Integrand size = 29, antiderivative size = 393 \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^2 \, dx=\frac {(b g-a h)^2 (a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (1+m,-n,-p,2+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^3 (1+m)}+\frac {2 h (b g-a h) (a+b x)^{2+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (2+m,-n,-p,3+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^3 (2+m)}+\frac {h^2 (a+b x)^{3+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (3+m,-n,-p,4+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^3 (3+m)} \]
(-a*h+b*g)^2*(b*x+a)^(1+m)*(d*x+c)^n*(f*x+e)^p*AppellF1(1+m,-n,-p,2+m,-d*( b*x+a)/(-a*d+b*c),-f*(b*x+a)/(-a*f+b*e))/b^3/(1+m)/((b*(d*x+c)/(-a*d+b*c)) ^n)/((b*(f*x+e)/(-a*f+b*e))^p)+2*h*(-a*h+b*g)*(b*x+a)^(2+m)*(d*x+c)^n*(f*x +e)^p*AppellF1(2+m,-n,-p,3+m,-d*(b*x+a)/(-a*d+b*c),-f*(b*x+a)/(-a*f+b*e))/ b^3/(2+m)/((b*(d*x+c)/(-a*d+b*c))^n)/((b*(f*x+e)/(-a*f+b*e))^p)+h^2*(b*x+a )^(3+m)*(d*x+c)^n*(f*x+e)^p*AppellF1(3+m,-n,-p,4+m,-d*(b*x+a)/(-a*d+b*c),- f*(b*x+a)/(-a*f+b*e))/b^3/(3+m)/((b*(d*x+c)/(-a*d+b*c))^n)/((b*(f*x+e)/(-a *f+b*e))^p)
\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^2 \, dx=\int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^2 \, dx \]
Time = 0.62 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.35, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {199, 177, 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.
| \(\displaystyle \int (g+h x)^2 (a+b x)^m (c+d x)^n (e+f x)^p \, dx\) |
| \(\Big \downarrow \) 199 |
| \(\displaystyle \frac {(b g-a h) \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)dx}{b}+\frac {h \int (a+b x)^{m+1} (c+d x)^n (e+f x)^p (g+h x)dx}{b}\) |
| \(\Big \downarrow \) 177 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b g-a h) \int (a+b x)^m (c+d x)^n (e+f x)^pdx}{b}+\frac {h \int (a+b x)^{m+1} (c+d x)^n (e+f x)^pdx}{b}\right )}{b}+\frac {h \left (\frac {(b g-a h) \int (a+b x)^{m+1} (c+d x)^n (e+f x)^pdx}{b}+\frac {h \int (a+b x)^{m+2} (c+d x)^n (e+f x)^pdx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 157 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b g-a h) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^pdx}{b}+\frac {h (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \int (a+b x)^{m+1} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^pdx}{b}\right )}{b}+\frac {h \left (\frac {(b g-a h) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \int (a+b x)^{m+1} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^pdx}{b}+\frac {h (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \int (a+b x)^{m+2} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^pdx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 156 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b g-a h) (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^pdx}{b}+\frac {h (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \int (a+b x)^{m+1} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^pdx}{b}\right )}{b}+\frac {h \left (\frac {(b g-a h) (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \int (a+b x)^{m+1} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^pdx}{b}+\frac {h (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \int (a+b x)^{m+2} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^pdx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 155 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b g-a h) (a+b x)^{m+1} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+1,-n,-p,m+2,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (m+1)}+\frac {h (a+b x)^{m+2} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+2,-n,-p,m+3,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (m+2)}\right )}{b}+\frac {h \left (\frac {(b g-a h) (a+b x)^{m+2} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+2,-n,-p,m+3,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (m+2)}+\frac {h (a+b x)^{m+3} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+3,-n,-p,m+4,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (m+3)}\right )}{b}\) |
((b*g - a*h)*(((b*g - a*h)*(a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^p*Appel lF1[1 + m, -n, -p, 2 + m, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b *e - a*f))])/(b^2*(1 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f*x))/(b* e - a*f))^p) + (h*(a + b*x)^(2 + m)*(c + d*x)^n*(e + f*x)^p*AppellF1[2 + m , -n, -p, 3 + m, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f) )])/(b^2*(2 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f*x))/(b*e - a*f)) ^p)))/b + (h*(((b*g - a*h)*(a + b*x)^(2 + m)*(c + d*x)^n*(e + f*x)^p*Appel lF1[2 + m, -n, -p, 3 + m, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b *e - a*f))])/(b^2*(2 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f*x))/(b* e - a*f))^p) + (h*(a + b*x)^(3 + m)*(c + d*x)^n*(e + f*x)^p*AppellF1[3 + m , -n, -p, 4 + m, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f) )])/(b^2*(3 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f*x))/(b*e - a*f)) ^p)))/b
3.2.40.3.1 Defintions of rubi rules used
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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h/b Int[(a + b*x)^(m + 1)*(c + d *x)^n*(e + f*x)^p, x], x] + Simp[(b*g - a*h)/b Int[(a + b*x)^m*(c + d*x)^ n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (Su mSimplerQ[m, 1] || ( !SumSimplerQ[n, 1] && !SumSimplerQ[p, 1]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_))^(q_), x_] :> Simp[h/b Int[(a + b*x)^(m + 1)*( c + d*x)^n*(e + f*x)^p*(g + h*x)^(q - 1), x], x] + Simp[(b*g - a*h)/b Int [(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^(q - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && IGtQ[q, 0] && (SumSimplerQ[m, 1] || ( !SumSimplerQ[n, 1] && !SumSimplerQ[p, 1]))
\[\int \left (b x +a \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{p} \left (h x +g \right )^{2}d x\]
\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^2 \, dx=\int { {\left (h x + g\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \]
Timed out. \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^2 \, dx=\text {Timed out} \]
\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^2 \, dx=\int { {\left (h x + g\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \]
\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^2 \, dx=\int { {\left (h x + g\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \]
Timed out. \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^2 \, dx=\int {\left (e+f\,x\right )}^p\,{\left (g+h\,x\right )}^2\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \]