3.134 \(\int (a+b x) (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx\)

Optimal. Leaf size=363 \[ \frac{(c+d x)^{-m-2} (e+f x)^{m+1} \left (a d f (c f h (m+1)+d (2 f g-e h (m+3)))+b \left (c^2 f^2 h \left (m^2+3 m+2\right )+c d f (m+1) (f g-2 e h (m+3))+d^2 (-e) (m+3) (f g-e h (m+2))\right )\right )}{d^2 f (m+2) (m+3) (d e-c f)^2}-\frac{(c+d x)^{-m-1} (e+f x)^{m+1} \left (a d f (c f h (m+1)+d (2 f g-e h (m+3)))+b \left (c^2 f^2 h \left (m^2+3 m+2\right )+c d f (m+1) (f g-2 e h (m+3))+d^2 (-e) (m+3) (f g-e h (m+2))\right )\right )}{d^2 (m+1) (m+2) (m+3) (d e-c f)^3}-\frac{(c+d x)^{-m-3} (e+f x)^{m+1} (a d f (d g-c h)-b c (c f h (m+2)+d (f g-e h (m+3)))+b d h (m+3) x (d e-c f))}{d^2 f (m+3) (d e-c f)} \]

[Out]

((b*(c^2*f^2*h*(2 + 3*m + m^2) - d^2*e*(3 + m)*(f*g - e*h*(2 + m)) + c*d*f*(1 +
m)*(f*g - 2*e*h*(3 + m))) + a*d*f*(c*f*h*(1 + m) + d*(2*f*g - e*h*(3 + m))))*(c
+ d*x)^(-2 - m)*(e + f*x)^(1 + m))/(d^2*f*(d*e - c*f)^2*(2 + m)*(3 + m)) - ((b*(
c^2*f^2*h*(2 + 3*m + m^2) - d^2*e*(3 + m)*(f*g - e*h*(2 + m)) + c*d*f*(1 + m)*(f
*g - 2*e*h*(3 + m))) + a*d*f*(c*f*h*(1 + m) + d*(2*f*g - e*h*(3 + m))))*(c + d*x
)^(-1 - m)*(e + f*x)^(1 + m))/(d^2*(d*e - c*f)^3*(1 + m)*(2 + m)*(3 + m)) - ((c
+ d*x)^(-3 - m)*(e + f*x)^(1 + m)*(a*d*f*(d*g - c*h) - b*c*(c*f*h*(2 + m) + d*(f
*g - e*h*(3 + m))) + b*d*(d*e - c*f)*h*(3 + m)*x))/(d^2*f*(d*e - c*f)*(3 + m))

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Rubi [A]  time = 1.12502, antiderivative size = 360, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{(c+d x)^{-m-2} (e+f x)^{m+1} \left (a d f (c f h (m+1)-d e h (m+3)+2 d f g)+b \left (c^2 f^2 h \left (m^2+3 m+2\right )+c d f (m+1) (f g-2 e h (m+3))+d^2 (-e) (m+3) (f g-e h (m+2))\right )\right )}{d^2 f (m+2) (m+3) (d e-c f)^2}-\frac{(c+d x)^{-m-1} (e+f x)^{m+1} \left (a d f (c f h (m+1)-d e h (m+3)+2 d f g)+b \left (c^2 f^2 h \left (m^2+3 m+2\right )+c d f (m+1) (f g-2 e h (m+3))+d^2 (-e) (m+3) (f g-e h (m+2))\right )\right )}{d^2 (m+1) (m+2) (m+3) (d e-c f)^3}-\frac{(c+d x)^{-m-3} (e+f x)^{m+1} (a d f (d g-c h)-b c (c f h (m+2)-d e h (m+3)+d f g)+b d h (m+3) x (d e-c f))}{d^2 f (m+3) (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(c + d*x)^(-4 - m)*(e + f*x)^m*(g + h*x),x]

[Out]

