3.3148 \(\int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-1+n} \, dx\)

Optimal. Leaf size=138 \[ \frac {(a+b x)^{m+1} (e+f x)^n (c+d x)^{-m-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{m+n} F_1\left (m+1;m+n,1-n;m+2;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {140, 139, 138} \[ \frac {(a+b x)^{m+1} (e+f x)^n (c+d x)^{-m-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{m+n} F_1\left (m+1;m+n,1-n;m+2;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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 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]

Rubi steps

\begin {align*} \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-1+n} \, dx &=\left ((c+d x)^{-m-n} \left (\frac {b (c+d x)}{b c-a d}\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 )^{-m-n} (e+f x)^{-1+n} \, dx\\ &=\frac {\left (b (c+d x)^{-m-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{m+n} (e+f x)^n \left (\frac {b (e+f x)}{b e-a f}\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 )^{-m-n} \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^{-1+n} \, dx}{b e-a f}\\ &=\frac {(a+b x)^{1+m} (c+d x)^{-m-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{m+n} (e+f x)^n \left (\frac {b (e+f x)}{b e-a f}\right )^{-n} F_1\left (1+m;m+n,1-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 = 0.16, size = 133, normalized size = 0.96 \[ \frac {(a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{1-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{m+n} F_1\left (m+1;m+n,1-n;m+2;\frac {d (a+b x)}{a d-b c},\frac {f (a+b x)}{a f-b e}\right )}{b (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 1.35, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - n} {\left (f x + e\right )}^{n - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - n} {\left (f x + e\right )}^{n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{m} \left (d x +c \right )^{-m -n} \left (f x +e \right )^{n -1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - n} {\left (f x + e\right )}^{n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e+f\,x\right )}^{n-1}\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+n}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________