Integrand size = 38, antiderivative size = 368 \[ \int (a+b x)^m (c+d x)^n (-b c f+2 a d f+b d f x)^n (g+h x) \, dx=\frac {(b g-a h) (a+b x)^{1+m} (c+d x)^n (-((b c-2 a d) f)+b d f x)^n \left (-c (b c-2 a d) f+2 a d^2 f x+b d^2 f x^2\right )^{-n} \left (1-\frac {d^2 (a+b x)^2}{(b c-a d)^2}\right )^{-n} \left (-\frac {(b c-a d)^2 f}{b}+\frac {d^2 f (a+b x)^2}{b}\right )^n \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-n,\frac {3+m}{2},\frac {d^2 (a+b x)^2}{(b c-a d)^2}\right )}{b^2 (1+m)}+\frac {h (a+b x)^{2+m} (c+d x)^n (-((b c-2 a d) f)+b d f x)^n \left (-c (b c-2 a d) f+2 a d^2 f x+b d^2 f x^2\right )^{-n} \left (1-\frac {d^2 (a+b x)^2}{(b c-a d)^2}\right )^{-n} \left (-\frac {(b c-a d)^2 f}{b}+\frac {d^2 f (a+b x)^2}{b}\right )^n \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-n,\frac {4+m}{2},\frac {d^2 (a+b x)^2}{(b c-a d)^2}\right )}{b^2 (2+m)} \] Output:
(-a*h+b*g)*(b*x+a)^(1+m)*(d*x+c)^n*(-(-2*a*d+b*c)*f+b*d*f*x)^n*(-(-a*d+b*c )^2*f/b+d^2*f*(b*x+a)^2/b)^n*hypergeom([-n, 1/2+1/2*m],[3/2+1/2*m],d^2*(b* x+a)^2/(-a*d+b*c)^2)/b^2/(1+m)/((-c*(-2*a*d+b*c)*f+2*a*d^2*f*x+b*d^2*f*x^2 )^n)/((1-d^2*(b*x+a)^2/(-a*d+b*c)^2)^n)+h*(b*x+a)^(2+m)*(d*x+c)^n*(-(-2*a* d+b*c)*f+b*d*f*x)^n*(-(-a*d+b*c)^2*f/b+d^2*f*(b*x+a)^2/b)^n*hypergeom([-n, 1+1/2*m],[2+1/2*m],d^2*(b*x+a)^2/(-a*d+b*c)^2)/b^2/(2+m)/((-c*(-2*a*d+b*c )*f+2*a*d^2*f*x+b*d^2*f*x^2)^n)/((1-d^2*(b*x+a)^2/(-a*d+b*c)^2)^n)
\[ \int (a+b x)^m (c+d x)^n (-b c f+2 a d f+b d f x)^n (g+h x) \, dx=\int (a+b x)^m (c+d x)^n (-b c f+2 a d f+b d f x)^n (g+h x) \, dx \] Input:
Integrate[(a + b*x)^m*(c + d*x)^n*(-(b*c*f) + 2*a*d*f + b*d*f*x)^n*(g + h* x),x]
Output:
Integrate[(a + b*x)^m*(c + d*x)^n*(-(b*c*f) + 2*a*d*f + b*d*f*x)^n*(g + h* x), x]
Time = 0.44 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {177, 146, 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.
