\(\int (a+b x)^m (c+d x)^{-4-m} (e+f x) (g+h x) \, dx\) [130]
Optimal result
Integrand size = 29, antiderivative size = 362 \[
\int (a+b x)^m (c+d x)^{-4-m} (e+f x) (g+h x) \, dx=\frac {\left (a^2 d^2 f h \left (6+5 m+m^2\right )-a b d (3+m) (d (f g+e h)+2 c f h (1+m))+b^2 \left (2 d^2 e g+c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{b d^2 (b c-a d)^2 (2+m) (3+m)}+\frac {\left (a^2 d^2 f h \left (6+5 m+m^2\right )-a b d (3+m) (d (f g+e h)+2 c f h (1+m))+b^2 \left (2 d^2 e g+c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^2 (b c-a d)^3 (1+m) (2+m) (3+m)}+\frac {(a+b x)^{1+m} (c+d x)^{-3-m} \left (a c d f h (3+m)+b \left (d^2 e g-c d (f g+e h)-c^2 f h (2+m)\right )-d (b c-a d) f h (3+m) x\right )}{b d^2 (b c-a d) (3+m)}
\]
[Out]
(a^2*d^2*f*h*(m^2+5*m+6)-a*b*d*(3+m)*(d*(e*h+f*g)+2*c*f*h*(1+m))+b^2*(2*d^2*e*g+c*d*(e*h+f*g)*(1+m)+c^2*f*h*(m
^2+3*m+2)))*(b*x+a)^(1+m)*(d*x+c)^(-2-m)/b/d^2/(-a*d+b*c)^2/(2+m)/(3+m)+(a^2*d^2*f*h*(m^2+5*m+6)-a*b*d*(3+m)*(
d*(e*h+f*g)+2*c*f*h*(1+m))+b^2*(2*d^2*e*g+c*d*(e*h+f*g)*(1+m)+c^2*f*h*(m^2+3*m+2)))*(b*x+a)^(1+m)*(d*x+c)^(-1-
m)/d^2/(-a*d+b*c)^3/(1+m)/(2+m)/(3+m)+(b*x+a)^(1+m)*(d*x+c)^(-3-m)*(a*c*d*f*h*(3+m)+b*(d^2*e*g-c*d*(e*h+f*g)-c
^2*f*h*(2+m))-d*(-a*d+b*c)*f*h*(3+m)*x)/b/d^2/(-a*d+b*c)/(3+m)
Rubi [A] (verified)
Time = 0.23 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.00, number
of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {151, 47, 37}
\[
\int (a+b x)^m (c+d x)^{-4-m} (e+f x) (g+h x) \, dx=\frac {(a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f h \left (m^2+5 m+6\right )-a b d (m+3) (2 c f h (m+1)+d (e h+f g))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )+c d (m+1) (e h+f g)+2 d^2 e g\right )\right )}{b d^2 (m+2) (m+3) (b c-a d)^2}+\frac {(a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f h \left (m^2+5 m+6\right )-a b d (m+3) (2 c f h (m+1)+d (e h+f g))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )+c d (m+1) (e h+f g)+2 d^2 e g\right )\right )}{d^2 (m+1) (m+2) (m+3) (b c-a d)^3}+\frac {(a+b x)^{m+1} (c+d x)^{-m-3} \left (-d f h (m+3) x (b c-a d)+a c d f h (m+3)+b \left (c^2 (-f) h (m+2)-c d (e h+f g)+d^2 e g\right )\right )}{b d^2 (m+3) (b c-a d)}
\]
[In]
Int[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x)*(g + h*x),x]
[Out]
((a^2*d^2*f*h*(6 + 5*m + m^2) - a*b*d*(3 + m)*(d*(f*g + e*h) + 2*c*f*h*(1 + m)) + b^2*(2*d^2*e*g + c*d*(f*g +
e*h)*(1 + m) + c^2*f*h*(2 + 3*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(b*d^2*(b*c - a*d)^2*(2 + m)*(3
+ m)) + ((a^2*d^2*f*h*(6 + 5*m + m^2) - a*b*d*(3 + m)*(d*(f*g + e*h) + 2*c*f*h*(1 + m)) + b^2*(2*d^2*e*g + c*
d*(f*g + e*h)*(1 + m) + c^2*f*h*(2 + 3*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/(d^2*(b*c - a*d)^3*(1
+ m)*(2 + m)*(3 + m)) + ((a + b*x)^(1 + m)*(c + d*x)^(-3 - m)*(a*c*d*f*h*(3 + m) + b*(d^2*e*g - c*d*(f*g + e*h
) - c^2*f*h*(2 + m)) - d*(b*c - a*d)*f*h*(3 + m)*x))/(b*d^2*(b*c - a*d)*(3 + m))
Rule 37
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]
Rule 47
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
Rule 151
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(
