Optimal. Leaf size=209 \[ -\frac{\left (4 a c-5 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{64 a^{7/2}}-\frac{x \left (4 a c-5 b^2 d\right ) \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{32 a^3}-\frac{5 b d^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{12 a^2 \left (\frac{d}{x}\right )^{3/2}}+\frac{x^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{2 a} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.691702, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{\left (4 a c-5 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{64 a^{7/2}}-\frac{x \left (4 a c-5 b^2 d\right ) \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{32 a^3}-\frac{5 b d^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{12 a^2 \left (\frac{d}{x}\right )^{3/2}}+\frac{x^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{2 a} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*Sqrt[d/x] + c/x]*x,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 49.0968, size = 172, normalized size = 0.82 \[ \frac{x^{2} \left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}}}{2 a} - \frac{5 b d^{2} \left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}}}{12 a^{2} \left (\frac{d}{x}\right )^{\frac{3}{2}}} - \frac{x \left (2 a + b \sqrt{\frac{d}{x}}\right ) \left (4 a c - 5 b^{2} d\right ) \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{32 a^{3}} - \frac{\left (4 a c - 5 b^{2} d\right ) \left (4 a c - b^{2} d\right ) \operatorname{atanh}{\left (\frac{2 a + b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{64 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(a+c/x+b*(d/x)**(1/2))**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.188334, size = 0, normalized size = 0. \[ \int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x \, dx \]
Verification is Not applicable to the result.
[In] Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]*x,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.038, size = 398, normalized size = 1.9 \[{\frac{1}{192}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }}\sqrt{x} \left ( 30\,{a}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3/2}{b}^{3}+60\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d\sqrt{x}{b}^{2}-15\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){d}^{2}a{b}^{4}+96\,\sqrt{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}{a}^{7/2}-80\,{a}^{5/2} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}\sqrt{{\frac{d}{x}}}\sqrt{x}b-48\,{a}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{x}c-24\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}\sqrt{x}bc+72\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) d{a}^{2}{b}^{2}c-48\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{3}{c}^{2} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{a}^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(a+c/x+b*(d/x)^(1/2))^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(d/x) + a + c/x)*x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [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*sqrt(d/x) + a + c/x)*x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(a+c/x+b*(d/x)**(1/2))**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.332584, size = 360, normalized size = 1.72 \[ \frac{1}{192} \,{\left (2 \, \sqrt{b \sqrt{d} \sqrt{x} + a x + c}{\left (2 \,{\left (4 \, \sqrt{x}{\left (\frac{b \sqrt{d}}{a} + 6 \, \sqrt{x}\right )} - \frac{5 \, a b^{2} d - 12 \, a^{2} c}{a^{3}}\right )} \sqrt{x} + \frac{15 \, b^{3} d^{\frac{3}{2}} - 52 \, a b c \sqrt{d}}{a^{3}}\right )} + \frac{3 \,{\left (5 \, b^{4} d^{2} - 24 \, a b^{2} c d + 16 \, a^{2} c^{2}\right )}{\rm ln}\left ({\left | -b \sqrt{d} - 2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{b \sqrt{d} \sqrt{x} + a x + c}\right )} \right |}\right )}{a^{\frac{7}{2}}}\right )}{\rm sign}\left (x\right ) - \frac{{\left (15 \, b^{4} d^{2}{\rm ln}\left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 72 \, a b^{2} c d{\rm ln}\left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 30 \, \sqrt{a} b^{3} \sqrt{c} d^{\frac{3}{2}} + 48 \, a^{2} c^{2}{\rm ln}\left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 104 \, a^{\frac{3}{2}} b c^{\frac{3}{2}} \sqrt{d}\right )}{\rm sign}\left (x\right )}{192 \, a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(d/x) + a + c/x)*x,x, algorithm="giac")
[Out]