| \(\displaystyle \int (g+h x) (a+b x)^m (c+d x)^n (2 a d f-b c f+b d f x)^n \, dx\) |
| \(\Big \downarrow \) 177 |
| \(\displaystyle \frac {(b g-a h) \int (a+b x)^m (c+d x)^n (b d f x-(b c-2 a d) f)^ndx}{b}+\frac {h \int (a+b x)^{m+1} (c+d x)^n (b d f x-(b c-2 a d) f)^ndx}{b}\) |
| \(\Big \downarrow \) 146 |
| \(\displaystyle \frac {(b g-a h) (c+d x)^n (b d f x-f (b c-2 a d))^n \left (-c f (b c-2 a d)+2 a d^2 f x+b d^2 f x^2\right )^{-n} \int (a+b x)^m \left (\frac {d^2 f (a+b x)^2}{b}-\frac {(b c-a d)^2 f}{b}\right )^nd(a+b x)}{b^2}+\frac {h (c+d x)^n (b d f x-f (b c-2 a d))^n \left (-c f (b c-2 a d)+2 a d^2 f x+b d^2 f x^2\right )^{-n} \int (a+b x)^{m+1} \left (\frac {d^2 f (a+b x)^2}{b}-\frac {(b c-a d)^2 f}{b}\right )^nd(a+b x)}{b^2}\) |
| \(\Big \downarrow \) 279 |
| \(\displaystyle \frac {(b g-a h) (c+d x)^n \left (1-\frac {d^2 (a+b x)^2}{(b c-a d)^2}\right )^{-n} (b d f x-f (b c-2 a d))^n \left (-c f (b c-2 a d)+2 a d^2 f x+b d^2 f x^2\right )^{-n} \left (\frac {d^2 f (a+b x)^2}{b}-\frac {f (b c-a d)^2}{b}\right )^n \int (a+b x)^m \left (1-\frac {d^2 (a+b x)^2}{(b c-a d)^2}\right )^nd(a+b x)}{b^2}+\frac {h (c+d x)^n \left (1-\frac {d^2 (a+b x)^2}{(b c-a d)^2}\right )^{-n} (b d f x-f (b c-2 a d))^n \left (-c f (b c-2 a d)+2 a d^2 f x+b d^2 f x^2\right )^{-n} \left (\frac {d^2 f (a+b x)^2}{b}-\frac {f (b c-a d)^2}{b}\right )^n \int (a+b x)^{m+1} \left (1-\frac {d^2 (a+b x)^2}{(b c-a d)^2}\right )^nd(a+b x)}{b^2}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {(b g-a h) (a+b x)^{m+1} (c+d x)^n \left (1-\frac {d^2 (a+b x)^2}{(b c-a d)^2}\right )^{-n} (b d f x-f (b c-2 a d))^n \left (-c f (b c-2 a d)+2 a d^2 f x+b d^2 f x^2\right )^{-n} \left (\frac {d^2 f (a+b x)^2}{b}-\frac {f (b c-a d)^2}{b}\right )^n \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-n,\frac {m+3}{2},\frac {d^2 (a+b x)^2}{(b c-a d)^2}\right )}{b^2 (m+1)}+\frac {h (a+b x)^{m+2} (c+d x)^n \left (1-\frac {d^2 (a+b x)^2}{(b c-a d)^2}\right )^{-n} (b d f x-f (b c-2 a d))^n \left (-c f (b c-2 a d)+2 a d^2 f x+b d^2 f x^2\right )^{-n} \left (\frac {d^2 f (a+b x)^2}{b}-\frac {f (b c-a d)^2}{b}\right )^n \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},-n,\frac {m+4}{2},\frac {d^2 (a+b x)^2}{(b c-a d)^2}\right )}{b^2 (m+2)}\) |
Input:
Int[(a + b*x)^m*(c + d*x)^n*(-(b*c*f) + 2*a*d*f + b*d*f*x)^n*(g + h*x),x]
Output:
((b*g - a*h)*(a + b*x)^(1 + m)*(c + d*x)^n*(-((b*c - 2*a*d)*f) + b*d*f*x)^ n*(-(((b*c - a*d)^2*f)/b) + (d^2*f*(a + b*x)^2)/b)^n*Hypergeometric2F1[(1 + m)/2, -n, (3 + m)/2, (d^2*(a + b*x)^2)/(b*c - a*d)^2])/(b^2*(1 + m)*(-(c *(b*c - 2*a*d)*f) + 2*a*d^2*f*x + b*d^2*f*x^2)^n*(1 - (d^2*(a + b*x)^2)/(b *c - a*d)^2)^n) + (h*(a + b*x)^(2 + m)*(c + d*x)^n*(-((b*c - 2*a*d)*f) + b *d*f*x)^n*(-(((b*c - a*d)^2*f)/b) + (d^2*f*(a + b*x)^2)/b)^n*Hypergeometri c2F1[(2 + m)/2, -n, (4 + m)/2, (d^2*(a + b*x)^2)/(b*c - a*d)^2])/(b^2*(2 + m)*(-(c*(b*c - 2*a*d)*f) + 2*a*d^2*f*x + b*d^2*f*x^2)^n*(1 - (d^2*(a + b* x)^2)/(b*c - a*d)^2)^n)
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]
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[((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])
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])
\[\int \left (b x +a \right )^{m} \left (x d +c \right )^{n} \left (f b d x +2 a d f -b c f \right )^{n} \left (h x +g \right )d x\]
Input:
int((b*x+a)^m*(d*x+c)^n*(b*d*f*x+2*a*d*f-b*c*f)^n*(h*x+g),x)
Output:
