\(\int (a-x)^m (f x)^p (c+d x)^n \, dx\) [996]

Optimal result

Integrand size = 20, antiderivative size = 79 \[ \int (a-x)^m (f x)^p (c+d x)^n \, dx=\frac {(a-x)^m (f x)^{1+p} \left (1-\frac {x}{a}\right )^{-m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+p,-m,-n,2+p,\frac {x}{a},-\frac {d x}{c}\right )}{f (1+p)} \]

[Out]

(a-x)^m*(f*x)^(p+1)*(d*x+c)^n*AppellF1(p+1,-m,-n,2+p,x/a,-d*x/c)/f/(p+1)/((1-x/a)^m)/((1+d*x/c)^n)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {140, 138} \[ \int (a-x)^m (f x)^p (c+d x)^n \, dx=\frac {(a-x)^m \left (1-\frac {x}{a}\right )^{-m} (f x)^{p+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (p+1,-m,-n,p+2,\frac {x}{a},-\frac {d x}{c}\right )}{f (p+1)} \]

[In]

Int[(a - x)^m*(f*x)^p*(c + d*x)^n,x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \left ((a-x)^m \left (1-\frac {x}{a}\right )^{-m}\right ) \int (f x)^p \left (1-\frac {x}{a}\right )^m (c+d x)^n \, dx \\ & = \left ((a-x)^m \left (1-\frac {x}{a}\right )^{-m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n}\right ) \int (f x)^p \left (1-\frac {x}{a}\right )^m \left (1+\frac {d x}{c}\right )^n \, dx \\ & = \frac {(a-x)^m (f x)^{1+p} \left (1-\frac {x}{a}\right )^{-m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} F_1\left (1+p;-m,-n;2+p;\frac {x}{a},-\frac {d x}{c}\right )}{f (1+p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97 \[ \int (a-x)^m (f x)^p (c+d x)^n \, dx=\frac {(a-x)^m \left (\frac {a-x}{a}\right )^{-m} x (f x)^p (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+p,-m,-n,2+p,\frac {x}{a},-\frac {d x}{c}\right )}{1+p} \]

[In]

Integrate[(a - x)^m*(f*x)^p*(c + d*x)^n,x]

[Out]

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

Maple [F]

\[\int \left (a -x \right )^{m} \left (f x \right )^{p} \left (d x +c \right )^{n}d x\]

[In]

int((a-x)^m*(f*x)^p*(d*x+c)^n,x)

[Out]

int((a-x)^m*(f*x)^p*(d*x+c)^n,x)

Fricas [F]

\[ \int (a-x)^m (f x)^p (c+d x)^n \, dx=\int { {\left (d x + c\right )}^{n} \left (f x\right )^{p} {\left (a - x\right )}^{m} \,d x } \]

[In]

integrate((a-x)^m*(f*x)^p*(d*x+c)^n,x, algorithm="fricas")

[Out]

integral((d*x + c)^n*(f*x)^p*(a - x)^m, x)

Sympy [F]

\[ \int (a-x)^m (f x)^p (c+d x)^n \, dx=\int \left (f x\right )^{p} \left (a - x\right )^{m} \left (c + d x\right )^{n}\, dx \]

[In]

integrate((a-x)**m*(f*x)**p*(d*x+c)**n,x)

[Out]

Integral((f*x)**p*(a - x)**m*(c + d*x)**n, x)

Maxima [F]

\[ \int (a-x)^m (f x)^p (c+d x)^n \, dx=\int { {\left (d x + c\right )}^{n} \left (f x\right )^{p} {\left (a - x\right )}^{m} \,d x } \]

[In]

integrate((a-x)^m*(f*x)^p*(d*x+c)^n,x, algorithm="maxima")

[Out]

integrate((d*x + c)^n*(f*x)^p*(a - x)^m, x)

Giac [F]

\[ \int (a-x)^m (f x)^p (c+d x)^n \, dx=\int { {\left (d x + c\right )}^{n} \left (f x\right )^{p} {\left (a - x\right )}^{m} \,d x } \]

[In]

integrate((a-x)^m*(f*x)^p*(d*x+c)^n,x, algorithm="giac")

[Out]

integrate((d*x + c)^n*(f*x)^p*(a - x)^m, x)

Mupad [F(-1)]

Timed out. \[ \int (a-x)^m (f x)^p (c+d x)^n \, dx=\int {\left (f\,x\right )}^p\,{\left (a-x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \]

[In]

int((f*x)^p*(a - x)^m*(c + d*x)^n,x)

[Out]

int((f*x)^p*(a - x)^m*(c + d*x)^n, x)