3.2767 \(\int \frac{(c x)^{2+n}}{a+b x^n} \, dx\)

Optimal. Leaf size=44 \[ \frac{(c x)^{n+3} \, _2F_1\left (1,\frac{n+3}{n};2+\frac{3}{n};-\frac{b x^n}{a}\right )}{a c (n+3)} \]

[Out]

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

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Rubi [A]  time = 0.0448661, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{(c x)^{n+3} \, _2F_1\left (1,\frac{n+3}{n};2+\frac{3}{n};-\frac{b x^n}{a}\right )}{a c (n+3)} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(2 + n)/(a + b*x^n),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.67701, size = 29, normalized size = 0.66 \[ \frac{\left (c x\right )^{n + 3}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n + 3}{n} \\ 2 + \frac{3}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a c \left (n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(2+n)/(a+b*x**n),x)

[Out]

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

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Mathematica [A]  time = 0.0371593, size = 47, normalized size = 1.07 \[ -\frac{c^2 x^{3-n} (c x)^n \left (\, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right )-1\right )}{3 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(2 + n)/(a + b*x^n),x]

[Out]

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

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{2+n}}{a+b{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(2+n)/(a+b*x^n),x)

[Out]

int((c*x)^(2+n)/(a+b*x^n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{c^{n + 2} x^{3}}{3 \, b} - a c^{n + 2} \int \frac{x^{2}}{b^{2} x^{n} + a b}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(n + 2)/(b*x^n + a),x, algorithm="maxima")

[Out]

1/3*c^(n + 2)*x^3/b - a*c^(n + 2)*integrate(x^2/(b^2*x^n + a*b), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (c x\right )^{n + 2}}{b x^{n} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(n + 2)/(b*x^n + a),x, algorithm="fricas")

[Out]

integral((c*x)^(n + 2)/(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)**(2+n)/(a+b*x**n),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{n + 2}}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(n + 2)/(b*x^n + a),x, algorithm="giac")

[Out]

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