3.2787 \(\int (c x)^{-n} \left (a+b x^n\right )^p \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0633778, antiderivative size = 64, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{(c x)^{1-n} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n}-1,-p;\frac{1}{n};-\frac{b x^n}{a}\right )}{c (1-n)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^p/(c*x)^n,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.32617, size = 48, normalized size = 0.91 \[ \frac{\left (c x\right )^{- n + 1} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{n - 1}{n} \\ \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{c \left (- n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.132383, size = 59, normalized size = 1.11 \[ -\frac{x (c x)^{-n} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n}-1,-p;\frac{1}{n};-\frac{b x^n}{a}\right )}{n-1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.097, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b{x}^{n} \right ) ^{p}}{ \left ( cx \right ) ^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^p/((c*x)^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^n + a)^p/(c*x)^n, x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^p/(c*x)^n, x)