3.2.36 \(\int \frac {(A+B x) (c+d x)^n (e+f x)^p}{a+b x} \, dx\) [136]

3.2.36.1 Optimal result

Integrand size = 27, antiderivative size = 177 \[ \int \frac {(A+B x) (c+d x)^n (e+f x)^p}{a+b x} \, dx=-\frac {(A b-a B) (c+d x)^{1+n} (e+f x)^p \left (\frac {d (e+f x)}{d e-c f}\right )^{-p} \operatorname {AppellF1}\left (1+n,1,-p,2+n,\frac {b (c+d x)}{b c-a d},-\frac {f (c+d x)}{d e-c f}\right )}{b (b c-a d) (1+n)}-\frac {B (c+d x)^{1+n} (e+f x)^{1+p} \operatorname {Hypergeometric2F1}\left (1,2+n+p,2+p,\frac {d (e+f x)}{d e-c f}\right )}{b (d e-c f) (1+p)} \]

output

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

3.2.36.2 Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.12 \[ \int \frac {(A+B x) (c+d x)^n (e+f x)^p}{a+b x} \, dx=\frac {(c+d x)^n (e+f x)^p \left (\frac {(A b-a B) \left (\frac {b (c+d x)}{d (a+b x)}\right )^{-n} \left (\frac {b (e+f x)}{f (a+b x)}\right )^{-p} \operatorname {AppellF1}\left (-n-p,-n,-p,1-n-p,\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )}{n+p}+\frac {b B \left (\frac {f (c+d x)}{-d e+c f}\right )^{-n} (e+f x) \operatorname {Hypergeometric2F1}\left (-n,1+p,2+p,\frac {d (e+f x)}{d e-c f}\right )}{f (1+p)}\right )}{b^2} \]

input

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

output

((c + d*x)^n*(e + f*x)^p*(((A*b - a*B)*AppellF1[-n - p, -n, -p, 1 - n - p, 
 (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a + b*x))])/((n + p)*((b 
*(c + d*x))/(d*(a + b*x)))^n*((b*(e + f*x))/(f*(a + b*x)))^p) + (b*B*(e + 
f*x)*Hypergeometric2F1[-n, 1 + p, 2 + p, (d*(e + f*x))/(d*e - c*f)])/(f*(1 
 + p)*((f*(c + d*x))/(-(d*e) + c*f))^n)))/b^2
 

3.2.36.3 Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {175, 80, 79, 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.

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

\(\Big \downarrow \) 175

\(\displaystyle \frac {(A b-a B) \int \frac {(c+d x)^n (e+f x)^p}{a+b x}dx}{b}+\frac {B \int (c+d x)^n (e+f x)^pdx}{b}\)

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {(A b-a B) \int \frac {(c+d x)^n (e+f x)^p}{a+b x}dx}{b}+\frac {B (c+d x)^{n+1} (e+f x)^p \left (\frac {d (e+f x)}{d e-c f}\right )^{-p} \operatorname {Hypergeometric2F1}\left (n+1,-p,n+2,-\frac {f (c+d x)}{d e-c f}\right )}{b d (n+1)}\)

\(\Big \downarrow \) 154

\(\displaystyle \frac {(A b-a B) (e+f x)^p \left (\frac {d (e+f x)}{d e-c f}\right )^{-p} \int \frac {(c+d x)^n \left (\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}\right )^p}{a+b x}dx}{b}+\frac {B (c+d x)^{n+1} (e+f x)^p \left (\frac {d (e+f x)}{d e-c f}\right )^{-p} \operatorname {Hypergeometric2F1}\left (n+1,-p,n+2,-\frac {f (c+d x)}{d e-c f}\right )}{b d (n+1)}\)

\(\Big \downarrow \) 153

\(\displaystyle \frac {B (c+d x)^{n+1} (e+f x)^p \left (\frac {d (e+f x)}{d e-c f}\right )^{-p} \operatorname {Hypergeometric2F1}\left (n+1,-p,n+2,-\frac {f (c+d x)}{d e-c f}\right )}{b d (n+1)}-\frac {(A b-a B) (c+d x)^{n+1} (e+f x)^p \left (\frac {d (e+f x)}{d e-c f}\right )^{-p} \operatorname {AppellF1}\left (n+1,-p,1,n+2,-\frac {f (c+d x)}{d e-c f},\frac {b (c+d x)}{b c-a d}\right )}{b (n+1) (b c-a d)}\)

input

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

output

-(((A*b - a*B)*(c + d*x)^(1 + n)*(e + f*x)^p*AppellF1[1 + n, -p, 1, 2 + n, 
 -((f*(c + d*x))/(d*e - c*f)), (b*(c + d*x))/(b*c - a*d)])/(b*(b*c - a*d)* 
(1 + n)*((d*(e + f*x))/(d*e - c*f))^p)) + (B*(c + d*x)^(1 + n)*(e + f*x)^p 
*Hypergeometric2F1[1 + n, -p, 2 + n, -((f*(c + d*x))/(d*e - c*f))])/(b*d*( 
1 + n)*((d*(e + f*x))/(d*e - c*f))^p)
 

3.2.36.3.1 Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(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*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

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]
 

rule 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

3.2.36.4 Maple [F]

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

input

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

output

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

3.2.36.5 Fricas [F]

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

input

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

output

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

3.2.36.6 Sympy [F(-1)]

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

input

integrate((B*x+A)*(d*x+c)**n*(f*x+e)**p/(b*x+a),x)
 

output

Timed out
 

3.2.36.7 Maxima [F]

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

input

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

output

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

3.2.36.8 Giac [F]

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

input

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

output

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

3.2.36.9 Mupad [F(-1)]

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

input

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

output

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