Optimal. Leaf size=79 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+2}}{b^2 (p+2)}-\frac{a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b^2 (p+1)} \]
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Rubi [A] time = 0.0875262, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+2}}{b^2 (p+2)}-\frac{a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b^2 (p+1)} \]
Antiderivative was successfully verified.
[In] Int[(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^p,x]
[Out]
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Rubi in Sympy [A] time = 21.9106, size = 65, normalized size = 0.82 \[ - \frac{a x \left (c x^{n}\right )^{- \frac{1}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 1}}{b^{2} \left (p + 1\right )} + \frac{x \left (c x^{n}\right )^{- \frac{1}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 2}}{b^{2} \left (p + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**n)**(1/n)*(a+b*(c*x**n)**(1/n))**p,x)
[Out]
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Mathematica [A] time = 0.263018, size = 88, normalized size = 1.11 \[ \frac{x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \left (a^2 \left (c x^n\right )^{-1/n} \left (\left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^{-p}-1\right )+a b p+b^2 (p+1) \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2 (p+1) (p+2)} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^p,x]
[Out]
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Maple [C] time = 0.421, size = 571, normalized size = 7.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^p,x)
[Out]
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Maxima [A] time = 1.77188, size = 89, normalized size = 1.13 \[ \frac{{\left (b^{2} c^{\frac{2}{n}}{\left (p + 1\right )} x^{2} + a b c^{\left (\frac{1}{n}\right )} p x - a^{2}\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} c^{-\frac{1}{n}}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)^p*(c*x^n)^(1/n),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226544, size = 104, normalized size = 1.32 \[ \frac{{\left (a b c^{\left (\frac{1}{n}\right )} p x +{\left (b^{2} p + b^{2}\right )} c^{\frac{2}{n}} x^{2} - a^{2}\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p}}{{\left (b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}\right )} c^{\left (\frac{1}{n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)^p*(c*x^n)^(1/n),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (c x^{n}\right )^{\frac{1}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**n)**(1/n)*(a+b*(c*x**n)**(1/n))**p,x)
[Out]
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GIAC/XCAS [A] time = 23.3487, size = 211, normalized size = 2.67 \[ \frac{b^{2} p x^{2} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{2 \,{\rm ln}\left (c\right )}{n}\right )} + b^{2} x^{2} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{2 \,{\rm ln}\left (c\right )}{n}\right )} + a b p x e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right ) + \frac{{\rm ln}\left (c\right )}{n}\right )} - a^{2} e^{\left (p{\rm ln}\left (b x e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a\right )\right )}}{b^{2} p^{2} e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + 3 \, b^{2} p e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + 2 \, b^{2} e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)^p*(c*x^n)^(1/n),x, algorithm="giac")
[Out]