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

Optimal. Leaf size=133 \[ \frac{d^2 \left (a+\frac{b}{x}\right )^{n+1}}{c^2 \left (\frac{c}{x}+d\right ) (a c-b d)}-\frac{d \left (a+\frac{b}{x}\right )^{n+1} (2 a c-b d (n+2)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c^2 (n+1) (a c-b d)^2}-\frac{\left (a+\frac{b}{x}\right )^{n+1}}{b c^2 (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.130979, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {514, 446, 89, 80, 68} \[ \frac{d^2 \left (a+\frac{b}{x}\right )^{n+1}}{c^2 \left (\frac{c}{x}+d\right ) (a c-b d)}-\frac{d \left (a+\frac{b}{x}\right )^{n+1} (2 a c-b d (n+2)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c^2 (n+1) (a c-b d)^2}-\frac{\left (a+\frac{b}{x}\right )^{n+1}}{b c^2 (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^n/(x^2*(c + d*x)^2),x]

[Out]

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

Rule 514

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^n}{x^2 (c+d x)^2} \, dx &=\int \frac{\left (a+\frac{b}{x}\right )^n}{\left (d+\frac{c}{x}\right )^2 x^4} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{x^2 (a+b x)^n}{(d+c x)^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{d^2 \left (a+\frac{b}{x}\right )^{1+n}}{c^2 (a c-b d) \left (d+\frac{c}{x}\right )}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^n (-d (a c-b d (1+n))+c (a c-b d) x)}{d+c x} \, dx,x,\frac{1}{x}\right )}{c^2 (a c-b d)}\\ &=-\frac{\left (a+\frac{b}{x}\right )^{1+n}}{b c^2 (1+n)}+\frac{d^2 \left (a+\frac{b}{x}\right )^{1+n}}{c^2 (a c-b d) \left (d+\frac{c}{x}\right )}+\frac{(d (2 a c-b d (2+n))) \operatorname{Subst}\left (\int \frac{(a+b x)^n}{d+c x} \, dx,x,\frac{1}{x}\right )}{c^2 (a c-b d)}\\ &=-\frac{\left (a+\frac{b}{x}\right )^{1+n}}{b c^2 (1+n)}+\frac{d^2 \left (a+\frac{b}{x}\right )^{1+n}}{c^2 (a c-b d) \left (d+\frac{c}{x}\right )}-\frac{d (2 a c-b d (2+n)) \left (a+\frac{b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c^2 (a c-b d)^2 (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0716438, size = 125, normalized size = 0.94 \[ -\frac{(a x+b) \left (a+\frac{b}{x}\right )^n \left (b d (c+d x) (2 a c-b d (n+2)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )+(a c-b d) (a c (c+d x)-b d (c+d (n+2) x))\right )}{b c^2 (n+1) x (c+d x) (a c-b d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.535, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( dx+c \right ) ^{2}} \left ( a+{\frac{b}{x}} \right ) ^{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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