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

Optimal. Leaf size=42 \[ \frac{c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a}+\frac{d \log \left (a+b x^n\right )}{b n} \]

[Out]

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

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Rubi [A]  time = 0.0304338, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1891, 245, 260} \[ \frac{c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a}+\frac{d \log \left (a+b x^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^(-1 + n))/(a + b*x^n),x]

[Out]

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

Rule 1891

Int[((A_) + (B_.)*(x_)^(m_.))*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[A, Int[(a + b*x^n)^p, x], x] +
Dist[B, Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, A, B, m, n, p}, x] && EqQ[m - n + 1, 0]

Rule 245

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

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

Mathematica [A]  time = 0.0561254, size = 42, normalized size = 1. \[ \frac{c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a}+\frac{d \log \left (a+b x^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.339, size = 0, normalized size = 0. \begin{align*} \int{\frac{c+d{x}^{-1+n}}{a+b{x}^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 9.52058, size = 65, normalized size = 1.55 \begin{align*} d \left (\begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = 0 \\\frac{x^{n}}{a n} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\\frac{\log{\left (\frac{a}{b} + x^{n} \right )}}{b n} & \text{otherwise} \end{cases}\right ) + \frac{c x \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a n^{2} \Gamma \left (1 + \frac{1}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*Piecewise((log(x)/a, Eq(b, 0) & Eq(n, 0)), (x**n/(a*n), Eq(b, 0)), (log(x)/(a + b), Eq(n, 0)), (log(a/b + x*
*n)/(b*n), True)) + c*x*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a*n**2*gamma(1 + 1/n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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