\(\int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx\) [139]

Optimal result

Integrand size = 29, antiderivative size = 530 \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\frac {(b g-a h)^3 (a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (1+m,-n,-p,2+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (1+m)}+\frac {3 h (b g-a h)^2 (a+b x)^{2+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (2+m,-n,-p,3+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (2+m)}+\frac {3 h^2 (b g-a h) (a+b x)^{3+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (3+m,-n,-p,4+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (3+m)}+\frac {h^3 (a+b x)^{4+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (4+m,-n,-p,5+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (4+m)} \]

[Out]

(-a*h+b*g)^3*(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^4/(1+m)/((b*(d*x+c)/(-a*d+b*c))^n)/((b*(f*x+e)/(-a*f+b*e))^p)+3*h*(-a*h+b*g)^2*(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^4/(2+m)/((b*(d*x+c)/(-a*d+b*c)
)^n)/((b*(f*x+e)/(-a*f+b*e))^p)+3*h^2*(-a*h+b*g)*(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^4/(3+m)/((b*(d*x+c)/(-a*d+b*c))^n)/((b*(f*x+e)/(-a*f+b*e))^p)+h^3*(
b*x+a)^(4+m)*(d*x+c)^n*(f*x+e)^p*AppellF1(4+m,-n,-p,5+m,-d*(b*x+a)/(-a*d+b*c),-f*(b*x+a)/(-a*f+b*e))/b^4/(4+m)
/((b*(d*x+c)/(-a*d+b*c))^n)/((b*(f*x+e)/(-a*f+b*e))^p)

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {187, 165, 145, 144, 143} \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\frac {3 h^2 (b g-a h) (a+b x)^{m+3} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+3,-n,-p,m+4,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+3)}+\frac {(b g-a h)^3 (a+b x)^{m+1} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+1,-n,-p,m+2,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+1)}+\frac {3 h (b g-a h)^2 (a+b x)^{m+2} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+2,-n,-p,m+3,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+2)}+\frac {h^3 (a+b x)^{m+4} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+4,-n,-p,m+5,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+4)} \]

[In]

Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^3,x]

[Out]

((b*g - a*h)^3*(a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^p*AppellF1[1 + m, -n, -p, 2 + m, -((d*(a + b*x))/(b*c -
a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b^4*(1 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f*x))/(b*e - a*f))^
p) + (3*h*(b*g - a*h)^2*(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^4*(2 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f*x))/(b*e
- a*f))^p) + (3*h^2*(b*g - a*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^4*(3 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f
*x))/(b*e - a*f))^p) + (h^3*(a + b*x)^(4 + m)*(c + d*x)^n*(e + f*x)^p*AppellF1[4 + m, -n, -p, 5 + m, -((d*(a +
b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b^4*(4 + m)*((b*(c + d*x))/(b*c - a*d))^n*((b*(e + f*x))/
(b*e - a*f))^p)

Rule 143

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

Rule 144

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

Rule 145

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^
FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*(b*(c/(b*c -
a*d)) + b*d*(x/(b*c - a*d)))^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[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x] &&  !SimplerQ[e +
f*x, a + b*x]

Rule 165

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Dist[h/b, Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(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] && (SumSimplerQ[m, 1] || ( !SumS
implerQ[n, 1] &&  !SumSimplerQ[p, 1]))

Rule 187

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_), x
_Symbol] :> Dist[h/b, Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^(q - 1), x], x] + Dist[(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]))

Rubi steps \begin{align*} \text {integral}& = \frac {h \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p (g+h x)^2 \, dx}{b}+\frac {(b g-a h) \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^2 \, dx}{b} \\ & = \frac {h^2 \int (a+b x)^{2+m} (c+d x)^n (e+f x)^p (g+h x) \, dx}{b^2}+2 \frac {(h (b g-a h)) \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p (g+h x) \, dx}{b^2}+\frac {(b g-a h)^2 \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x) \, dx}{b^2} \\ & = \frac {h^3 \int (a+b x)^{3+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+\frac {\left (h^2 (b g-a h)\right ) \int (a+b x)^{2+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2\right ) \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+2 \left (\frac {\left (h^2 (b g-a h)\right ) \int (a+b x)^{2+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2\right ) \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p \, dx}{b^3}\right )+\frac {(b g-a h)^3 \int (a+b x)^m (c+d x)^n (e+f x)^p \, dx}{b^3} \\ & = \frac {\left (h^3 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{3+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}+\frac {\left (h^2 (b g-a h) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{2+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{1+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}+2 \left (\frac {\left (h^2 (b g-a h) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{2+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{1+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b^3}\right )+\frac {\left ((b g-a h)^3 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \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)^p \, dx}{b^3} \\ & = \frac {\left (h^3 (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}\right ) \int (a+b x)^{3+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 )^p \, dx}{b^3}+\frac {\left (h^2 (b g-a h) (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}\right ) \int (a+b x)^{2+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 )^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2 (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}\right ) \int (a+b x)^{1+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 )^p \, dx}{b^3}+2 \left (\frac {\left (h^2 (b g-a h) (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}\right ) \int (a+b x)^{2+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 )^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2 (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}\right ) \int (a+b x)^{1+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 )^p \, dx}{b^3}\right )+\frac {\left ((b g-a h)^3 (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}\right ) \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 )^p \, dx}{b^3} \\ & = \frac {(b g-a h)^3 (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} F_1\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^4 (1+m)}+\frac {h (b g-a h)^2 (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} F_1\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^4 (2+m)}+\frac {h^2 (b g-a h) (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} F_1\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^4 (3+m)}+2 \left (\frac {h (b g-a h)^2 (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} F_1\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^4 (2+m)}+\frac {h^2 (b g-a h) (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} F_1\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^4 (3+m)}\right )+\frac {h^3 (a+b x)^{4+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} F_1\left (4+m;-n,-p;5+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (4+m)} \\ \end{align*}

Mathematica [F]

\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx \]

[In]

Integrate[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^3,x]

[Out]

Integrate[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^3, x]

Maple [F]

\[\int \left (b x +a \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{p} \left (h x +g \right )^{3}d x\]

[In]

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^p*(h*x+g)^3,x)

[Out]

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^p*(h*x+g)^3,x)

Fricas [F]

\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\int { {\left (h x + g\right )}^{3} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \]

[In]

integrate((b*x+a)^m*(d*x+c)^n*(f*x+e)^p*(h*x+g)^3,x, algorithm="fricas")

[Out]

integral((h^3*x^3 + 3*g*h^2*x^2 + 3*g^2*h*x + g^3)*(b*x + a)^m*(d*x + c)^n*(f*x + e)^p, x)

Sympy [F(-1)]

Timed out. \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**m*(d*x+c)**n*(f*x+e)**p*(h*x+g)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\int { {\left (h x + g\right )}^{3} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \]

[In]

integrate((b*x+a)^m*(d*x+c)^n*(f*x+e)^p*(h*x+g)^3,x, algorithm="maxima")

[Out]

integrate((h*x + g)^3*(b*x + a)^m*(d*x + c)^n*(f*x + e)^p, x)

Giac [F(-1)]

Timed out. \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)^m*(d*x+c)^n*(f*x+e)^p*(h*x+g)^3,x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\int {\left (e+f\,x\right )}^p\,{\left (g+h\,x\right )}^3\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \]

[In]

int((e + f*x)^p*(g + h*x)^3*(a + b*x)^m*(c + d*x)^n,x)

[Out]

int((e + f*x)^p*(g + h*x)^3*(a + b*x)^m*(c + d*x)^n, x)