3.2465 \(\int \frac{x}{a+b x^n} \, dx\)

Optimal. Leaf size=33 \[ \frac{x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0312275, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^n),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.66271, size = 20, normalized size = 0.61 \[ \frac{x^{2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a+b*x**n),x)

[Out]

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

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Mathematica [A]  time = 0.0141384, size = 33, normalized size = 1. \[ \frac{x^2 \, _2F_1\left (1,\frac{2}{n};1+\frac{2}{n};-\frac{b x^n}{a}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x^n),x]

[Out]

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

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int{\frac{x}{a+b{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^n + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^n + a), x)