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

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

[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]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 = {1905, 251, 267} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 [B] Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(46)=92\).
time = 0.11, size = 108, normalized size = 2.35 \begin {gather*} \frac {x \left (c+d x^{-1+n}\right ) \left (\frac {a^2 n (-a d+b c x)}{b \left (a+b x^n\right )^2}+\frac {a c (-1+2 n) x}{a+b x^n}+c \left (1-3 n+2 n^2\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )\right )}{2 a^3 n^2 \left (c x+d x^n\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {c +d \,x^{-1+n}}{\left (a +b \,x^{n}\right )^{3}}\, dx\]

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)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [F]
time = 0.35, size = 47, normalized size = 1.02 \begin {gather*} {\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 {gather*}

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)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 5.41, size = 59, normalized size = 1.28 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________