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

Optimal. Leaf size=73 \[ -\frac{x^{m-1} \left (a+\frac{b}{x}\right )^n \left (\frac{b}{a x}+1\right )^{-n} F_1\left (1-m;-n,2;2-m;-\frac{b}{a x},-\frac{c}{d x}\right )}{d^2 (1-m)} \]

[Out]

-(((a + b/x)^n*x^(-1 + m)*AppellF1[1 - m, -n, 2, 2 - m, -(b/(a*x)), -(c/(d*x))])/(d^2*(1 - m)*(1 + b/(a*x))^n)
)

________________________________________________________________________________________

Rubi [A]  time = 0.0708469, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {514, 497, 135, 133} \[ -\frac{x^{m-1} \left (a+\frac{b}{x}\right )^n \left (\frac{b}{a x}+1\right )^{-n} F_1\left (1-m;-n,2;2-m;-\frac{b}{a x},-\frac{c}{d x}\right )}{d^2 (1-m)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b/x)^n*x^m)/(c + d*x)^2,x]

[Out]

-(((a + b/x)^n*x^(-1 + m)*AppellF1[1 - m, -n, 2, 2 - m, -(b/(a*x)), -(c/(d*x))])/(d^2*(1 - m)*(1 + b/(a*x))^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 497

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Dist[(e*x)^m*
(x^(-1))^m, Subst[Int[((a + b/x^n)^p*(c + d/x^n)^q)/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, m, p,
q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0] &&  !RationalQ[m]

Rule 135

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c^IntPart[n]*(c +
d*x)^FracPart[n])/(1 + (d*x)/c)^FracPart[n], Int[(b*x)^m*(1 + (d*x)/c)^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d
, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 0]

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^n x^m}{(c+d x)^2} \, dx &=\int \frac{\left (a+\frac{b}{x}\right )^n x^{-2+m}}{\left (d+\frac{c}{x}\right )^2} \, dx\\ &=-\left (\left (\left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-m} (a+b x)^n}{(d+c x)^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\left (\left (\left (a+\frac{b}{x}\right )^n \left (1+\frac{b}{a x}\right )^{-n} \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-m} \left (1+\frac{b x}{a}\right )^n}{(d+c x)^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{\left (a+\frac{b}{x}\right )^n \left (1+\frac{b}{a x}\right )^{-n} x^{-1+m} F_1\left (1-m;-n,2;2-m;-\frac{b}{a x},-\frac{c}{d x}\right )}{d^2 (1-m)}\\ \end{align*}

Mathematica [F]  time = 0.0712447, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^n x^m}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.561, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \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^m/(d*x+c)^2,x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*(a + b/x)**n/(c + d*x)**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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