3.582 \(\int \frac{c+d x^{-1+n}}{(a+b x^n)^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0288111, antiderivative size = 44, 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, 261} \[ \frac{c x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2}-\frac{d}{b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 261

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

Rubi steps

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

Mathematica [A]  time = 0.0754271, size = 44, normalized size = 1. \[ \frac{c x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2}-\frac{d}{a b n+b^2 n x^n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.36, size = 0, normalized size = 0. \begin{align*} \int{\frac{c+d{x}^{-1+n}}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*x^(-1+n))/(a+b*x^n)^2,x, algorithm="maxima")

[Out]

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

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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^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, 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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 22.7527, size = 313, normalized size = 7.11 \begin{align*} c \left (\frac{n x \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )} + \frac{n x \Gamma \left (\frac{1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )} - \frac{x \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )} + \frac{b n x x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )} - \frac{b x x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )}\right ) + d \left (\begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac{x^{- n}}{b^{2} n} & \text{for}\: a = 0 \\\frac{\tilde{\infty } x^{n}}{n} & \text{for}\: b = - a x^{- n} \\\frac{\log{\left (x \right )}}{\left (a + b\right )^{2}} & \text{for}\: n = 0 \\\frac{x^{n}}{a^{2} n + a b n x^{n}} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(n*x*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1 +
1/n))) + n*x*gamma(1/n)/(a*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1 + 1/n))) - x*lerchphi(b*x**n*exp_pola
r(I*pi)/a, 1, 1/n)*gamma(1/n)/(a*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1 + 1/n))) + b*n*x*x**n*lerchphi(b
*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a**2*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1 + 1/n))) - b*x*
x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a**2*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1
+ 1/n)))) + d*Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (-x**(-n)/(b**2*n), Eq(a, 0)), (zoo*x**n
/n, Eq(b, -a*x**(-n))), (log(x)/(a + b)**2, Eq(n, 0)), (x**n/(a**2*n + a*b*n*x**n), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x^{n - 1} + c}{{\left (b x^{n} + a\right )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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