Optimal. Leaf size=49 \[ -\frac{\left (a+b x^n\right )^{7/2} \, _2F_1\left (1,\frac{7}{2}-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x} \]
[Out]
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Rubi [A] time = 0.060704, antiderivative size = 61, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{5}{2},-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{x \sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(5/2)/x^2,x]
[Out]
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Rubi in Sympy [A] time = 7.22326, size = 48, normalized size = 0.98 \[ - \frac{a^{2} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{x \sqrt{1 + \frac{b x^{n}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**(5/2)/x**2,x)
[Out]
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Mathematica [B] time = 0.236969, size = 141, normalized size = 2.88 \[ \frac{2 \left (a+b x^n\right ) \left (a^2 \left (23 n^2-18 n+4\right )+a b \left (11 n^2-26 n+8\right ) x^n+b^2 \left (3 n^2-8 n+4\right ) x^{2 n}\right )-15 a^3 n^3 \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right )}{(n-2) (3 n-2) (5 n-2) x \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^(5/2)/x^2,x]
[Out]
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Maple [F] time = 0.067, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( a+b{x}^{n} \right ) ^{{\frac{5}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^(5/2)/x^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(5/2)/x^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)^(5/2)/x^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((a+b*x**n)**(5/2)/x**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(5/2)/x^2,x, algorithm="giac")
[Out]