Optimal. Leaf size=137 \[ \frac{256 b^4 \sqrt{a x+b x^{2/3}}}{105 a^5 \sqrt [3]{x}}-\frac{128 b^3 \sqrt{a x+b x^{2/3}}}{105 a^4}+\frac{32 b^2 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{35 a^3}-\frac{16 b x^{2/3} \sqrt{a x+b x^{2/3}}}{21 a^2}+\frac{2 x \sqrt{a x+b x^{2/3}}}{3 a} \]
[Out]
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Rubi [A] time = 0.307186, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{256 b^4 \sqrt{a x+b x^{2/3}}}{105 a^5 \sqrt [3]{x}}-\frac{128 b^3 \sqrt{a x+b x^{2/3}}}{105 a^4}+\frac{32 b^2 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{35 a^3}-\frac{16 b x^{2/3} \sqrt{a x+b x^{2/3}}}{21 a^2}+\frac{2 x \sqrt{a x+b x^{2/3}}}{3 a} \]
Antiderivative was successfully verified.
[In] Int[x/Sqrt[b*x^(2/3) + a*x],x]
[Out]
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Rubi in Sympy [A] time = 26.4664, size = 128, normalized size = 0.93 \[ \frac{2 x \sqrt{a x + b x^{\frac{2}{3}}}}{3 a} - \frac{16 b x^{\frac{2}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}{21 a^{2}} + \frac{32 b^{2} \sqrt [3]{x} \sqrt{a x + b x^{\frac{2}{3}}}}{35 a^{3}} - \frac{128 b^{3} \sqrt{a x + b x^{\frac{2}{3}}}}{105 a^{4}} + \frac{256 b^{4} \sqrt{a x + b x^{\frac{2}{3}}}}{105 a^{5} \sqrt [3]{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x**(2/3)+a*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0424941, size = 74, normalized size = 0.54 \[ \frac{2 \sqrt{a x+b x^{2/3}} \left (35 a^4 x^{4/3}-40 a^3 b x+48 a^2 b^2 x^{2/3}-64 a b^3 \sqrt [3]{x}+128 b^4\right )}{105 a^5 \sqrt [3]{x}} \]
Antiderivative was successfully verified.
[In] Integrate[x/Sqrt[b*x^(2/3) + a*x],x]
[Out]
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Maple [A] time = 0.007, size = 68, normalized size = 0.5 \[{\frac{2}{105\,{a}^{5}}\sqrt [3]{x} \left ( b+a\sqrt [3]{x} \right ) \left ( 35\,{x}^{4/3}{a}^{4}-40\,x{a}^{3}b+48\,{x}^{2/3}{a}^{2}{b}^{2}-64\,\sqrt [3]{x}a{b}^{3}+128\,{b}^{4} \right ){\frac{1}{\sqrt{b{x}^{{\frac{2}{3}}}+ax}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x^(2/3)+a*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.46307, size = 109, normalized size = 0.8 \[ \frac{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}}}{3 \, a^{5}} - \frac{24 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b}{7 \, a^{5}} + \frac{36 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{2}}{5 \, a^{5}} - \frac{8 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{3}}{a^{5}} + \frac{6 \, \sqrt{a x^{\frac{1}{3}} + b} b^{4}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(a*x + b*x^(2/3)),x, algorithm="maxima")
[Out]
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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/sqrt(a*x + b*x^(2/3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{a x + b x^{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x**(2/3)+a*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.224448, size = 142, normalized size = 1.04 \[ -\frac{256 \, b^{\frac{9}{2}}{\rm sign}\left (x^{\frac{1}{3}}\right )}{105 \, a^{5}} + \frac{2 \,{\left (35 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{32} - 180 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{32} b + 378 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{32} b^{2} - 420 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{32} b^{3} + 315 \, \sqrt{a x^{\frac{1}{3}} + b} a^{32} b^{4}\right )}}{105 \, a^{37}{\rm sign}\left (x^{\frac{1}{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(a*x + b*x^(2/3)),x, algorithm="giac")
[Out]