3.2769 \(\int \frac{(c x)^n}{a+b x^n} \, dx\)

Optimal. Leaf size=40 \[ \frac{(c x)^{n+1} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{b x^n}{a}\right )}{a c (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0403035, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{(c x)^{n+1} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{b x^n}{a}\right )}{a c (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^n/(a + b*x^n),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.74888, size = 29, normalized size = 0.72 \[ \frac{\left (c x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n + 1}{n} \\ 2 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a c \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.021484, size = 38, normalized size = 0.95 \[ -\frac{x^{1-n} (c x)^n \left (\, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )-1\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{n}}{a+b{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^n/(a+b*x^n),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -a c^{n} \int \frac{1}{b^{2} x^{n} + a b}\,{d x} + \frac{c^{n} x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*c^n*integrate(1/(b^2*x^n + a*b), x) + c^n*x/b

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^n/(b*x^n + a), x)