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

Optimal. Leaf size=80 \[ \frac{x \left (a+\frac{b}{x}\right )^m \left (\frac{a x}{b}+1\right )^{-m} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (1-m;-m,-n;2-m;-\frac{a x}{b},-\frac{d x}{c}\right )}{1-m} \]

[Out]

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

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Rubi [A]  time = 0.0591799, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {435, 135, 133} \[ \frac{x \left (a+\frac{b}{x}\right )^m \left (\frac{a x}{b}+1\right )^{-m} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (1-m;-m,-n;2-m;-\frac{a x}{b},-\frac{d x}{c}\right )}{1-m} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^m*(c + d*x)^n,x]

[Out]

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

Rule 435

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

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

Mathematica [F]  time = 0.0647359, size = 0, normalized size = 0. \[ \int \left (a+\frac{b}{x}\right )^m (c+d x)^n \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b/x)^m*(c + d*x)^n,x]

[Out]

Integrate[(a + b/x)^m*(c + d*x)^n, x]

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Maple [F]  time = 0.091, size = 0, normalized size = 0. \begin{align*} \int \left ( a+{\frac{b}{x}} \right ) ^{m} \left ( dx+c \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^n*(a + b/x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^n*(a + b/x)^m, x)