3.3044 \(\int \frac{\left (a+b \left (c x^q\right )^n\right )^p}{x^2} \, dx\)

Optimal. Leaf size=71 \[ -\frac{\left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-\frac{1}{n q};1-\frac{1}{n q};-\frac{b \left (c x^q\right )^n}{a}\right )}{x} \]

[Out]

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

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Rubi [A]  time = 0.0747531, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{\left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-\frac{1}{n q};1-\frac{1}{n q};-\frac{b \left (c x^q\right )^n}{a}\right )}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c*x^q)^n)^p/x^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0928376, size = 71, normalized size = 1. \[ -\frac{\left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-\frac{1}{n q};1-\frac{1}{n q};-\frac{b \left (c x^q\right )^n}{a}\right )}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.332, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b \left ( c{x}^{q} \right ) ^{n} \right ) ^{p}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^q)^n)^p/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x^q)^n*b + a)^p/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(((c*x^q)^n*b + a)^p/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x^q)^n*b + a)^p/x^2, x)