3.583 \(\int \frac{c+d x^{-1+n}}{(a+b x^n)^3} \, dx\)

Optimal. Leaf size=46 \[ \frac{c x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3}-\frac{d}{2 b n \left (a+b x^n\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0295255, antiderivative size = 46, 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} \[ \frac{c x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3}-\frac{d}{2 b n \left (a+b x^n\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 )^3} \, dx &=c \int \frac{1}{\left (a+b x^n\right )^3} \, dx+d \int \frac{x^{-1+n}}{\left (a+b x^n\right )^3} \, dx\\ &=-\frac{d}{2 b n \left (a+b x^n\right )^2}+\frac{c x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3}\\ \end{align*}

Mathematica [A]  time = 0.0744282, size = 63, normalized size = 1.37 \[ \frac{2 b c n x \left (a+b x^n\right )^2 \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )-a^3 d}{2 a^3 b n \left (a+b x^n\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.355, size = 0, normalized size = 0. \begin{align*} \int{\frac{c+d{x}^{-1+n}}{ \left ( a+b{x}^{n} \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*x^(-1+n))/(a+b*x^n)^3,x)

[Out]

int((c+d*x^(-1+n))/(a+b*x^n)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (2 \, n^{2} - 3 \, n + 1\right )} c \int \frac{1}{2 \,{\left (a^{2} b n^{2} x^{n} + a^{3} n^{2}\right )}}\,{d x} + \frac{b^{2} c{\left (2 \, n - 1\right )} x x^{n} + a b c{\left (3 \, n - 1\right )} x - a^{2} d n}{2 \,{\left (a^{2} b^{3} n^{2} x^{2 \, n} + 2 \, a^{3} b^{2} n^{2} x^{n} + a^{4} b n^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*x^(-1+n))/(a+b*x^n)^3,x, algorithm="maxima")

[Out]

(2*n^2 - 3*n + 1)*c*integrate(1/2/(a^2*b*n^2*x^n + a^3*n^2), x) + 1/2*(b^2*c*(2*n - 1)*x*x^n + a*b*c*(3*n - 1)
*x - a^2*d*n)/(a^2*b^3*n^2*x^(2*n) + 2*a^3*b^2*n^2*x^n + a^4*b*n^2)

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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^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, 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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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