\(\int (a+b x)^m (c+d x)^n (2 b c f-a d f+b d f x)^m (g+h x) \, dx\) [239]

Optimal result

Integrand size = 38, antiderivative size = 368 \[ \int (a+b x)^m (c+d x)^n (2 b c f-a d f+b d f x)^m (g+h x) \, dx=\frac {(d g-c h) (a+b x)^m (c+d x)^{1+n} ((2 b c-a d) f+b d f x)^m \left (a (2 b c-a d) f+2 b^2 c f x+b^2 d f x^2\right )^{-m} \left (1-\frac {b^2 (c+d x)^2}{(b c-a d)^2}\right )^{-m} \left (-\frac {(b c-a d)^2 f}{d}+\frac {b^2 f (c+d x)^2}{d}\right )^m \operatorname {Hypergeometric2F1}\left (-m,\frac {1+n}{2},\frac {3+n}{2},\frac {b^2 (c+d x)^2}{(b c-a d)^2}\right )}{d^2 (1+n)}+\frac {h (a+b x)^m (c+d x)^{2+n} ((2 b c-a d) f+b d f x)^m \left (a (2 b c-a d) f+2 b^2 c f x+b^2 d f x^2\right )^{-m} \left (1-\frac {b^2 (c+d x)^2}{(b c-a d)^2}\right )^{-m} \left (-\frac {(b c-a d)^2 f}{d}+\frac {b^2 f (c+d x)^2}{d}\right )^m \operatorname {Hypergeometric2F1}\left (-m,\frac {2+n}{2},\frac {4+n}{2},\frac {b^2 (c+d x)^2}{(b c-a d)^2}\right )}{d^2 (2+n)} \] Output:

(-c*h+d*g)*(b*x+a)^m*(d*x+c)^(1+n)*((-a*d+2*b*c)*f+b*d*f*x)^m*(-(-a*d+b*c) 
^2*f/d+b^2*f*(d*x+c)^2/d)^m*hypergeom([-m, 1/2+1/2*n],[3/2+1/2*n],b^2*(d*x 
+c)^2/(-a*d+b*c)^2)/d^2/(1+n)/((a*(-a*d+2*b*c)*f+2*b^2*c*f*x+b^2*d*f*x^2)^ 
m)/((1-b^2*(d*x+c)^2/(-a*d+b*c)^2)^m)+h*(b*x+a)^m*(d*x+c)^(2+n)*((-a*d+2*b 
*c)*f+b*d*f*x)^m*(-(-a*d+b*c)^2*f/d+b^2*f*(d*x+c)^2/d)^m*hypergeom([-m, 1+ 
1/2*n],[2+1/2*n],b^2*(d*x+c)^2/(-a*d+b*c)^2)/d^2/(2+n)/((a*(-a*d+2*b*c)*f+ 
2*b^2*c*f*x+b^2*d*f*x^2)^m)/((1-b^2*(d*x+c)^2/(-a*d+b*c)^2)^m)
 

Mathematica [F]

\[ \int (a+b x)^m (c+d x)^n (2 b c f-a d f+b d f x)^m (g+h x) \, dx=\int (a+b x)^m (c+d x)^n (2 b c f-a d f+b d f x)^m (g+h x) \, dx \] Input:

Integrate[(a + b*x)^m*(c + d*x)^n*(2*b*c*f - a*d*f + b*d*f*x)^m*(g + h*x), 
x]
 

Output:

Integrate[(a + b*x)^m*(c + d*x)^n*(2*b*c*f - a*d*f + b*d*f*x)^m*(g + h*x), 
 x]
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.51 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {177, 146, 157, 156, 155, 279, 278}

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.

