Optimal. Leaf size=174 \[ -\frac{63 b^5 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{64 a^{11/2}}+\frac{63 b^4 \sqrt{a x+b \sqrt{x}}}{64 a^5}-\frac{21 b^3 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{32 a^4}+\frac{21 b^2 x \sqrt{a x+b \sqrt{x}}}{40 a^3}-\frac{9 b x^{3/2} \sqrt{a x+b \sqrt{x}}}{20 a^2}+\frac{2 x^2 \sqrt{a x+b \sqrt{x}}}{5 a} \]
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Rubi [A] time = 0.334354, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{63 b^5 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{64 a^{11/2}}+\frac{63 b^4 \sqrt{a x+b \sqrt{x}}}{64 a^5}-\frac{21 b^3 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{32 a^4}+\frac{21 b^2 x \sqrt{a x+b \sqrt{x}}}{40 a^3}-\frac{9 b x^{3/2} \sqrt{a x+b \sqrt{x}}}{20 a^2}+\frac{2 x^2 \sqrt{a x+b \sqrt{x}}}{5 a} \]
Antiderivative was successfully verified.
[In] Int[x^2/Sqrt[b*Sqrt[x] + a*x],x]
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Rubi in Sympy [A] time = 33.2406, size = 163, normalized size = 0.94 \[ \frac{2 x^{2} \sqrt{a x + b \sqrt{x}}}{5 a} - \frac{9 b x^{\frac{3}{2}} \sqrt{a x + b \sqrt{x}}}{20 a^{2}} + \frac{21 b^{2} x \sqrt{a x + b \sqrt{x}}}{40 a^{3}} - \frac{21 b^{3} \sqrt{x} \sqrt{a x + b \sqrt{x}}}{32 a^{4}} + \frac{63 b^{4} \sqrt{a x + b \sqrt{x}}}{64 a^{5}} - \frac{63 b^{5} \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b \sqrt{x}}} \right )}}{64 a^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x**(1/2)+a*x)**(1/2),x)
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Mathematica [A] time = 0.163727, size = 115, normalized size = 0.66 \[ \frac{2 \sqrt{a} \sqrt{a x+b \sqrt{x}} \left (128 a^4 x^2-144 a^3 b x^{3/2}+168 a^2 b^2 x-210 a b^3 \sqrt{x}+315 b^4\right )-315 b^5 \log \left (2 \sqrt{a} \sqrt{a x+b \sqrt{x}}+2 a \sqrt{x}+b\right )}{640 a^{11/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/Sqrt[b*Sqrt[x] + a*x],x]
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Maple [A] time = 0.023, size = 227, normalized size = 1.3 \[{\frac{1}{640}\sqrt{b\sqrt{x}+ax} \left ( -544\,b\sqrt{x} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{15/2}+256\,x \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{17/2}+880\,{b}^{2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{13/2}-1300\,{b}^{3}\sqrt{b\sqrt{x}+ax}\sqrt{x}{a}^{13/2}+1280\,{b}^{4}\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{a}^{11/2}-650\,{b}^{4}\sqrt{b\sqrt{x}+ax}{a}^{11/2}+325\,{b}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5}-640\,{b}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{a}^{-{\frac{21}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x^(1/2)+a*x)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(a*x + b*sqrt(x)),x, algorithm="maxima")
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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(x^2/sqrt(a*x + b*sqrt(x)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a x + b \sqrt{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x**(1/2)+a*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.279174, size = 150, normalized size = 0.86 \[ \frac{1}{320} \, \sqrt{a x + b \sqrt{x}}{\left (2 \,{\left (4 \,{\left (2 \, \sqrt{x}{\left (\frac{8 \, \sqrt{x}}{a} - \frac{9 \, b}{a^{2}}\right )} + \frac{21 \, b^{2}}{a^{3}}\right )} \sqrt{x} - \frac{105 \, b^{3}}{a^{4}}\right )} \sqrt{x} + \frac{315 \, b^{4}}{a^{5}}\right )} + \frac{63 \, b^{5}{\rm ln}\left ({\left | -2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} - b \right |}\right )}{128 \, a^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(a*x + b*sqrt(x)),x, algorithm="giac")
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