3.3043 \(\int \frac{\left (a+b \left (c x^q\right )^n\right )^p}{x} \, dx\)

Optimal. Leaf size=53 \[ -\frac{\left (a+b \left (c x^q\right )^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \left (c x^q\right )^n}{a}+1\right )}{a n (p+1) q} \]

[Out]

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

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Rubi [A]  time = 0.0841715, antiderivative size = 53, 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 \[ -\frac{\left (a+b \left (c x^q\right )^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \left (c x^q\right )^n}{a}+1\right )}{a n (p+1) q} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c*x^q)^n)^p/x,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*(c*x**q)**n)**p/x, x)

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Mathematica [A]  time = 0.104769, size = 70, normalized size = 1.32 \[ \frac{\left (\frac{a \left (c x^q\right )^{-n}}{b}+1\right )^{-p} \left (a+b \left (c x^q\right )^n\right )^p \, _2F_1\left (-p,-p;1-p;-\frac{a \left (c x^q\right )^{-n}}{b}\right )}{n p q} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.472, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b \left ( c{x}^{q} \right ) ^{n} \right ) ^{p}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^q)^n)^p/x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x^q)^n*b + a)^p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(((c*x^q)^n*b + a)^p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*(c*x**q)**n)**p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x^q)^n*b + a)^p/x, x)