3.2921 \(\int \frac{\left (a+b (c x)^n\right )^p}{x^2} \, dx\)

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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

integrate(((c*x)^n*b + a)^p/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(((c*x)^n*b + a)^p/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*(c*x)**n)**p/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x)^n*b + a)^p/x^2, x)