Optimal. Leaf size=45 \[ -\frac{2 (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-\frac{3}{2},-n;-\frac{1}{2};-\frac{b x}{a}\right )}{3 x^{3/2}} \]
[Out]
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Rubi [A] time = 0.0277944, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2 (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-\frac{3}{2},-n;-\frac{1}{2};-\frac{b x}{a}\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^n/x^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 5.37066, size = 41, normalized size = 0.91 \[ - \frac{2 \left (1 + \frac{b x}{a}\right )^{- n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, - \frac{3}{2} \\ - \frac{1}{2} \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{3 x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n/x**(5/2),x)
[Out]
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Mathematica [A] time = 0.0244816, size = 45, normalized size = 1. \[ -\frac{2 (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-\frac{3}{2},-n;-\frac{1}{2};-\frac{b x}{a}\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^n/x^(5/2),x]
[Out]
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Maple [F] time = 0.025, size = 0, normalized size = 0. \[ \int{ \left ( bx+a \right ) ^{n}{x}^{-{\frac{5}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n/x^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{x^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n/x^(5/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{x^{\frac{5}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n/x^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 90.6005, size = 32, normalized size = 0.71 \[ - \frac{2 a^{n}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - n \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{3 x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**n/x**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{x^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n/x^(5/2),x, algorithm="giac")
[Out]