Optimal. Leaf size=39 \[ \frac{x \, _2F_1\left (1,\frac{1}{n}-\frac{1}{2};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a \sqrt{a+b x^n}} \]
[Out]
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Rubi [A] time = 0.0338792, antiderivative size = 51, normalized size of antiderivative = 1.31, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{3}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(-3/2),x]
[Out]
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Rubi in Sympy [A] time = 3.81049, size = 42, normalized size = 1.08 \[ \frac{x \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} \sqrt{1 + \frac{b x^{n}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*x**n)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0672204, size = 60, normalized size = 1.54 \[ \frac{x \left ((n-2) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )+2\right )}{a n \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^(-3/2),x]
[Out]
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Maple [F] time = 0.04, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*x^n)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((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((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(1/(a+b*x**n)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(-3/2),x, algorithm="giac")
[Out]