\(\int \frac {x^m (e+f x)^n}{(a+b x) (c+d x)} \, dx\) [123]

Optimal result

Integrand size = 25, antiderivative size = 140 \[ \int \frac {x^m (e+f x)^n}{(a+b x) (c+d x)} \, dx=\frac {b x^{1+m} (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {f x}{e},-\frac {b x}{a}\right )}{a (b c-a d) (1+m)}-\frac {d x^{1+m} (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {f x}{e},-\frac {d x}{c}\right )}{c (b c-a d) (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {186, 140, 138} \[ \int \frac {x^m (e+f x)^n}{(a+b x) (c+d x)} \, dx=\frac {b x^{m+1} (e+f x)^n \left (\frac {f x}{e}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,1,m+2,-\frac {f x}{e},-\frac {b x}{a}\right )}{a (m+1) (b c-a d)}-\frac {d x^{m+1} (e+f x)^n \left (\frac {f x}{e}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,1,m+2,-\frac {f x}{e},-\frac {d x}{c}\right )}{c (m+1) (b c-a d)} \]

[In]

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

[Out]

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

Rule 186

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_), x
_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && IntegersQ[p, q]

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.74 \[ \int \frac {x^m (e+f x)^n}{(a+b x) (c+d x)} \, dx=\frac {x^{1+m} (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n} \left (-b c \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {f x}{e},-\frac {b x}{a}\right )+a d \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {f x}{e},-\frac {d x}{c}\right )\right )}{a c (-b c+a d) (1+m)} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {x^{m} \left (f x +e \right )^{n}}{\left (b x +a \right ) \left (d x +c \right )}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x^m (e+f x)^n}{(a+b x) (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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