\(\int \frac {c+d x^{-1+n}}{(a+b x^n)^3} \, dx\) [583]

Optimal result

Integrand size = 19, antiderivative size = 46 \[ \int \frac {c+d x^{-1+n}}{\left (a+b x^n\right )^3} \, dx=-\frac {d}{2 b n \left (a+b x^n\right )^2}+\frac {c x \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1905, 251, 267} \[ \int \frac {c+d x^{-1+n}}{\left (a+b x^n\right )^3} \, dx=\frac {c x \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^3}-\frac {d}{2 b n \left (a+b x^n\right )^2} \]

[In]

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

[Out]

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

Rule 251

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 267

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]

Rule 1905

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]

Rubi steps \begin{align*} \text {integral}& = 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] (verified)

Time = 0.13 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.37 \[ \int \frac {c+d x^{-1+n}}{\left (a+b x^n\right )^3} \, dx=\frac {-a^3 d+2 b c n x \left (a+b x^n\right )^2 \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{2 a^3 b n \left (a+b x^n\right )^2} \]

[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)

Maple [F]

\[\int \frac {c +d \,x^{-1+n}}{\left (a +b \,x^{n}\right )^{3}}d x\]

[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)

Fricas [F]

\[ \int \frac {c+d x^{-1+n}}{\left (a+b x^n\right )^3} \, dx=\int { \frac {d x^{n - 1} + c}{{\left (b x^{n} + a\right )}^{3}} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^{-1+n}}{\left (a+b x^n\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {c+d x^{-1+n}}{\left (a+b x^n\right )^3} \, dx=\int { \frac {d x^{n - 1} + c}{{\left (b x^{n} + a\right )}^{3}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {c+d x^{-1+n}}{\left (a+b x^n\right )^3} \, dx=\int { \frac {d x^{n - 1} + c}{{\left (b x^{n} + a\right )}^{3}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 10.81 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28 \[ \int \frac {c+d x^{-1+n}}{\left (a+b x^n\right )^3} \, dx=\frac {c\,x\,{{}}_2{\mathrm {F}}_1\left (3,\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{a^3}-\frac {d}{2\,b\,\left (a^2\,n+b^2\,n\,x^{2\,n}+2\,a\,b\,n\,x^n\right )} \]

[In]

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

[Out]

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