3.2789 \(\int (c x)^{-3 n} \left (a+b x^n\right )^p \, dx\)

Optimal. Leaf size=56 \[ \frac{(c x)^{1-3 n} \left (a+b x^n\right )^{p+1} \, _2F_1\left (1,p+\frac{1}{n}-2;\frac{1}{n}-2;-\frac{b x^n}{a}\right )}{a c (1-3 n)} \]

[Out]

((c*x)^(1 - 3*n)*(a + b*x^n)^(1 + p)*Hypergeometric2F1[1, -2 + n^(-1) + p, -2 +
n^(-1), -((b*x^n)/a)])/(a*c*(1 - 3*n))

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Rubi [A]  time = 0.0679794, antiderivative size = 66, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{(c x)^{1-3 n} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n}-3,-p;\frac{1}{n}-2;-\frac{b x^n}{a}\right )}{c (1-3 n)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^p/(c*x)^(3*n),x]

[Out]

((c*x)^(1 - 3*n)*(a + b*x^n)^p*Hypergeometric2F1[-3 + n^(-1), -p, -2 + n^(-1), -
((b*x^n)/a)])/(c*(1 - 3*n)*(1 + (b*x^n)/a)^p)

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Rubi in Sympy [A]  time = 9.52405, size = 51, normalized size = 0.91 \[ \frac{\left (c x\right )^{- 3 n + 1} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, -3 + \frac{1}{n} \\ -2 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{c \left (- 3 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**p/((c*x)**(3*n)),x)

[Out]

(c*x)**(-3*n + 1)*(1 + b*x**n/a)**(-p)*(a + b*x**n)**p*hyper((-p, -3 + 1/n), (-2
+ 1/n,), -b*x**n/a)/(c*(-3*n + 1))

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Mathematica [A]  time = 0.32161, size = 63, normalized size = 1.12 \[ -\frac{x (c x)^{-3 n} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n}-3,-p;\frac{1}{n}-2;-\frac{b x^n}{a}\right )}{3 n-1} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^p/(c*x)^(3*n),x]

[Out]

-((x*(a + b*x^n)^p*Hypergeometric2F1[-3 + n^(-1), -p, -2 + n^(-1), -((b*x^n)/a)]
)/((-1 + 3*n)*(c*x)^(3*n)*(1 + (b*x^n)/a)^p))

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Maple [F]  time = 0.102, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b{x}^{n} \right ) ^{p}}{ \left ( cx \right ) ^{3\,n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^p/((c*x)^(3*n)),x)

[Out]

int((a+b*x^n)^p/((c*x)^(3*n)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{p} \left (c x\right )^{-3 \, n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^p/(c*x)^(3*n),x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^p*(c*x)^(-3*n), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{n} + a\right )}^{p}}{\left (c x\right )^{3 \, n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^p/(c*x)^(3*n),x, algorithm="fricas")

[Out]

integral((b*x^n + a)^p/(c*x)^(3*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)**(3*n)),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{p}}{\left (c x\right )^{3 \, n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^p/(c*x)^(3*n),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^p/(c*x)^(3*n), x)