3.2775 \(\int \frac{(c x)^{1+n}}{a+b x^n} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0130062, 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, Rules used = {364} \[ \frac{(c x)^{n+2} \, _2F_1\left (1,\frac{n+2}{n};2 \left (1+\frac{1}{n}\right );-\frac{b x^n}{a}\right )}{a c (n+2)} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(1 + n)/(a + b*x^n),x]

[Out]

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{(c x)^{1+n}}{a+b x^n} \, dx &=\frac{(c x)^{2+n} \, _2F_1\left (1,\frac{2+n}{n};2 \left (1+\frac{1}{n}\right );-\frac{b x^n}{a}\right )}{a c (2+n)}\\ \end{align*}

Mathematica [A]  time = 0.0099471, size = 45, normalized size = 1.02 \[ \frac{c x^2 (c x)^n \, _2F_1\left (1,\frac{n+2}{n};\frac{n+2}{n}+1;-\frac{b x^n}{a}\right )}{a (n+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( cx \right ) ^{1+n}}{a+b{x}^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(1+n)/(a+b*x^n),x)

[Out]

int((c*x)^(1+n)/(a+b*x^n),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(1+n)/(a+b*x^n),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (c x\right )^{n + 1}}{b x^{n} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(1+n)/(a+b*x^n),x, algorithm="fricas")

[Out]

integral((c*x)^(n + 1)/(b*x^n + a), x)

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Sympy [C]  time = 2.00279, size = 94, normalized size = 2.14 \begin{align*} \frac{c c^{n} x^{2} x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, 1 + \frac{2}{n}\right ) \Gamma \left (1 + \frac{2}{n}\right )}{a n \Gamma \left (2 + \frac{2}{n}\right )} + \frac{2 c c^{n} x^{2} x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, 1 + \frac{2}{n}\right ) \Gamma \left (1 + \frac{2}{n}\right )}{a n^{2} \Gamma \left (2 + \frac{2}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*c**n*x**2*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1 + 2/n)*gamma(1 + 2/n)/(a*n*gamma(2 + 2/n)) + 2*c*c**n
*x**2*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1 + 2/n)*gamma(1 + 2/n)/(a*n**2*gamma(2 + 2/n))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{n + 1}}{b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(1+n)/(a+b*x^n),x, algorithm="giac")

[Out]

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