3*(3*(a + 3*c)*sqrt(a)*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 3*(3*a
+ c)*sqrt(c)*log((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x) - 2*(4*b*x + 7*a + 9*
c)*sqrt(b*x + a) + 2*(4*b*x + 9*a + 7*c)*sqrt(b*x + c))/(a^3 - 3*a^2*c + 3*a*c^2
- c^3), 1/3*(6*(3*a + c)*sqrt(-c)*arctan(sqrt(b*x + c)/sqrt(-c)) - 3*(a + 3*c)*
sqrt(a)*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(4*b*x + 7*a + 9*c)*sqr
t(b*x + a) - 2*(4*b*x + 9*a + 7*c)*sqrt(b*x + c))/(a^3 - 3*a^2*c + 3*a*c^2 - c^3
), -1/3*(6*sqrt(-a)*(a + 3*c)*arctan(sqrt(b*x + a)/sqrt(-a)) + 3*(3*a + c)*sqrt(
c)*log((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x) - 2*(4*b*x + 7*a + 9*c)*sqrt(b*x
+ a) + 2*(4*b*x + 9*a + 7*c)*sqrt(b*x + c))/(a^3 - 3*a^2*c + 3*a*c^2 - c^3), -2
/3*(3*sqrt(-a)*(a + 3*c)*arctan(sqrt(b*x + a)/sqrt(-a)) - 3*(3*a + c)*sqrt(-c)*a
rctan(sqrt(b*x + c)/sqrt(-c)) - (4*b*x + 7*a + 9*c)*sqrt(b*x + a) + (4*b*x + 9*a
+ 7*c)*sqrt(b*x + c))/(a^3 - 3*a^2*c + 3*a*c^2 - c^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out