b*c - a*d)*(m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Dist[
(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m +
1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d*(b*c - a*d)*(m +
1)*(m + n + 3)), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((Ge
Q[m, -2] && LtQ[m, -1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]
Rubi steps \begin{align*}
\text {integral}& = \frac {(a+b x)^{1+m} (c+d x)^{-3-m} \left (a c d f h (3+m)+b \left (d^2 e g-c d (f g+e h)-c^2 f h (2+m)\right )-d (b c-a d) f h (3+m) x\right )}{b d^2 (b c-a d) (3+m)}+\frac {\left (a^2 d^2 f h \left (6+5 m+m^2\right )-a b d (3+m) (d (f g+e h)+2 c f h (1+m))+b^2 \left (2 d^2 e g+c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) \int (a+b x)^m (c+d x)^{-3-m} \, dx}{b d^2 (b c-a d) (3+m)} \\ & = \frac {\left (a^2 d^2 f h \left (6+5 m+m^2\right )-a b d (3+m) (d (f g+e h)+2 c f h (1+m))+b^2 \left (2 d^2 e g+c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{b d^2 (b c-a d)^2 (2+m) (3+m)}+\frac {(a+b x)^{1+m} (c+d x)^{-3-m} \left (a c d f h (3+m)+b \left (d^2 e g-c d (f g+e h)-c^2 f h (2+m)\right )-d (b c-a d) f h (3+m) x\right )}{b d^2 (b c-a d) (3+m)}+\frac {\left (a^2 d^2 f h \left (6+5 m+m^2\right )-a b d (3+m) (d (f g+e h)+2 c f h (1+m))+b^2 \left (2 d^2 e g+c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d^2 (b c-a d)^2 (2+m) (3+m)} \\ & = \frac {\left (a^2 d^2 f h \left (6+5 m+m^2\right )-a b d (3+m) (d (f g+e h)+2 c f h (1+m))+b^2 \left (2 d^2 e g+c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{b d^2 (b c-a d)^2 (2+m) (3+m)}+\frac {\left (a^2 d^2 f h \left (6+5 m+m^2\right )-a b d (3+m) (d (f g+e h)+2 c f h (1+m))+b^2 \left (2 d^2 e g+c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^2 (b c-a d)^3 (1+m) (2+m) (3+m)}+\frac {(a+b x)^{1+m} (c+d x)^{-3-m} \left (a c d f h (3+m)+b \left (d^2 e g-c d (f g+e h)-c^2 f h (2+m)\right )-d (b c-a d) f h (3+m) x\right )}{b d^2 (b c-a d) (3+m)} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.36 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.61
\[
\int (a+b x)^m (c+d x)^{-4-m} (e+f x) (g+h x) \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-3-m} \left (a d f h (3+m) (c+d x)+\frac {\left (a^2 d^2 f h \left (6+5 m+m^2\right )-a b d (3+m) (d (f g+e h)+2 c f h (1+m))+b^2 \left (2 d^2 e g+c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (c+d x) (-a d (1+m)+b c (2+m)+b d x)}{(b c-a d)^2 (1+m) (2+m)}+b \left (d^2 e g-c^2 f h (2+m)-c d (e h+f (g+h (3+m) x))\right )\right )}{b d^2 (b c-a d) (3+m)}
\]
[In]
Integrate[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x)*(g + h*x),x]
[Out]
((a + b*x)^(1 + m)*(c + d*x)^(-3 - m)*(a*d*f*h*(3 + m)*(c + d*x) + ((a^2*d^2*f*h*(6 + 5*m + m^2) - a*b*d*(3 +
m)*(d*(f*g + e*h) + 2*c*f*h*(1 + m)) + b^2*(2*d^2*e*g + c*d*(f*g + e*h)*(1 + m) + c^2*f*h*(2 + 3*m + m^2)))*(c
+ d*x)*(-(a*d*(1 + m)) + b*c*(2 + m) + b*d*x))/((b*c - a*d)^2*(1 + m)*(2 + m)) + b*(d^2*e*g - c^2*f*h*(2 + m)
- c*d*(e*h + f*(g + h*(3 + m)*x)))))/(b*d^2*(b*c - a*d)*(3 + m))
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(893\) vs. \(2(362)=724\).