\(\displaystyle \int (g+h x) (a+b x)^m (c+d x)^n (-a d f+2 b c f+b d f x)^m \, dx\)

\(\Big \downarrow \) 177

\(\displaystyle \frac {(b g-a h) \int (a+b x)^m (c+d x)^n ((2 b c-a d) f+b d x f)^mdx}{b}+\frac {h \int (a+b x)^{m+1} (c+d x)^n ((2 b c-a d) f+b d x f)^mdx}{b}\)

\(\Big \downarrow \) 146

\(\displaystyle \frac {(b g-a h) (a+b x)^m (f (2 b c-a d)+b d f x)^m \left (a f (2 b c-a d)+2 b^2 c f x+b^2 d f x^2\right )^{-m} \int (c+d x)^n \left (\frac {b^2 f (c+d x)^2}{d}-\frac {(b c-a d)^2 f}{d}\right )^md(c+d x)}{b d}+\frac {h \int (a+b x)^{m+1} (c+d x)^n ((2 b c-a d) f+b d x f)^mdx}{b}\)

\(\Big \downarrow \) 157

\(\displaystyle \frac {(b g-a h) (a+b x)^m (f (2 b c-a d)+b d f x)^m \left (a f (2 b c-a d)+2 b^2 c f x+b^2 d f x^2\right )^{-m} \int (c+d x)^n \left (\frac {b^2 f (c+d x)^2}{d}-\frac {(b c-a d)^2 f}{d}\right )^md(c+d x)}{b d}+\frac {h (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \int (a+b x)^{m+1} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n ((2 b c-a d) f+b d x f)^mdx}{b}\)

\(\Big \downarrow \) 156

\(\displaystyle \frac {(b g-a h) (a+b x)^m (f (2 b c-a d)+b d f x)^m \left (a f (2 b c-a d)+2 b^2 c f x+b^2 d f x^2\right )^{-m} \int (c+d x)^n \left (\frac {b^2 f (c+d x)^2}{d}-\frac {(b c-a d)^2 f}{d}\right )^md(c+d x)}{b d}+\frac {h 2^m (c+d x)^n \left (\frac {-a d+2 b c+b d x}{b c-a d}\right )^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (f (2 b c-a d)+b d f x)^m \int (a+b x)^{m+1} \left (\frac {2 b c-a d}{2 (b c-a d)}+\frac {b d x}{2 (b c-a d)}\right )^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^ndx}{b}\)

\(\Big \downarrow \) 155

\(\displaystyle \frac {(b g-a h) (a+b x)^m (f (2 b c-a d)+b d f x)^m \left (a f (2 b c-a d)+2 b^2 c f x+b^2 d f x^2\right )^{-m} \int (c+d x)^n \left (\frac {b^2 f (c+d x)^2}{d}-\frac {(b c-a d)^2 f}{d}\right )^md(c+d x)}{b d}+\frac {h 2^m (a+b x)^{m+2} (c+d x)^n \left (\frac {-a d+2 b c+b d x}{b c-a d}\right )^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (f (2 b c-a d)+b d f x)^m \operatorname {AppellF1}\left (m+2,-m,-n,m+3,-\frac {d (a+b x)}{2 (b c-a d)},-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (m+2)}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {(b g-a h) (a+b x)^m \left (1-\frac {b^2 (c+d x)^2}{(b c-a d)^2}\right )^{-m} (f (2 b c-a d)+b d f x)^m \left (a f (2 b c-a d)+2 b^2 c f x+b^2 d f x^2\right )^{-m} \left (\frac {b^2 f (c+d x)^2}{d}-\frac {f (b c-a d)^2}{d}\right )^m \int (c+d x)^n \left (1-\frac {b^2 (c+d x)^2}{(b c-a d)^2}\right )^md(c+d x)}{b d}+\frac {h 2^m (a+b x)^{m+2} (c+d x)^n \left (\frac {-a d+2 b c+b d x}{b c-a d}\right )^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (f (2 b c-a d)+b d f x)^m \operatorname {AppellF1}\left (m+2,-m,-n,m+3,-\frac {d (a+b x)}{2 (b c-a d)},-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (m+2)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {h 2^m (a+b x)^{m+2} (c+d x)^n \left (\frac {-a d+2 b c+b d x}{b c-a d}\right )^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (f (2 b c-a d)+b d f x)^m \operatorname {AppellF1}\left (m+2,-m,-n,m+3,-\frac {d (a+b x)}{2 (b c-a d)},-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (m+2)}+\frac {(b g-a h) (a+b x)^m (c+d x)^{n+1} \left (1-\frac {b^2 (c+d x)^2}{(b c-a d)^2}\right )^{-m} (f (2 b c-a d)+b d f x)^m \left (a f (2 b c-a d)+2 b^2 c f x+b^2 d f x^2\right )^{-m} \left (\frac {b^2 f (c+d x)^2}{d}-\frac {f (b c-a d)^2}{d}\right )^m \operatorname {Hypergeometric2F1}\left (-m,\frac {n+1}{2},\frac {n+3}{2},\frac {b^2 (c+d x)^2}{(b c-a d)^2}\right )}{b d (n+1)}\)

