Optimal. Leaf size=51 \[ -\frac{\left (a+b x^n\right )^{7/2} \, _2F_1\left (1,\frac{7}{2}-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a x^2} \]
[Out]
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Rubi [A] time = 0.0661571, antiderivative size = 63, 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 \[ -\frac{a^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{5}{2},-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 x^2 \sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(5/2)/x^3,x]
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Rubi in Sympy [A] time = 7.13268, size = 51, normalized size = 1. \[ - \frac{a^{2} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 x^{2} \sqrt{1 + \frac{b x^{n}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**(5/2)/x**3,x)
[Out]
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Mathematica [B] time = 0.253055, size = 144, normalized size = 2.82 \[ \frac{4 \left (a+b x^n\right ) \left (a^2 \left (23 n^2-36 n+16\right )+a b \left (11 n^2-52 n+32\right ) x^n+b^2 \left (3 n^2-16 n+16\right ) x^{2 n}\right )-15 a^3 n^3 \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right )}{2 (n-4) (3 n-4) (5 n-4) x^2 \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^(5/2)/x^3,x]
[Out]
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Maple [F] time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( a+b{x}^{n} \right ) ^{{\frac{5}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^(5/2)/x^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(5/2)/x^3,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^3,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**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(5/2)/x^3,x, algorithm="giac")
[Out]