Time = 2.25 (sec) , antiderivative size = 894, normalized size of antiderivative = 2.47
| | |
| method | result | size |
| | |
| gosper |
\(-\frac {\left (b x +a \right )^{1+m} \left (d x +c \right )^{-3-m} \left (a^{2} d^{2} f h \,m^{2} x^{2}-2 a b c d f h \,m^{2} x^{2}+b^{2} c^{2} f h \,m^{2} x^{2}+a^{2} d^{2} e h \,m^{2} x +a^{2} d^{2} f g \,m^{2} x +5 a^{2} d^{2} f h m \,x^{2}-2 a b c d e h \,m^{2} x -2 a b c d f g \,m^{2} x -8 a b c d f h m \,x^{2}-a b \,d^{2} e h m \,x^{2}-a b \,d^{2} f g m \,x^{2}+b^{2} c^{2} e h \,m^{2} x +b^{2} c^{2} f g \,m^{2} x +3 b^{2} c^{2} f h m \,x^{2}+b^{2} c d e h m \,x^{2}+b^{2} c d f g m \,x^{2}+2 a^{2} c d f h m x +a^{2} d^{2} e g \,m^{2}+4 a^{2} d^{2} e h m x +4 a^{2} d^{2} f g m x +6 a^{2} d^{2} f h \,x^{2}-2 a b \,c^{2} f h m x -2 a b c d e g \,m^{2}-8 a b c d e h m x -8 a b c d f g m x -6 a b c d f h \,x^{2}-2 a b \,d^{2} e g m x -3 a b \,d^{2} e h \,x^{2}-3 a b \,d^{2} f g \,x^{2}+b^{2} c^{2} e g \,m^{2}+4 b^{2} c^{2} e h m x +4 b^{2} c^{2} f g m x +2 b^{2} c^{2} f h \,x^{2}+2 b^{2} c d e g m x +b^{2} c d e h \,x^{2}+b^{2} c d f g \,x^{2}+2 b^{2} d^{2} e g \,x^{2}+a^{2} c d e h m +a^{2} c d f g m +6 a^{2} c d f h x +3 a^{2} d^{2} e g m +3 a^{2} d^{2} e h x +3 a^{2} d^{2} f g x -a b \,c^{2} e h m -a b \,c^{2} f g m -2 a b \,c^{2} f h x -8 a b c d e g m -10 a b c d e h x -10 a b c d f g x -2 a b \,d^{2} e g x +5 b^{2} c^{2} e g m +3 b^{2} c^{2} e h x +3 b^{2} c^{2} f g x +6 b^{2} c d e g x +2 a^{2} c^{2} f h +a^{2} c d e h +a^{2} c d f g +2 a^{2} d^{2} e g -3 a b \,c^{2} e h -3 a b \,c^{2} f g -6 a b c d e g +6 b^{2} c^{2} e g \right )}{a^{3} d^{3} m^{3}-3 a^{2} b c \,d^{2} m^{3}+3 a \,b^{2} c^{2} d \,m^{3}-b^{3} c^{3} m^{3}+6 a^{3} d^{3} m^{2}-18 a^{2} b c \,d^{2} m^{2}+18 a \,b^{2} c^{2} d \,m^{2}-6 b^{3} c^{3} m^{2}+11 a^{3} d^{3} m -33 a^{2} b c \,d^{2} m +33 a \,b^{2} c^{2} d m -11 b^{3} c^{3} m +6 a^{3} d^{3}-18 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -6 b^{3} c^{3}}\) |