Input:

Int[(a + b*x)^m*(c + d*x)^n*(2*b*c*f - a*d*f + b*d*f*x)^m*(g + h*x),x]
 

Output:

(2^m*h*(a + b*x)^(2 + m)*(c + d*x)^n*((2*b*c - a*d)*f + b*d*f*x)^m*AppellF 
1[2 + m, -m, -n, 3 + m, -1/2*(d*(a + b*x))/(b*c - a*d), -((d*(a + b*x))/(b 
*c - a*d))])/(b^2*(2 + m)*((b*(c + d*x))/(b*c - a*d))^n*((2*b*c - a*d + b* 
d*x)/(b*c - a*d))^m) + ((b*g - a*h)*(a + b*x)^m*(c + d*x)^(1 + n)*((2*b*c 
- a*d)*f + b*d*f*x)^m*(-(((b*c - a*d)^2*f)/d) + (b^2*f*(c + d*x)^2)/d)^m*H 
ypergeometric2F1[-m, (1 + n)/2, (3 + n)/2, (b^2*(c + d*x)^2)/(b*c - a*d)^2 
])/(b*d*(1 + n)*(a*(2*b*c - a*d)*f + 2*b^2*c*f*x + b^2*d*f*x^2)^m*(1 - (b^ 
2*(c + d*x)^2)/(b*c - a*d)^2)^m)
 

Defintions of rubi rules used

rule 146

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(c + d*x)^n*((e + f*x)^p/(b*(c*e + (d*e + c*f)*x + d*f* 
x^2)^n))   Subst[Int[x^m*(c*e - (d*e + c*f)^2/(4*d*f) + d*f*(x^2/b^2))^n, x 
], x, a + b*x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[p, n] && 
 EqQ[b*d*e + b*c*f - 2*a*d*f, 0]
 

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 177

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]))
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

Maple [F]

\[\int \left (b x +a \right )^{m} \left (x d +c \right )^{n} \left (f b d x -a d f +2 b c f \right )^{m} \left (h x +g \right )d x\]

Input:

int((b*x+a)^m*(d*x+c)^n*(b*d*f*x-a*d*f+2*b*c*f)^m*(h*x+g),x)
 

Output:

int((b*x+a)^m*(d*x+c)^n*(b*d*f*x-a*d*f+2*b*c*f)^m*(h*x+g),x)
 

Fricas [F]

\[ \int (a+b x)^m (c+d x)^n (2 b c f-a d f+b d f x)^m (g+h x) \, dx=\int { {\left (h x + g\right )} {\left (b d f x + 2 \, b c f - a d f\right )}^{m} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^n*(b*d*f*x-a*d*f+2*b*c*f)^m*(h*x+g),x, algorit 
hm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int (a+b x)^m (c+d x)^n (2 b c f-a d f+b d f x)^m (g+h x) \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**m*(d*x+c)**n*(b*d*f*x-a*d*f+2*b*c*f)**m*(h*x+g),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (a+b x)^m (c+d x)^n (2 b c f-a d f+b d f x)^m (g+h x) \, dx=\int { {\left (h x + g\right )} {\left (b d f x + 2 \, b c f - a d f\right )}^{m} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^n*(b*d*f*x-a*d*f+2*b*c*f)^m*(h*x+g),x, algorit 
hm="maxima")
 

Output:

integrate((h*x + g)*(b*d*f*x + 2*b*c*f - a*d*f)^m*(b*x + a)^m*(d*x + c)^n, 
 x)
 

Giac [F]

\[ \int (a+b x)^m (c+d x)^n (2 b c f-a d f+b d f x)^m (g+h x) \, dx=\int { {\left (h x + g\right )} {\left (b d f x + 2 \, b c f - a d f\right )}^{m} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^n*(b*d*f*x-a*d*f+2*b*c*f)^m*(h*x+g),x, algorit 
hm="giac")
 

