3.2920 \(\int \frac{\left (a+b (c x)^n\right )^p}{x} \, dx\)

Optimal. Leaf size=46 \[ -\frac{\left (a+b (c x)^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b (c x)^n}{a}+1\right )}{a n (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.100896, antiderivative size = 46, 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 \[ -\frac{\left (a+b (c x)^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b (c x)^n}{a}+1\right )}{a n (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c*x)^n)^p/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.80263, size = 34, normalized size = 0.74 \[ - \frac{\left (a + b \left (c x\right )^{n}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{1 + \frac{b \left (c x\right )^{n}}{a}} \right )}}{a n \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0485229, size = 61, normalized size = 1.33 \[ \frac{\left (\frac{a (c x)^{-n}}{b}+1\right )^{-p} \left (a+b (c x)^n\right )^p \, _2F_1\left (-p,-p;1-p;-\frac{a (c x)^{-n}}{b}\right )}{n p} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b \left ( cx \right ) ^{n} \right ) ^{p}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x)^n)^p/x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x)^n*b + a)^p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(((c*x)^n*b + a)^p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*(c*x)**n)**p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x)^n*b + a)^p/x, x)