\(894\) |
| parallelrisch |
\(\text {Expression too large to display}\) |
\(4792\) |
| | |
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[In]
int((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)*(h*x+g),x,method=_RETURNVERBOSE)
[Out]
-(b*x+a)^(1+m)*(d*x+c)^(-3-m)/(a^3*d^3*m^3-3*a^2*b*c*d^2*m^3+3*a*b^2*c^2*d*m^3-b^3*c^3*m^3+6*a^3*d^3*m^2-18*a^
2*b*c*d^2*m^2+18*a*b^2*c^2*d*m^2-6*b^3*c^3*m^2+11*a^3*d^3*m-33*a^2*b*c*d^2*m+33*a*b^2*c^2*d*m-11*b^3*c^3*m+6*a
^3*d^3-18*a^2*b*c*d^2+18*a*b^2*c^2*d-6*b^3*c^3)*(a^2*d^2*f*h*m^2*x^2-2*a*b*c*d*f*h*m^2*x^2+b^2*c^2*f*h*m^2*x^2
+a^2*d^2*e*h*m^2*x+a^2*d^2*f*g*m^2*x+5*a^2*d^2*f*h*m*x^2-2*a*b*c*d*e*h*m^2*x-2*a*b*c*d*f*g*m^2*x-8*a*b*c*d*f*h
*m*x^2-a*b*d^2*e*h*m*x^2-a*b*d^2*f*g*m*x^2+b^2*c^2*e*h*m^2*x+b^2*c^2*f*g*m^2*x+3*b^2*c^2*f*h*m*x^2+b^2*c*d*e*h
*m*x^2+b^2*c*d*f*g*m*x^2+2*a^2*c*d*f*h*m*x+a^2*d^2*e*g*m^2+4*a^2*d^2*e*h*m*x+4*a^2*d^2*f*g*m*x+6*a^2*d^2*f*h*x
^2-2*a*b*c^2*f*h*m*x-2*a*b*c*d*e*g*m^2-8*a*b*c*d*e*h*m*x-8*a*b*c*d*f*g*m*x-6*a*b*c*d*f*h*x^2-2*a*b*d^2*e*g*m*x
-3*a*b*d^2*e*h*x^2-3*a*b*d^2*f*g*x^2+b^2*c^2*e*g*m^2+4*b^2*c^2*e*h*m*x+4*b^2*c^2*f*g*m*x+2*b^2*c^2*f*h*x^2+2*b
^2*c*d*e*g*m*x+b^2*c*d*e*h*x^2+b^2*c*d*f*g*x^2+2*b^2*d^2*e*g*x^2+a^2*c*d*e*h*m+a^2*c*d*f*g*m+6*a^2*c*d*f*h*x+3
*a^2*d^2*e*g*m+3*a^2*d^2*e*h*x+3*a^2*d^2*f*g*x-a*b*c^2*e*h*m-a*b*c^2*f*g*m-2*a*b*c^2*f*h*x-8*a*b*c*d*e*g*m-10*
a*b*c*d*e*h*x-10*a*b*c*d*f*g*x-2*a*b*d^2*e*g*x+5*b^2*c^2*e*g*m+3*b^2*c^2*e*h*x+3*b^2*c^2*f*g*x+6*b^2*c*d*e*g*x
+2*a^2*c^2*f*h+a^2*c*d*e*h+a^2*c*d*f*g+2*a^2*d^2*e*g-3*a*b*c^2*e*h-3*a*b*c^2*f*g-6*a*b*c*d*e*g+6*b^2*c^2*e*g)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1659 vs. \(2 (362) = 724\).