Output:

integrate((h*x + g)*(b*d*f*x + 2*b*c*f - a*d*f)^m*(b*x + a)^m*(d*x + c)^n, 
 x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^m (c+d x)^n (2 b c f-a d f+b d f x)^m (g+h x) \, dx=\int \left (g+h\,x\right )\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n\,{\left (2\,b\,c\,f-a\,d\,f+b\,d\,f\,x\right )}^m \,d x \] Input:

int((g + h*x)*(a + b*x)^m*(c + d*x)^n*(2*b*c*f - a*d*f + b*d*f*x)^m,x)
 

Output:

int((g + h*x)*(a + b*x)^m*(c + d*x)^n*(2*b*c*f - a*d*f + b*d*f*x)^m, x)
 

Reduce [F]

\[ \int (a+b x)^m (c+d x)^n (2 b c f-a d f+b d f x)^m (g+h x) \, dx=\text {too large to display} \] Input:

int((b*x+a)^m*(d*x+c)^n*(b*d*f*x-a*d*f+2*b*c*f)^m*(h*x+g),x)
 

Output:

( - 2*(c + d*x)**n*(a + b*x)**m*( - a*d*f + 2*b*c*f + b*d*f*x)**m*a**2*c*d 
**2*h*m**2 - (c + d*x)**n*(a + b*x)**m*( - a*d*f + 2*b*c*f + b*d*f*x)**m*a 
**2*c*d**2*h*m*n - 2*(c + d*x)**n*(a + b*x)**m*( - a*d*f + 2*b*c*f + b*d*f 
*x)**m*a**2*d**3*g*m**2 - (c + d*x)**n*(a + b*x)**m*( - a*d*f + 2*b*c*f + 
b*d*f*x)**m*a**2*d**3*g*m*n - 2*(c + d*x)**n*(a + b*x)**m*( - a*d*f + 2*b* 
c*f + b*d*f*x)**m*a**2*d**3*g*m + 4*(c + d*x)**n*(a + b*x)**m*( - a*d*f + 
2*b*c*f + b*d*f*x)**m*a*b*c**2*d*h*m**2 + 2*(c + d*x)**n*(a + b*x)**m*( - 
a*d*f + 2*b*c*f + b*d*f*x)**m*a*b*c**2*d*h*m*n + 4*(c + d*x)**n*(a + b*x)* 
*m*( - a*d*f + 2*b*c*f + b*d*f*x)**m*a*b*c*d**2*g*m**2 + 2*(c + d*x)**n*(a 
 + b*x)**m*( - a*d*f + 2*b*c*f + b*d*f*x)**m*a*b*c*d**2*g*m*n + 4*(c + d*x 
)**n*(a + b*x)**m*( - a*d*f + 2*b*c*f + b*d*f*x)**m*a*b*c*d**2*g*m - 2*(c 
+ d*x)**n*(a + b*x)**m*( - a*d*f + 2*b*c*f + b*d*f*x)**m*b**2*c**3*h*m**2 
- (c + d*x)**n*(a + b*x)**m*( - a*d*f + 2*b*c*f + b*d*f*x)**m*b**2*c**3*h* 
m*n - 2*(c + d*x)**n*(a + b*x)**m*( - a*d*f + 2*b*c*f + b*d*f*x)**m*b**2*c 
**3*h*m - (c + d*x)**n*(a + b*x)**m*( - a*d*f + 2*b*c*f + b*d*f*x)**m*b**2 
*c**3*h*n + 2*(c + d*x)**n*(a + b*x)**m*( - a*d*f + 2*b*c*f + b*d*f*x)**m* 
b**2*c**2*d*g*m**2 + 3*(c + d*x)**n*(a + b*x)**m*( - a*d*f + 2*b*c*f + b*d 
*f*x)**m*b**2*c**2*d*g*m*n + 2*(c + d*x)**n*(a + b*x)**m*( - a*d*f + 2*b*c 
*f + b*d*f*x)**m*b**2*c**2*d*g*m + (c + d*x)**n*(a + b*x)**m*( - a*d*f + 2 
*b*c*f + b*d*f*x)**m*b**2*c**2*d*g*n**2 + 2*(c + d*x)**n*(a + b*x)**m*(...