3.6.87 \(\int (a+\frac {b}{x})^m (c+d x)^n \, dx\) [587]

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

[Out]

(a+b/x)^m*x*(d*x+c)^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 = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 = {446, 140, 138} \begin {gather*} \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} \end {gather*}

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 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 446

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]

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*}

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Mathematica [F]
time = 0.12, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+\frac {b}{x}\right )^m (c+d x)^n \, dx \end {gather*}

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

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

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 = 0.00, 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)^m*(d*x+c)^n,x, algorithm="giac")

[Out]

integrate((d*x + c)^n*(a + b/x)^m, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+\frac {b}{x}\right )}^m\,{\left (c+d\,x\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^m*(c + d*x)^n,x)

[Out]

int((a + b/x)^m*(c + d*x)^n, x)

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