3.3155 \(\int (a+b x)^m (c+d x)^n (e+f x)^{-1-m-n} \, dx\)

Optimal. Leaf size=137 \[ \frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+1;-n,m+n+1;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)} \]

[Out]

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

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Rubi [A]  time = 0.0849411, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {140, 139, 138} \[ \frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+1;-n,m+n+1;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^(-1 - m - n),x]

[Out]

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

Rule 140

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^
FracPart[n]/((b/(b*c - a*d))^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*((b*c)/(b*c
- a*d) + (b*d*x)/(b*c - a*d))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] &&  !GtQ[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x] &&  !SimplerQ[e +
f*x, a + b*x]

Rule 139

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((b/(b*e - a*f))^IntPart[p]*((b*(e + f*x))/(b*e - a*f))^FracPart[p]), Int[(a + b*x)^m*(c + d*x)^n*
((b*e)/(b*e - a*f) + (b*f*x)/(b*e - a*f))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 138

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b*(m + 1
)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rubi steps

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

Mathematica [A]  time = 0.148743, size = 130, normalized size = 0.95 \[ \frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{m+n+1} F_1\left (m+1;-n,m+n+1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m*(c + d*x)^n*(e + f*x)^(-1 - m - n),x]

[Out]

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

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Maple [F]  time = 0.138, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{-1-m-n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-1-m-n),x)

[Out]

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-1-m-n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-1-m-n),x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-1-m-n),x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 1), 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((b*x+a)**m*(d*x+c)**n*(f*x+e)**(-1-m-n),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-1-m-n),x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 1), x)