3.3.84 \(\int \frac {(a+\frac {b}{x})^n x^m}{c+d x} \, dx\) [284]

Optimal. Leaf size=64 \[ \frac {\left (a+\frac {b}{x}\right )^n \left (1+\frac {b}{a x}\right )^{-n} x^m F_1\left (-m;-n,1;1-m;-\frac {b}{a x},-\frac {c}{d x}\right )}{d m} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {528, 511, 140, 138} \begin {gather*} \frac {x^m \left (a+\frac {b}{x}\right )^n \left (\frac {b}{a x}+1\right )^{-n} F_1\left (-m;-n,1;1-m;-\frac {b}{a x},-\frac {c}{d x}\right )}{d m} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b/x)^n*x^m)/(c + d*x),x]

[Out]

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

Rule 138

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

Rule 140

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 511

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 528

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])

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]
time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+\frac {b}{x}\right )^n x^m}{c+d x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (a +\frac {b}{x}\right )^{n} x^{m}}{d x +c}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+1/x*b)^n*x^m/(d*x+c),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((a+b/x)^n*x^m/(d*x+c),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{m} \left (a + \frac {b}{x}\right )^{n}}{c + d x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^m\,{\left (a+\frac {b}{x}\right )}^n}{c+d\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^m*(a + b/x)^n)/(c + d*x),x)

[Out]

int((x^m*(a + b/x)^n)/(c + d*x), x)

________________________________________________________________________________________