int((b*x+a)^m*(d*x+c)^n*(b*d*f*x+2*a*d*f-b*c*f)^n*(h*x+g),x)
\[ \int (a+b x)^m (c+d x)^n (-b c f+2 a d f+b d f x)^n (g+h x) \, dx=\int { {\left (h x + g\right )} {\left (b d f x - b c f + 2 \, a d f\right )}^{n} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((b*x+a)^m*(d*x+c)^n*(b*d*f*x+2*a*d*f-b*c*f)^n*(h*x+g),x, algorit hm="fricas")
Output:
integral((h*x + g)*(b*d*f*x - (b*c - 2*a*d)*f)^n*(b*x + a)^m*(d*x + c)^n, x)
Timed out. \[ \int (a+b x)^m (c+d x)^n (-b c f+2 a d f+b d f x)^n (g+h x) \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**m*(d*x+c)**n*(b*d*f*x+2*a*d*f-b*c*f)**n*(h*x+g),x)
Output:
Timed out
\[ \int (a+b x)^m (c+d x)^n (-b c f+2 a d f+b d f x)^n (g+h x) \, dx=\int { {\left (h x + g\right )} {\left (b d f x - b c f + 2 \, a d f\right )}^{n} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((b*x+a)^m*(d*x+c)^n*(b*d*f*x+2*a*d*f-b*c*f)^n*(h*x+g),x, algorit hm="maxima")
Output:
integrate((h*x + g)*(b*d*f*x - b*c*f + 2*a*d*f)^n*(b*x + a)^m*(d*x + c)^n, x)
\[ \int (a+b x)^m (c+d x)^n (-b c f+2 a d f+b d f x)^n (g+h x) \, dx=\int { {\left (h x + g\right )} {\left (b d f x - b c f + 2 \, a d f\right )}^{n} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((b*x+a)^m*(d*x+c)^n*(b*d*f*x+2*a*d*f-b*c*f)^n*(h*x+g),x, algorit hm="giac")
Output:
integrate((h*x + g)*(b*d*f*x - b*c*f + 2*a*d*f)^n*(b*x + a)^m*(d*x + c)^n, x)
Timed out. \[ \int (a+b x)^m (c+d x)^n (-b c f+2 a d f+b d f x)^n (g+h x) \, dx=\int \left (g+h\,x\right )\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n\,{\left (2\,a\,d\,f-b\,c\,f+b\,d\,f\,x\right )}^n \,d x \] Input:
int((g + h*x)*(a + b*x)^m*(c + d*x)^n*(2*a*d*f - b*c*f + b*d*f*x)^n,x)
Output:
int((g + h*x)*(a + b*x)^m*(c + d*x)^n*(2*a*d*f - b*c*f + b*d*f*x)^n, x)
\[ \int (a+b x)^m (c+d x)^n (-b c f+2 a d f+b d f x)^n (g+h x) \, dx=\text {too large to display} \] Input:
int((b*x+a)^m*(d*x+c)^n*(b*d*f*x+2*a*d*f-b*c*f)^n*(h*x+g),x)
Output:
( - (c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a**3*d**2*h*m *n - (c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a**3*d**2*h* m - 2*(c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a**3*d**2*h *n**2 - 2*(c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a**3*d* *2*h*n + 2*(c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a**2*b *c*d*h*m*n + 4*(c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a* *2*b*c*d*h*n**2 + (c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n *a**2*b*d**2*g*m**2 + 3*(c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f *x)**n*a**2*b*d**2*g*m*n + 2*(c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a**2*b*d**2*g*m + 2*(c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a**2*b*d**2*g*n**2 + 2*(c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a**2*b*d**2*g*n + (c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a**2*b*d**2*h*m**2*x + 4*(c + d*x)**n*(a + b*x)**m*( 2*a*d*f - b*c*f + b*d*f*x)**n*a**2*b*d**2*h*m*n*x + 4*(c + d*x)**n*(a + b* x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a**2*b*d**2*h*n**2*x - (c + d*x)**n*( a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a*b**2*c**2*h*m*n - 2*(c + d*x) **n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a*b**2*c**2*h*n**2 + 2*(c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a*b**2*c*d*g*m*n + 4 *(c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a*b**2*c*d*g*n** 2 + 4*(c + d*x)**n*(a + b*x)**m*(2*a*d*f - b*c*f + b*d*f*x)**n*a*b**2*c...