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

Optimal. Leaf size=79 \[ \frac{(a-x)^m \left (1-\frac{x}{a}\right )^{-m} (f x)^{p+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (p+1;-m,-n;p+2;\frac{x}{a},-\frac{d x}{c}\right )}{f (p+1)} \]

[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)

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Rubi [A]  time = 0.0456763, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {135, 133} \[ \frac{(a-x)^m \left (1-\frac{x}{a}\right )^{-m} (f x)^{p+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (p+1;-m,-n;p+2;\frac{x}{a},-\frac{d x}{c}\right )}{f (p+1)} \]

Antiderivative was successfully verified.

[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 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 (a-x)^m (f x)^p (c+d x)^n \, dx &=\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]  time = 0.0986884, size = 77, normalized size = 0.97 \[ \frac{x (a-x)^m \left (\frac{a-x}{a}\right )^{-m} (f x)^p (c+d x)^n \left (\frac{c+d x}{c}\right )^{-n} F_1\left (p+1;-m,-n;p+2;\frac{x}{a},-\frac{d x}{c}\right )}{p+1} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.125, size = 0, normalized size = 0. \begin{align*} \int \left ( a-x \right ) ^{m} \left ( fx \right ) ^{p} \left ( dx+c \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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 (f x\right )^{p}{\left (a - x\right )}^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[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)