Optimal. Leaf size=62 \[ -\frac{\left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{n},-p;-\frac{1-n}{n};-\frac{b (c x)^n}{a}\right )}{x} \]
[Out]
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Rubi [A] time = 0.0905104, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{n},-p;-\frac{1-n}{n};-\frac{b (c x)^n}{a}\right )}{x} \]
Antiderivative was successfully verified.
[In] Int[(a + b*(c*x)^n)^p/x^2,x]
[Out]
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Rubi in Sympy [A] time = 10.0767, size = 44, normalized size = 0.71 \[ - \frac{\left (1 + \frac{b \left (c x\right )^{n}}{a}\right )^{- p} \left (a + b \left (c x\right )^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b \left (c x\right )^{n}}{a}} \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(c*x)**n)**p/x**2,x)
[Out]
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Mathematica [A] time = 0.0436399, size = 59, normalized size = 0.95 \[ -\frac{\left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{n},-p;1-\frac{1}{n};-\frac{b (c x)^n}{a}\right )}{x} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*(c*x)^n)^p/x^2,x]
[Out]
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Maple [F] time = 0.071, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b \left ( cx \right ) ^{n} \right ) ^{p}}{{x}^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(c*x)^n)^p/x^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (\left (c x\right )^{n} b + a\right )}^{p}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x)^n*b + a)^p/x^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (\left (c x\right )^{n} b + a\right )}^{p}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x)^n*b + a)^p/x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b \left (c x\right )^{n}\right )^{p}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(c*x)**n)**p/x**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (\left (c x\right )^{n} b + a\right )}^{p}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x)^n*b + a)^p/x^2,x, algorithm="giac")
[Out]