\(\int \frac {(f+g x)^n (a d+(b d+a e) x+b e x^2)^p}{d+e x} \, dx\) [314]

Optimal result

Integrand size = 36, antiderivative size = 142 \[ \int \frac {(f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p}{d+e x} \, dx=\frac {(a+b x) \left (\frac {b (d+e x)}{b d-a e}\right )^{-p} (f+g x)^n \left (\frac {b (f+g x)}{b f-a g}\right )^{-n} \left (a d+(b d+a e) x+b e x^2\right )^p \operatorname {AppellF1}\left (1+p,1-p,-n,2+p,-\frac {e (a+b x)}{b d-a e},-\frac {g (a+b x)}{b f-a g}\right )}{(b d-a e) (1+p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.93 \[ \int \frac {(f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p}{d+e x} \, dx=\frac {(a+b x) \left (\frac {b (d+e x)}{b d-a e}\right )^{-p} ((a+b x) (d+e x))^p (f+g x)^n \left (\frac {b (f+g x)}{b f-a g}\right )^{-n} \operatorname {AppellF1}\left (1+p,1-p,-n,2+p,\frac {e (a+b x)}{-b d+a e},\frac {g (a+b x)}{-b f+a g}\right )}{(b d-a e) (1+p)} \] Input:

Integrate[((f + g*x)^n*(a*d + (b*d + a*e)*x + b*e*x^2)^p)/(d + e*x),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1268, 157, 156, 155}

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.

\(\displaystyle \int \frac {(f+g x)^n \left (x (a e+b d)+a d+b e x^2\right )^p}{d+e x} \, dx\)

\(\Big \downarrow \) 1268

\(\displaystyle (a+b x)^{-p} (d+e x)^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (a+b x)^p (d+e x)^{p-1} (f+g x)^ndx\)

\(\Big \downarrow \) 157

\(\displaystyle \frac {b (a+b x)^{-p} \left (\frac {b (d+e x)}{b d-a e}\right )^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (a+b x)^p \left (\frac {b d}{b d-a e}+\frac {b e x}{b d-a e}\right )^{p-1} (f+g x)^ndx}{b d-a e}\)

\(\Big \downarrow \) 156

\(\displaystyle \frac {b (a+b x)^{-p} (f+g x)^n \left (\frac {b (d+e x)}{b d-a e}\right )^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \left (\frac {b (f+g x)}{b f-a g}\right )^{-n} \int (a+b x)^p \left (\frac {b d}{b d-a e}+\frac {b e x}{b d-a e}\right )^{p-1} \left (\frac {b f}{b f-a g}+\frac {b g x}{b f-a g}\right )^ndx}{b d-a e}\)

\(\Big \downarrow \) 155

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

Input:

Int[((f + g*x)^n*(a*d + (b*d + a*e)*x + b*e*x^2)^p)/(d + e*x),x]
 

Output:

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

Defintions of rubi rules used

rule 155

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

rule 156

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

rule 157

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, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] & 
&  !GtQ[Simplify[b/(b*c - a*d)], 0] &&  !SimplerQ[c + d*x, a + b*x] &&  !Si 
mplerQ[e + f*x, a + b*x]
 

rule 1268

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/((d 
 + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p])   Int[(d + e*x)^(m + p)*(f 
 + g*x)^n*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
 

Maple [F]

\[\int \frac {\left (g x +f \right )^{n} \left (a d +\left (a e +b d \right ) x +b e \,x^{2}\right )^{p}}{e x +d}d x\]

Input:

int((g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p/(e*x+d),x)
 

Output:

int((g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p/(e*x+d),x)
 

Fricas [F]

\[ \int \frac {(f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (b e x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} {\left (g x + f\right )}^{n}}{e x + d} \,d x } \] Input:

integrate((g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p/(e*x+d),x, algorithm="fric 
as")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p}{d+e x} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)**n*(a*d+(a*e+b*d)*x+b*e*x**2)**p/(e*x+d),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (b e x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} {\left (g x + f\right )}^{n}}{e x + d} \,d x } \] Input:

integrate((g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p/(e*x+d),x, algorithm="maxi 
ma")
 

Output:

integrate((b*e*x^2 + a*d + (b*d + a*e)*x)^p*(g*x + f)^n/(e*x + d), x)
 

Giac [F]

\[ \int \frac {(f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (b e x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} {\left (g x + f\right )}^{n}}{e x + d} \,d x } \] Input:

integrate((g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p/(e*x+d),x, algorithm="giac 
")
 

Output:

integrate((b*e*x^2 + a*d + (b*d + a*e)*x)^p*(g*x + f)^n/(e*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p}{d+e x} \, dx=\int \frac {{\left (f+g\,x\right )}^n\,{\left (b\,e\,x^2+\left (a\,e+b\,d\right )\,x+a\,d\right )}^p}{d+e\,x} \,d x \] Input:

int(((f + g*x)^n*(a*d + x*(a*e + b*d) + b*e*x^2)^p)/(d + e*x),x)
 

Output:

int(((f + g*x)^n*(a*d + x*(a*e + b*d) + b*e*x^2)^p)/(d + e*x), x)
 

Reduce [F]

\[ \int \frac {(f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p}{d+e x} \, dx=\int \frac {\left (g x +f \right )^{n} \left (a d +\left (a e +b d \right ) x +b e \,x^{2}\right )^{p}}{e x +d}d x \] Input:

int((g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p/(e*x+d),x)
 

Output:

int((g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p/(e*x+d),x)