\(\int (d+e x)^2 (f+g x)^n (d^2-e^2 x^2)^p \, dx\) [135]

Optimal result

Integrand size = 29, antiderivative size = 123 \[ \int (d+e x)^2 (f+g x)^n \left (d^2-e^2 x^2\right )^p \, dx=-\frac {2^{2+p} d^2 (d-e x) \left (\frac {d+e x}{d}\right )^{-p} (f+g x)^n \left (\frac {e (f+g x)}{e f+d g}\right )^{-n} \left (d^2-e^2 x^2\right )^p \operatorname {AppellF1}\left (1+p,-2-p,-n,2+p,\frac {d-e x}{2 d},\frac {g (d-e x)}{e f+d g}\right )}{e (1+p)} \] Output:

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

Mathematica [F]

\[ \int (d+e x)^2 (f+g x)^n \left (d^2-e^2 x^2\right )^p \, dx=\int (d+e x)^2 (f+g x)^n \left (d^2-e^2 x^2\right )^p \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {718, 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.

\(\displaystyle \int (d+e x)^2 \left (d^2-e^2 x^2\right )^p (f+g x)^n \, dx\)

\(\Big \downarrow \) 718

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

\(\Big \downarrow \) 157

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

\(\Big \downarrow \) 156

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

\(\Big \downarrow \) 155

\(\displaystyle \frac {2^p (d+e x)^3 \left (\frac {d-e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} \operatorname {AppellF1}\left (p+3,-p,-n,p+4,\frac {d+e x}{2 d},-\frac {g (d+e x)}{e f-d g}\right )}{e (p+3)}\)

Input:

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

Output:

(2^p*(d + e*x)^3*(f + g*x)^n*(d^2 - e^2*x^2)^p*AppellF1[3 + p, -p, -n, 4 + 
 p, (d + e*x)/(2*d), -((g*(d + e*x))/(e*f - d*g))])/(e*(3 + p)*((d - e*x)/ 
d)^p*((e*(f + g*x))/(e*f - d*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 718

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

Maple [F]

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

Input:

int((e*x+d)^2*(g*x+f)^n*(-e^2*x^2+d^2)^p,x)
 

Output:

int((e*x+d)^2*(g*x+f)^n*(-e^2*x^2+d^2)^p,x)
 

Fricas [F]

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

integrate((e*x+d)^2*(g*x+f)^n*(-e^2*x^2+d^2)^p,x, algorithm="fricas")
 

Output:

integral((e^2*x^2 + 2*d*e*x + d^2)*(-e^2*x^2 + d^2)^p*(g*x + f)^n, x)
 

Sympy [F]

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

integrate((e*x+d)**2*(g*x+f)**n*(-e**2*x**2+d**2)**p,x)
 

Output:

Integral((-(-d + e*x)*(d + e*x))**p*(d + e*x)**2*(f + g*x)**n, x)
 

Maxima [F]

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

integrate((e*x+d)^2*(g*x+f)^n*(-e^2*x^2+d^2)^p,x, algorithm="maxima")
 

Output:

integrate((e*x + d)^2*(-e^2*x^2 + d^2)^p*(g*x + f)^n, x)
 

Giac [F]

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

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

Output:

integrate((e*x + d)^2*(-e^2*x^2 + d^2)^p*(g*x + f)^n, x)
 

Mupad [F(-1)]

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

int((f + g*x)^n*(d^2 - e^2*x^2)^p*(d + e*x)^2,x)
 

Output:

int((f + g*x)^n*(d^2 - e^2*x^2)^p*(d + e*x)^2, x)
 

Reduce [F]

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

int((e*x+d)^2*(g*x+f)^n*(-e^2*x^2+d^2)^p,x)
 

Output:

( - 2*(f + g*x)**n*(d**2 - e**2*x**2)**p*d**4*g**3*n**2 - 6*(f + g*x)**n*( 
d**2 - e**2*x**2)**p*d**4*g**3*n*p - 8*(f + g*x)**n*(d**2 - e**2*x**2)**p* 
d**4*g**3*n - 4*(f + g*x)**n*(d**2 - e**2*x**2)**p*d**4*g**3*p**2 - 12*(f 
+ g*x)**n*(d**2 - e**2*x**2)**p*d**4*g**3*p - 8*(f + g*x)**n*(d**2 - e**2* 
x**2)**p*d**4*g**3 - 4*(f + g*x)**n*(d**2 - e**2*x**2)**p*d**3*e*f*g**2*n* 
*2 - 12*(f + g*x)**n*(d**2 - e**2*x**2)**p*d**3*e*f*g**2*n*p - 14*(f + g*x 
)**n*(d**2 - e**2*x**2)**p*d**3*e*f*g**2*n - 8*(f + g*x)**n*(d**2 - e**2*x 
**2)**p*d**3*e*f*g**2*p**2 - 16*(f + g*x)**n*(d**2 - e**2*x**2)**p*d**3*e* 
f*g**2*p - 6*(f + g*x)**n*(d**2 - e**2*x**2)**p*d**3*e*f*g**2 - (f + g*x)* 
*n*(d**2 - e**2*x**2)**p*d**2*e**2*f**2*g*n**2 + (f + g*x)**n*(d**2 - e**2 
*x**2)**p*d**2*e**2*f**2*g*n + (f + g*x)**n*(d**2 - e**2*x**2)**p*d**2*e** 
2*f*g**2*n**2*x + 2*(f + g*x)**n*(d**2 - e**2*x**2)**p*d**2*e**2*f*g**2*n* 
p*x + 5*(f + g*x)**n*(d**2 - e**2*x**2)**p*d**2*e**2*f*g**2*n*x + 6*(f + g 
*x)**n*(d**2 - e**2*x**2)**p*d**2*e**2*f*g**2*p*x + 6*(f + g*x)**n*(d**2 - 
 e**2*x**2)**p*d**2*e**2*f*g**2*x + 2*(f + g*x)**n*(d**2 - e**2*x**2)**p*d 
*e**3*f**2*g*n**2*x + 4*(f + g*x)**n*(d**2 - e**2*x**2)**p*d*e**3*f**2*g*n 
*p*x + 6*(f + g*x)**n*(d**2 - e**2*x**2)**p*d*e**3*f**2*g*n*x + 2*(f + g*x 
)**n*(d**2 - e**2*x**2)**p*d*e**3*f*g**2*n**2*x**2 + 8*(f + g*x)**n*(d**2 
- e**2*x**2)**p*d*e**3*f*g**2*n*p*x**2 + 8*(f + g*x)**n*(d**2 - e**2*x**2) 
**p*d*e**3*f*g**2*n*x**2 + 8*(f + g*x)**n*(d**2 - e**2*x**2)**p*d*e**3*...