Time = 0.34 (sec) , antiderivative size = 1659, normalized size of antiderivative = 4.58
\[
\int (a+b x)^m (c+d x)^{-4-m} (e+f x) (g+h x) \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)*(h*x+g),x, algorithm="fricas")
[Out]
((a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*e*g*m^2 + ((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*f*h*m^2 + (2*b^3*d
^3*e + (b^3*c*d^2 - 3*a*b^2*d^3)*f)*g + ((b^3*c*d^2 - 3*a*b^2*d^3)*e + 2*(b^3*c^2*d - 3*a*b^2*c*d^2 + 3*a^2*b*
d^3)*f)*h + ((b^3*c*d^2 - a*b^2*d^3)*f*g + ((b^3*c*d^2 - a*b^2*d^3)*e + (3*b^3*c^2*d - 8*a*b^2*c*d^2 + 5*a^2*b
*d^3)*f)*h)*m)*x^4 + (((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*f*g + ((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*
e + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*f)*h)*m^2 + 4*(2*b^3*c*d^2*e + (b^3*c^2*d - 3*a*b^2*c*d^2)
*f)*g + 2*(2*(b^3*c^2*d - 3*a*b^2*c*d^2)*e + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 3*a^3*d^3)*f)*h + ((2*
(b^3*c*d^2 - a*b^2*d^3)*e + (5*b^3*c^2*d - 8*a*b^2*c*d^2 + 3*a^2*b*d^3)*f)*g + ((5*b^3*c^2*d - 8*a*b^2*c*d^2 +
3*a^2*b*d^3)*e + (3*b^3*c^3 - 7*a*b^2*c^2*d - a^2*b*c*d^2 + 5*a^3*d^3)*f)*h)*m)*x^3 + ((((b^3*c^2*d - 2*a*b^2
*c*d^2 + a^2*b*d^3)*e + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*f)*g + ((b^3*c^3 - a*b^2*c^2*d - a^2*b
*c*d^2 + a^3*d^3)*e + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f)*h)*m^2 + 3*(4*b^3*c^2*d*e + (b^3*c^3 - 3*a*b^
2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*f)*g + 3*(4*a^3*c*d^2*f + (b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^
3)*e)*h + (((7*b^3*c^2*d - 8*a*b^2*c*d^2 + a^2*b*d^3)*e + 4*(b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*f)
*g + (4*(b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*e + (a*b^2*c^3 - 8*a^2*b*c^2*d + 7*a^3*c*d^2)*f)*h)*m)
*x^2 + (2*(3*a*b^2*c^3 - 3*a^2*b*c^2*d + a^3*c*d^2)*e - (3*a^2*b*c^3 - a^3*c^2*d)*f)*g + (2*a^3*c^3*f - (3*a^2
*b*c^3 - a^3*c^2*d)*e)*h - ((a^2*b*c^3 - a^3*c^2*d)*e*h - ((5*a*b^2*c^3 - 8*a^2*b*c^2*d + 3*a^3*c*d^2)*e - (a^
2*b*c^3 - a^3*c^2*d)*f)*g)*m + (((a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*e*h + ((b^3*c^3 - a*b^2*c^2*d - a^2*b
*c*d^2 + a^3*d^3)*e + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f)*g)*m^2 + 2*((3*b^3*c^3 + 3*a*b^2*c^2*d - 3*a^
2*b*c*d^2 + a^3*d^3)*e - 2*(3*a^2*b*c^2*d - a^3*c*d^2)*f)*g + 4*(2*a^3*c^2*d*f - (3*a^2*b*c^2*d - a^3*c*d^2)*e
