\(\int \frac {(g x)^n (a d+(b d+a e) x+b e x^2)^p}{d+e x} \, dx\) [149]

Optimal result

Integrand size = 34, antiderivative size = 93 \[ \int \frac {(g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p}{d+e x} \, dx=\frac {(g x)^{1+n} \left (1+\frac {b x}{a}\right )^{-p} \left (1+\frac {e x}{d}\right )^{-p} \left (a d+(b d+a e) x+b e x^2\right )^p \operatorname {AppellF1}\left (1+n,-p,1-p,2+n,-\frac {b x}{a},-\frac {e x}{d}\right )}{d g (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int \frac {(g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p}{d+e x} \, dx=\frac {x (g x)^n \left (1+\frac {b x}{a}\right )^{-p} ((a+b x) (d+e x))^p \left (1+\frac {e x}{d}\right )^{-p} \operatorname {AppellF1}\left (1+n,-p,1-p,2+n,-\frac {b x}{a},-\frac {e x}{d}\right )}{d+d n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1268, 152, 152, 150}

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.

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

\(\Big \downarrow \) 1268

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

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> 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] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 152

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

integrate((g*x)^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)^n/(e*x + d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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