Optimal. Leaf size=142 \[ -\frac{512 a^4 \sqrt{a x+b \sqrt{x}}}{315 b^5 \sqrt{x}}+\frac{256 a^3 \sqrt{a x+b \sqrt{x}}}{315 b^4 x}-\frac{64 a^2 \sqrt{a x+b \sqrt{x}}}{105 b^3 x^{3/2}}+\frac{32 a \sqrt{a x+b \sqrt{x}}}{63 b^2 x^2}-\frac{4 \sqrt{a x+b \sqrt{x}}}{9 b x^{5/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.348738, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{512 a^4 \sqrt{a x+b \sqrt{x}}}{315 b^5 \sqrt{x}}+\frac{256 a^3 \sqrt{a x+b \sqrt{x}}}{315 b^4 x}-\frac{64 a^2 \sqrt{a x+b \sqrt{x}}}{105 b^3 x^{3/2}}+\frac{32 a \sqrt{a x+b \sqrt{x}}}{63 b^2 x^2}-\frac{4 \sqrt{a x+b \sqrt{x}}}{9 b x^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*Sqrt[b*Sqrt[x] + a*x]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 32.0236, size = 131, normalized size = 0.92 \[ - \frac{512 a^{4} \sqrt{a x + b \sqrt{x}}}{315 b^{5} \sqrt{x}} + \frac{256 a^{3} \sqrt{a x + b \sqrt{x}}}{315 b^{4} x} - \frac{64 a^{2} \sqrt{a x + b \sqrt{x}}}{105 b^{3} x^{\frac{3}{2}}} + \frac{32 a \sqrt{a x + b \sqrt{x}}}{63 b^{2} x^{2}} - \frac{4 \sqrt{a x + b \sqrt{x}}}{9 b x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**(1/2)+a*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0443692, size = 72, normalized size = 0.51 \[ -\frac{4 \sqrt{a x+b \sqrt{x}} \left (128 a^4 x^2-64 a^3 b x^{3/2}+48 a^2 b^2 x-40 a b^3 \sqrt{x}+35 b^4\right )}{315 b^5 x^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*Sqrt[b*Sqrt[x] + a*x]),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.016, size = 254, normalized size = 1.8 \[{\frac{1}{315\,{b}^{6}}\sqrt{b\sqrt{x}+ax} \left ( 315\,{a}^{9/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ) b{x}^{11/2}-315\,{a}^{9/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ) b{x}^{11/2}+630\,{a}^{5}\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{11/2}-1260\,{a}^{4} \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{9/2}+630\,{a}^{5}\sqrt{b\sqrt{x}+ax}{x}^{11/2}-492\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{7/2}{a}^{2}{b}^{2}-140\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{5/2}{b}^{4}+748\,{a}^{3} \left ( b\sqrt{x}+ax \right ) ^{3/2}b{x}^{4}+300\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{3}a{b}^{3} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{x}^{-{\frac{11}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^(1/2)+a*x)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 5.31708, size = 116, normalized size = 0.82 \[ -\frac{4 \,{\left (\frac{315 \, \sqrt{a \sqrt{x} + b} a^{4}}{x^{\frac{1}{4}}} - \frac{420 \,{\left (a \sqrt{x} + b\right )}^{\frac{3}{2}} a^{3}}{x^{\frac{3}{4}}} + \frac{378 \,{\left (a \sqrt{x} + b\right )}^{\frac{5}{2}} a^{2}}{x^{\frac{5}{4}}} - \frac{180 \,{\left (a \sqrt{x} + b\right )}^{\frac{7}{2}} a}{x^{\frac{7}{4}}} + \frac{35 \,{\left (a \sqrt{x} + b\right )}^{\frac{9}{2}}}{x^{\frac{9}{4}}}\right )}}{315 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a*x + b*sqrt(x))*x^3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.269194, size = 86, normalized size = 0.61 \[ \frac{4 \,{\left (64 \, a^{3} b x^{2} + 40 \, a b^{3} x -{\left (128 \, a^{4} x^{2} + 48 \, a^{2} b^{2} x + 35 \, b^{4}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{315 \, b^{5} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a*x + b*sqrt(x))*x^3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{a x + b \sqrt{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(b*x**(1/2)+a*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.226749, size = 197, normalized size = 1.39 \[ \frac{4 \,{\left (1008 \, a^{2}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{4} + 1680 \, a^{\frac{3}{2}} b{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{3} + 1080 \, a b^{2}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{2} + 315 \, \sqrt{a} b^{3}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} + 35 \, b^{4}\right )}}{315 \,{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a*x + b*sqrt(x))*x^3),x, algorithm="giac")
[Out]