)*h + (((5*b^3*c^3 - a*b^2*c^2*d - 7*a^2*b*c*d^2 + 3*a^3*d^3)*e + (3*a*b^2*c^3 - 8*a^2*b*c^2*d + 5*a^3*c*d^2)*
f)*g + ((3*a*b^2*c^3 - 8*a^2*b*c^2*d + 5*a^3*c*d^2)*e - 2*(a^2*b*c^3 - a^3*c^2*d)*f)*h)*m)*x)*(b*x + a)^m*(d*x
+ c)^(-m - 4)/(6*b^3*c^3 - 18*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 6*a^3*d^3 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c
*d^2 - a^3*d^3)*m^3 + 6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*m^2 + 11*(b^3*c^3 - 3*a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - a^3*d^3)*m)
Sympy [F(-2)]
Exception generated. \[
\int (a+b x)^m (c+d x)^{-4-m} (e+f x) (g+h x) \, dx=\text {Exception raised: HeuristicGCDFailed}
\]
[In]
integrate((b*x+a)**m*(d*x+c)**(-4-m)*(f*x+e)*(h*x+g),x)
[Out]
Exception raised: HeuristicGCDFailed >> no luck
Maxima [F]
\[
\int (a+b x)^m (c+d x)^{-4-m} (e+f x) (g+h x) \, dx=\int { {\left (f x + e\right )} {\left (h x + g\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4} \,d x }
\]
[In]
integrate((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)*(h*x+g),x, algorithm="maxima")
[Out]
integrate((f*x + e)*(h*x + g)*(b*x + a)^m*(d*x + c)^(-m - 4), x)
Giac [F]
\[
\int (a+b x)^m (c+d x)^{-4-m} (e+f x) (g+h x) \, dx=\int { {\left (f x + e\right )} {\left (h x + g\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4} \,d x }
\]
[In]
integrate((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)*(h*x+g),x, algorithm="giac")
[Out]
integrate((f*x + e)*(h*x + g)*(b*x + a)^m*(d*x + c)^(-m - 4), x)
Mupad [B] (verification not implemented)
Time = 4.85 (sec) , antiderivative size = 1895, normalized size of antiderivative = 5.23
\[
\int (a+b x)^m (c+d x)^{-4-m} (e+f x) (g+h x) \, dx=\text {Too large to display}
\]
[In]
int(((e + f*x)*(g + h*x)*(a + b*x)^m)/(c + d*x)^(m + 4),x)
[Out]
- ((a + b*x)^m*(2*a^3*c^3*f*h + 6*a*b^2*c^3*e*g - 3*a^2*b*c^3*e*h - 3*a^2*b*c^3*f*g + 2*a^3*c*d^2*e*g + a^3*c^
2*d*e*h + a^3*c^2*d*f*g - 6*a^2*b*c^2*d*e*g + 5*a*b^2*c^3*e*g*m - a^2*b*c^3*e*h*m - a^2*b*c^3*f*g*m + 3*a^3*c*
d^2*e*g*m + a^3*c^2*d*e*h*m + a^3*c^2*d*f*g*m + a*b^2*c^3*e*g*m^2 + a^3*c*d^2*e*g*m^2 - 2*a^2*b*c^2*d*e*g*m^2
- 8*a^2*b*c^2*d*e*g*m))/((a*d - b*c)^3*(c + d*x)^(m + 4)*(11*m + 6*m^2 + m^3 + 6)) - (x^3*(a + b*x)^m*(6*a^3*d
^3*f*h + 2*b^3*c^3*f*h + 8*b^3*c*d^2*e*g + 4*b^3*c^2*d*e*h + 4*b^3*c^2*d*f*g + 5*a^3*d^3*f*h*m + 3*b^3*c^3*f*h
*m + a^3*d^3*f*h*m^2 + b^3*c^3*f*h*m^2 - 12*a*b^2*c*d^2*e*h - 12*a*b^2*c*d^2*f*g - 6*a*b^2*c^2*d*f*h + 6*a^2*b
