Integrand size = 36, antiderivative size = 145 \[ \int (d+e x)^2 (f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {(b d-a e)^2 (a+b x) \left (\frac {b (d+e x)}{b d-a e}\right )^{-p} (f+g x)^n \left (\frac {b (f+g x)}{b f-a g}\right )^{-n} \left (a d+(b d+a e) x+b e x^2\right )^p \operatorname {AppellF1}\left (1+p,-2-p,-n,2+p,-\frac {e (a+b x)}{b d-a e},-\frac {g (a+b x)}{b f-a g}\right )}{b^3 (1+p)} \] Output:
(-a*e+b*d)^2*(b*x+a)*(g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p*AppellF1(p+1,-2 -p,-n,2+p,-e*(b*x+a)/(-a*e+b*d),-g*(b*x+a)/(-a*g+b*f))/b^3/(p+1)/((b*(e*x+ d)/(-a*e+b*d))^p)/((b*(g*x+f)/(-a*g+b*f))^n)
Time = 0.79 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.93 \[ \int (d+e x)^2 (f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {(b d-a e)^2 (a+b x) \left (\frac {b (d+e x)}{b d-a e}\right )^{-p} ((a+b x) (d+e x))^p (f+g x)^n \left (\frac {b (f+g x)}{b f-a g}\right )^{-n} \operatorname {AppellF1}\left (1+p,-2-p,-n,2+p,\frac {e (a+b x)}{-b d+a e},\frac {g (a+b x)}{-b f+a g}\right )}{b^3 (1+p)} \] Input:
Integrate[(d + e*x)^2*(f + g*x)^n*(a*d + (b*d + a*e)*x + b*e*x^2)^p,x]
Output:
((b*d - a*e)^2*(a + b*x)*((a + b*x)*(d + e*x))^p*(f + g*x)^n*AppellF1[1 + p, -2 - p, -n, 2 + p, (e*(a + b*x))/(-(b*d) + a*e), (g*(a + b*x))/(-(b*f) + a*g)])/(b^3*(1 + p)*((b*(d + e*x))/(b*d - a*e))^p*((b*(f + g*x))/(b*f - a*g))^n)
Time = 0.36 (sec) , antiderivative size = 145, 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.
\(\displaystyle \int (d+e x)^2 (f+g x)^n \left (x (a e+b d)+a d+b e x^2\right )^p \, dx\) |
\(\Big \downarrow \) 1268 |
\(\displaystyle (a+b x)^{-p} (d+e x)^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (a+b x)^p (d+e x)^{p+2} (f+g x)^ndx\) |
\(\Big \downarrow \) 157 |
\(\displaystyle \frac {(b d-a e)^2 (a+b x)^{-p} \left (\frac {b (d+e x)}{b d-a e}\right )^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (a+b x)^p \left (\frac {b d}{b d-a e}+\frac {b e x}{b d-a e}\right )^{p+2} (f+g x)^ndx}{b^2}\) |
\(\Big \downarrow \) 156 |
\(\displaystyle \frac {(b d-a e)^2 (a+b x)^{-p} (f+g x)^n \left (\frac {b (d+e x)}{b d-a e}\right )^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \left (\frac {b (f+g x)}{b f-a g}\right )^{-n} \int (a+b x)^p \left (\frac {b d}{b d-a e}+\frac {b e x}{b d-a e}\right )^{p+2} \left (\frac {b f}{b f-a g}+\frac {b g x}{b f-a g}\right )^ndx}{b^2}\) |
\(\Big \downarrow \) 155 |
\(\displaystyle \frac {(a+b x) (b d-a e)^2 (f+g x)^n \left (\frac {b (d+e x)}{b d-a e}\right )^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \left (\frac {b (f+g x)}{b f-a g}\right )^{-n} \operatorname {AppellF1}\left (p+1,-p-2,-n,p+2,-\frac {e (a+b x)}{b d-a e},-\frac {g (a+b x)}{b f-a g}\right )}{b^3 (p+1)}\) |
Input:
Int[(d + e*x)^2*(f + g*x)^n*(a*d + (b*d + a*e)*x + b*e*x^2)^p,x]
Output:
((b*d - a*e)^2*(a + b*x)*(f + g*x)^n*(a*d + (b*d + a*e)*x + b*e*x^2)^p*App ellF1[1 + p, -2 - p, -n, 2 + p, -((e*(a + b*x))/(b*d - a*e)), -((g*(a + b* x))/(b*f - a*g))])/(b^3*(1 + p)*((b*(d + e*x))/(b*d - a*e))^p*((b*(f + g*x ))/(b*f - a*g))^n)
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])
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]
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]
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]
\[\int \left (e x +d \right )^{2} \left (g x +f \right )^{n} \left (a d +\left (a e +b d \right ) x +b e \,x^{2}\right )^{p}d x\]
Input:
int((e*x+d)^2*(g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x)
Output:
int((e*x+d)^2*(g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x)
\[ \int (d+e x)^2 (f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b e x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} {\left (g x + f\right )}^{n} \,d x } \] Input:
integrate((e*x+d)^2*(g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x, algorithm="fr icas")
Output:
integral((e^2*x^2 + 2*d*e*x + d^2)*(b*e*x^2 + a*d + (b*d + a*e)*x)^p*(g*x + f)^n, x)
Timed out. \[ \int (d+e x)^2 (f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**2*(g*x+f)**n*(a*d+(a*e+b*d)*x+b*e*x**2)**p,x)
Output:
Timed out
\[ \int (d+e x)^2 (f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b e x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} {\left (g x + f\right )}^{n} \,d x } \] Input:
integrate((e*x+d)^2*(g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x, algorithm="ma xima")
Output:
integrate((e*x + d)^2*(b*e*x^2 + a*d + (b*d + a*e)*x)^p*(g*x + f)^n, x)
\[ \int (d+e x)^2 (f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b e x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} {\left (g x + f\right )}^{n} \,d x } \] Input:
integrate((e*x+d)^2*(g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x, algorithm="gi ac")
Output:
integrate((e*x + d)^2*(b*e*x^2 + a*d + (b*d + a*e)*x)^p*(g*x + f)^n, x)
Timed out. \[ \int (d+e x)^2 (f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\int {\left (f+g\,x\right )}^n\,{\left (d+e\,x\right )}^2\,{\left (b\,e\,x^2+\left (a\,e+b\,d\right )\,x+a\,d\right )}^p \,d x \] Input:
int((f + g*x)^n*(d + e*x)^2*(a*d + x*(a*e + b*d) + b*e*x^2)^p,x)
Output:
int((f + g*x)^n*(d + e*x)^2*(a*d + x*(a*e + b*d) + b*e*x^2)^p, x)
\[ \int (d+e x)^2 (f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\int \left (e x +d \right )^{2} \left (g x +f \right )^{n} \left (a d +\left (a e +b d \right ) x +b e \,x^{2}\right )^{p}d x \] Input:
int((e*x+d)^2*(g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x)
Output:
int((e*x+d)^2*(g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x)