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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]