*c*d^2*f*h - 2*a*b^2*d^3*e*g*m + 3*a^2*b*d^3*e*h*m + 3*a^2*b*d^3*f*g*m + 2*b^3*c*d^2*e*g*m + 5*b^3*c^2*d*e*h*m
+ 5*b^3*c^2*d*f*g*m + a^2*b*d^3*e*h*m^2 + a^2*b*d^3*f*g*m^2 + b^3*c^2*d*e*h*m^2 + b^3*c^2*d*f*g*m^2 - 2*a*b^2
*c*d^2*e*h*m^2 - 2*a*b^2*c*d^2*f*g*m^2 - a*b^2*c^2*d*f*h*m^2 - a^2*b*c*d^2*f*h*m^2 - 8*a*b^2*c*d^2*e*h*m - 8*a
*b^2*c*d^2*f*g*m - 7*a*b^2*c^2*d*f*h*m - a^2*b*c*d^2*f*h*m))/((a*d - b*c)^3*(c + d*x)^(m + 4)*(11*m + 6*m^2 +
m^3 + 6)) - (x*(a + b*x)^m*(2*a^3*d^3*e*g + 6*b^3*c^3*e*g + 4*a^3*c*d^2*e*h + 4*a^3*c*d^2*f*g + 8*a^3*c^2*d*f*
h + 3*a^3*d^3*e*g*m + 5*b^3*c^3*e*g*m + a^3*d^3*e*g*m^2 + b^3*c^3*e*g*m^2 + 6*a*b^2*c^2*d*e*g - 6*a^2*b*c*d^2*
e*g - 12*a^2*b*c^2*d*e*h - 12*a^2*b*c^2*d*f*g + 3*a*b^2*c^3*e*h*m + 3*a*b^2*c^3*f*g*m - 2*a^2*b*c^3*f*h*m + 5*
a^3*c*d^2*e*h*m + 5*a^3*c*d^2*f*g*m + 2*a^3*c^2*d*f*h*m + a*b^2*c^3*e*h*m^2 + a*b^2*c^3*f*g*m^2 + a^3*c*d^2*e*
h*m^2 + a^3*c*d^2*f*g*m^2 - a*b^2*c^2*d*e*g*m^2 - a^2*b*c*d^2*e*g*m^2 - 2*a^2*b*c^2*d*e*h*m^2 - 2*a^2*b*c^2*d*
f*g*m^2 - a*b^2*c^2*d*e*g*m - 7*a^2*b*c*d^2*e*g*m - 8*a^2*b*c^2*d*e*h*m - 8*a^2*b*c^2*d*f*g*m))/((a*d - b*c)^3
*(c + d*x)^(m + 4)*(11*m + 6*m^2 + m^3 + 6)) - (x^2*(a + b*x)^m*(3*a^3*d^3*e*h + 3*a^3*d^3*f*g + 3*b^3*c^3*e*h
+ 3*b^3*c^3*f*g + 12*b^3*c^2*d*e*g + 12*a^3*c*d^2*f*h + 4*a^3*d^3*e*h*m + 4*a^3*d^3*f*g*m + 4*b^3*c^3*e*h*m +
4*b^3*c^3*f*g*m + a^3*d^3*e*h*m^2 + a^3*d^3*f*g*m^2 + b^3*c^3*e*h*m^2 + b^3*c^3*f*g*m^2 - 9*a*b^2*c^2*d*e*h -
9*a*b^2*c^2*d*f*g - 9*a^2*b*c*d^2*e*h - 9*a^2*b*c*d^2*f*g + a^2*b*d^3*e*g*m + a*b^2*c^3*f*h*m + 7*b^3*c^2*d*e
*g*m + 7*a^3*c*d^2*f*h*m + a^2*b*d^3*e*g*m^2 + a*b^2*c^3*f*h*m^2 + b^3*c^2*d*e*g*m^2 + a^3*c*d^2*f*h*m^2 - 2*a
*b^2*c*d^2*e*g*m^2 - a*b^2*c^2*d*e*h*m^2 - a*b^2*c^2*d*f*g*m^2 - a^2*b*c*d^2*e*h*m^2 - a^2*b*c*d^2*f*g*m^2 - 2
*a^2*b*c^2*d*f*h*m^2 - 8*a*b^2*c*d^2*e*g*m - 4*a*b^2*c^2*d*e*h*m - 4*a*b^2*c^2*d*f*g*m - 4*a^2*b*c*d^2*e*h*m -
4*a^2*b*c*d^2*f*g*m - 8*a^2*b*c^2*d*f*h*m))/((a*d - b*c)^3*(c + d*x)^(m + 4)*(11*m + 6*m^2 + m^3 + 6)) - (x^4
*(a + b*x)^m*(2*b^3*d^3*e*g - 3*a*b^2*d^3*e*h - 3*a*b^2*d^3*f*g + 6*a^2*b*d^3*f*h + b^3*c*d^2*e*h + b^3*c*d^2*
f*g + 2*b^3*c^2*d*f*h - 6*a*b^2*c*d^2*f*h - a*b^2*d^3*e*h*m - a*b^2*d^3*f*g*m + 5*a^2*b*d^3*f*h*m + b^3*c*d^2*
e*h*m + b^3*c*d^2*f*g*m + 3*b^3*c^2*d*f*h*m + a^2*b*d^3*f*h*m^2 + b^3*c^2*d*f*h*m^2 - 2*a*b^2*c*d^2*f*h*m^2 -
8*a*b^2*c*d^2*f*h*m))/((a*d - b*c)^3*(c + d*x)^(m + 4)*(11*m + 6*m^2 + m^3 + 6))