Optimal. Leaf size=50 \[ \frac{3 b \sqrt [3]{a+b x^{3/2}}}{2 a^2 \sqrt{x}}-\frac{\sqrt [3]{a+b x^{3/2}}}{2 a x^2} \]
[Out]
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Rubi [A] time = 0.0479379, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{3 b \sqrt [3]{a+b x^{3/2}}}{2 a^2 \sqrt{x}}-\frac{\sqrt [3]{a+b x^{3/2}}}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*x^(3/2))^(2/3)),x]
[Out]
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Rubi in Sympy [A] time = 4.46244, size = 42, normalized size = 0.84 \[ - \frac{\sqrt [3]{a + b x^{\frac{3}{2}}}}{2 a x^{2}} + \frac{3 b \sqrt [3]{a + b x^{\frac{3}{2}}}}{2 a^{2} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(a+b*x**(3/2))**(2/3),x)
[Out]
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Mathematica [A] time = 0.0226849, size = 33, normalized size = 0.66 \[ -\frac{\left (a-3 b x^{3/2}\right ) \sqrt [3]{a+b x^{3/2}}}{2 a^2 x^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a + b*x^(3/2))^(2/3)),x]
[Out]
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Maple [F] time = 0.023, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( a+b{x}^{{\frac{3}{2}}} \right ) ^{-{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(a+b*x^(3/2))^(2/3),x)
[Out]
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Maxima [A] time = 1.43183, size = 47, normalized size = 0.94 \[ \frac{\frac{4 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}} b}{\sqrt{x}} - \frac{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{4}{3}}}{x^{2}}}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^(3/2) + a)^(2/3)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.442881, size = 36, normalized size = 0.72 \[ \frac{{\left (3 \, b x^{\frac{3}{2}} - a\right )}{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}}}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^(3/2) + a)^(2/3)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 25.7729, size = 76, normalized size = 1.52 \[ - \frac{2 \sqrt [3]{b} \sqrt [3]{\frac{a}{b x^{\frac{3}{2}}} + 1} \Gamma \left (- \frac{4}{3}\right )}{9 a x^{\frac{3}{2}} \Gamma \left (\frac{2}{3}\right )} + \frac{2 b^{\frac{4}{3}} \sqrt [3]{\frac{a}{b x^{\frac{3}{2}}} + 1} \Gamma \left (- \frac{4}{3}\right )}{3 a^{2} \Gamma \left (\frac{2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(a+b*x**(3/2))**(2/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^(3/2) + a)^(2/3)*x^3),x, algorithm="giac")
[Out]