Optimal. Leaf size=48 \[ \frac{x^2 \, _2F_1\left (1,\frac{2}{n}-\frac{1}{2};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a \sqrt{a+b x^n}} \]
[Out]
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Rubi [A] time = 0.0564546, antiderivative size = 60, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^2 \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{3}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[x/(a + b*x^n)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 6.56893, size = 46, normalized size = 0.96 \[ \frac{x^{2} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{2} \sqrt{1 + \frac{b x^{n}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a+b*x**n)**(3/2),x)
[Out]
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Mathematica [A] time = 0.101374, size = 69, normalized size = 1.44 \[ \frac{x^2 \left ((n-4) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )+4\right )}{2 a n \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a + b*x^n)^(3/2),x]
[Out]
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Maple [F] time = 0.042, size = 0, normalized size = 0. \[ \int{x \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a+b*x^n)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^n + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^n + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a+b*x**n)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^n + a)^(3/2),x, algorithm="giac")
[Out]