3.2790 \(\int (c x)^{-1-n p} \left (a+b x^n\right )^p \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0656724, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{(c x)^{-n p} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-p;1-p;-\frac{b x^n}{a}\right )}{c n p} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.53604, size = 46, normalized size = 0.73 \[ - \frac{\left (c x\right )^{- n p} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - p \\ - p + 1 \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{c n p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.059904, size = 63, normalized size = 1. \[ -\frac{x (c x)^{-n p-1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (-p,-p;1-p;-\frac{b x^n}{a}\right )}{n p} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.104, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{-np-1} \left ( a+b{x}^{n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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