\(\int \frac {(a+b x)^p (c+d x)^q}{e+f x} \, dx\) [14]

Optimal result

Integrand size = 22, antiderivative size = 100 \[ \int \frac {(a+b x)^p (c+d x)^q}{e+f x} \, dx=\frac {(a+b x)^{1+p} (c+d x)^q \left (\frac {b (c+d x)}{b c-a d}\right )^{-q} \operatorname {AppellF1}\left (1+p,-q,1,2+p,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{(b e-a f) (1+p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^p (c+d x)^q}{e+f x} \, dx=\frac {(a+b x)^{1+p} (c+d x)^q \left (\frac {b (c+d x)}{b c-a d}\right )^{-q} \operatorname {AppellF1}\left (1+p,-q,1,2+p,\frac {d (a+b x)}{-b c+a d},\frac {f (a+b x)}{-b e+a f}\right )}{(b e-a f) (1+p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {154, 153}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b x)^p (c+d x)^q}{e+f x} \, dx\)

\(\Big \downarrow \) 154

\(\displaystyle (c+d x)^q \left (\frac {b (c+d x)}{b c-a d}\right )^{-q} \int \frac {(a+b x)^p \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^q}{e+f x}dx\)

\(\Big \downarrow \) 153

\(\displaystyle \frac {(a+b x)^{p+1} (c+d x)^q \left (\frac {b (c+d x)}{b c-a d}\right )^{-q} \operatorname {AppellF1}\left (p+1,-q,1,p+2,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{(p+1) (b e-a f)}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 153

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

rule 154

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

Maple [F]

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

Input:

int((b*x+a)^p*(d*x+c)^q/(f*x+e),x)
 

Output:

int((b*x+a)^p*(d*x+c)^q/(f*x+e),x)
 

Fricas [F]

\[ \int \frac {(a+b x)^p (c+d x)^q}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q}}{f x + e} \,d x } \] Input:

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

Output:

integral((b*x + a)^p*(d*x + c)^q/(f*x + e), x)
 

Sympy [F]

\[ \int \frac {(a+b x)^p (c+d x)^q}{e+f x} \, dx=\int \frac {\left (a + b x\right )^{p} \left (c + d x\right )^{q}}{e + f x}\, dx \] Input:

integrate((b*x+a)**p*(d*x+c)**q/(f*x+e),x)
 

Output:

Integral((a + b*x)**p*(c + d*x)**q/(e + f*x), x)
 

Maxima [F]

\[ \int \frac {(a+b x)^p (c+d x)^q}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q}}{f x + e} \,d x } \] Input:

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

Output:

integrate((b*x + a)^p*(d*x + c)^q/(f*x + e), x)
 

Giac [F]

\[ \int \frac {(a+b x)^p (c+d x)^q}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q}}{f x + e} \,d x } \] Input:

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

Output:

integrate((b*x + a)^p*(d*x + c)^q/(f*x + e), x)
 

Mupad [F(-1)]

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

int(((a + b*x)^p*(c + d*x)^q)/(e + f*x),x)
 

Output:

int(((a + b*x)^p*(c + d*x)^q)/(e + f*x), x)
 

Reduce [F]

\[ \int \frac {(a+b x)^p (c+d x)^q}{e+f x} \, dx=\int \frac {\left (b x +a \right )^{p} \left (d x +c \right )^{q}}{f x +e}d x \] Input:

int((b*x+a)^p*(d*x+c)^q/(f*x+e),x)
 

Output:

int((b*x+a)^p*(d*x+c)^q/(f*x+e),x)