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

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 = 0.134097, 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 \[ \frac{(a-x)^m \left (1-\frac{x}{a}\right )^{-m} (f x)^{p+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (p+1;-m,-n;p+2;\frac{x}{a},-\frac{d x}{c}\right )}{f (p+1)} \]

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)

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Rubi in Sympy [A]  time = 18.8258, size = 54, normalized size = 0.68 \[ \frac{\left (f x\right )^{p + 1} \left (1 - \frac{x}{a}\right )^{- m} \left (1 + \frac{d x}{c}\right )^{- n} \left (a - x\right )^{m} \left (c + d x\right )^{n} \operatorname{appellf_{1}}{\left (p + 1,- m,- n,p + 2,\frac{x}{a},- \frac{d x}{c} \right )}}{f \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.464028, size = 154, normalized size = 1.95 \[ \frac{a c (p+2) x (a-x)^m (f x)^p (c+d x)^n F_1\left (p+1;-m,-n;p+2;\frac{x}{a},-\frac{d x}{c}\right )}{(p+1) \left (a c (p+2) F_1\left (p+1;-m,-n;p+2;\frac{x}{a},-\frac{d x}{c}\right )-c m x F_1\left (p+2;1-m,-n;p+3;\frac{x}{a},-\frac{d x}{c}\right )+a d n x F_1\left (p+2;-m,1-n;p+3;\frac{x}{a},-\frac{d x}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(a*c*(2 + p)*(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*c*(2 + p)*AppellF1[1 + p, -m, -n, 2 + p, x/a, -((d*x)/
c)] - c*m*x*AppellF1[2 + p, 1 - m, -n, 3 + p, x/a, -((d*x)/c)] + a*d*n*x*AppellF
1[2 + p, -m, 1 - n, 3 + p, x/a, -((d*x)/c)]))

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Maple [F]  time = 0.2, size = 0, normalized size = 0. \[ \int \left ( a-x \right ) ^{m} \left ( fx \right ) ^{p} \left ( dx+c \right ) ^{n}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^n*(f*x)^p*(a - x)^m,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. \[{\rm integral}\left ({\left (d x + c\right )}^{n} \left (f x\right )^{p}{\left (a - x\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^n*(f*x)^p*(a - x)^m,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. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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