3.2663 \(\int \frac{x^m}{a+b x^n} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0356912, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x^n),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.65345, size = 27, normalized size = 0.68 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0253583, size = 41, normalized size = 1.02 \[ \frac{x^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+1}{n}+1;-\frac{b x^n}{a}\right )}{a (m+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(a + b*x^n),x]

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{a+b{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(a+b*x^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x^n + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x^n + a), x)