3.3020 \(\int \frac{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p}{x^2} \, dx\)

Optimal. Leaf size=63 \[ \frac{b \left (c x^n\right )^{\frac{1}{n}} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )}{a^2 (p+1) x} \]

[Out]

(b*(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^(1 + p)*Hypergeometric2F1[2, 1 + p, 2 +
p, 1 + (b*(c*x^n)^n^(-1))/a])/(a^2*(1 + p)*x)

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Rubi [A]  time = 0.0564332, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{b \left (c x^n\right )^{\frac{1}{n}} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )}{a^2 (p+1) x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c*x^n)^n^(-1))^p/x^2,x]

[Out]

(b*(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^(1 + p)*Hypergeometric2F1[2, 1 + p, 2 +
p, 1 + (b*(c*x^n)^n^(-1))/a])/(a^2*(1 + p)*x)

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Rubi in Sympy [A]  time = 6.91521, size = 51, normalized size = 0.81 \[ \frac{b \left (c x^{n}\right )^{\frac{1}{n}} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, p + 1 \\ p + 2 \end{matrix}\middle |{1 + \frac{b \left (c x^{n}\right )^{\frac{1}{n}}}{a}} \right )}}{a^{2} x \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**n)**(1/n))**p/x**2,x)

[Out]

b*(c*x**n)**(1/n)*(a + b*(c*x**n)**(1/n))**(p + 1)*hyper((2, p + 1), (p + 2,), 1
+ b*(c*x**n)**(1/n)/a)/(a**2*x*(p + 1))

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Mathematica [A]  time = 0.111656, size = 77, normalized size = 1.22 \[ \frac{\left (\frac{a \left (c x^n\right )^{-1/n}}{b}+1\right )^{-p} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \, _2F_1\left (1-p,-p;2-p;-\frac{a \left (c x^n\right )^{-1/n}}{b}\right )}{(p-1) x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*(c*x^n)^n^(-1))^p/x^2,x]

[Out]

((a + b*(c*x^n)^n^(-1))^p*Hypergeometric2F1[1 - p, -p, 2 - p, -(a/(b*(c*x^n)^n^(
-1)))])/((-1 + p)*x*(1 + a/(b*(c*x^n)^n^(-1)))^p)

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Maple [F]  time = 0.334, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b\sqrt [n]{c{x}^{n}} \right ) ^{p}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^n)^(1/n))^p/x^2,x)

[Out]

int((a+b*(c*x^n)^(1/n))^p/x^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{p}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(1/n)*b + a)^p/x^2,x, algorithm="maxima")

[Out]

integrate(((c*x^n)^(1/n)*b + a)^p/x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{p}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(1/n)*b + a)^p/x^2,x, algorithm="fricas")

[Out]

integral(((c*x^n)^(1/n)*b + a)^p/x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**n)**(1/n))**p/x**2,x)

[Out]

Integral((a + b*(c*x**n)**(1/n))**p/x**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{p}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(1/n)*b + a)^p/x^2,x, algorithm="giac")

[Out]

integrate(((c*x^n)^(1/n)*b + a)^p/x^2, x)