((a*d*f*(2*d*f*g + c*f*h*(1 + m) - d*e*h*(3 + m)) + b*(c^2*f^2*h*(2 + 3*m + m^2)
- d^2*e*(3 + m)*(f*g - e*h*(2 + m)) + c*d*f*(1 + m)*(f*g - 2*e*h*(3 + m))))*(c
+ d*x)^(-2 - m)*(e + f*x)^(1 + m))/(d^2*f*(d*e - c*f)^2*(2 + m)*(3 + m)) - ((a*d
*f*(2*d*f*g + c*f*h*(1 + m) - d*e*h*(3 + m)) + b*(c^2*f^2*h*(2 + 3*m + m^2) - d^
2*e*(3 + m)*(f*g - e*h*(2 + m)) + c*d*f*(1 + m)*(f*g - 2*e*h*(3 + m))))*(c + d*x
)^(-1 - m)*(e + f*x)^(1 + m))/(d^2*(d*e - c*f)^3*(1 + m)*(2 + m)*(3 + m)) - ((c
+ d*x)^(-3 - m)*(e + f*x)^(1 + m)*(a*d*f*(d*g - c*h) - b*c*(d*f*g + c*f*h*(2 + m
) - d*e*h*(3 + m)) + b*d*(d*e - c*f)*h*(3 + m)*x))/(d^2*f*(d*e - c*f)*(3 + m))

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Rubi in Sympy [A]  time = 125.324, size = 337, normalized size = 0.93 \[ \frac{\left (c + d x\right )^{- m - 1} \left (e + f x\right )^{m + 1} \left (b c^{2} f^{2} h \left (m + 1\right ) \left (m + 2\right ) + c d f \left (m + 1\right ) \left (- 2 b e h \left (m + 3\right ) + f \left (a h + b g\right )\right ) + d^{2} \left (2 a f^{2} g + b e^{2} h \left (m + 2\right ) \left (m + 3\right ) - e f \left (m + 3\right ) \left (a h + b g\right )\right )\right )}{d^{2} \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right ) \left (c f - d e\right )^{3}} - \frac{\left (c + d x\right )^{- m - 3} \left (e + f x\right )^{m + 1} \left (- a d^{2} f g + b c^{2} f h \left (m + 2\right ) + b d h x \left (m + 3\right ) \left (c f - d e\right ) + c d \left (- b e h \left (m + 3\right ) + f \left (a h + b g\right )\right )\right )}{d^{2} f \left (m + 3\right ) \left (c f - d e\right )} + \frac{\left (c + d x\right )^{- m - 2} \left (e + f x\right )^{m + 1} \left (b c^{2} f^{2} h \left (m + 1\right ) \left (m + 2\right ) + c d f \left (m + 1\right ) \left (- 2 b e h \left (m + 3\right ) + f \left (a h + b g\right )\right ) + d^{2} \left (2 a f^{2} g + b e^{2} h \left (m + 2\right ) \left (m + 3\right ) - e f \left (m + 3\right ) \left (a h + b g\right )\right )\right )}{d^{2} f \left (m + 2\right ) \left (m + 3\right ) \left (c f - d e\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(d*x+c)**(-4-m)*(f*x+e)**m*(h*x+g),x)

[Out]

(c + d*x)**(-m - 1)*(e + f*x)**(m + 1)*(b*c**2*f**2*h*(m + 1)*(m + 2) + c*d*f*(m
+ 1)*(-2*b*e*h*(m + 3) + f*(a*h + b*g)) + d**2*(2*a*f**2*g + b*e**2*h*(m + 2)*(
m + 3) - e*f*(m + 3)*(a*h + b*g)))/(d**2*(m + 1)*(m + 2)*(m + 3)*(c*f - d*e)**3)
- (c + d*x)**(-m - 3)*(e + f*x)**(m + 1)*(-a*d**2*f*g + b*c**2*f*h*(m + 2) + b*
d*h*x*(m + 3)*(c*f - d*e) + c*d*(-b*e*h*(m + 3) + f*(a*h + b*g)))/(d**2*f*(m + 3
)*(c*f - d*e)) + (c + d*x)**(-m - 2)*(e + f*x)**(m + 1)*(b*c**2*f**2*h*(m + 1)*(
m + 2) + c*d*f*(m + 1)*(-2*b*e*h*(m + 3) + f*(a*h + b*g)) + d**2*(2*a*f**2*g + b
*e**2*h*(m + 2)*(m + 3) - e*f*(m + 3)*(a*h + b*g)))/(d**2*f*(m + 2)*(m + 3)*(c*f
- d*e)**2)

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Mathematica [A]  time = 2.35457, size = 321, normalized size = 0.88 \[ \frac{(c+d x)^{-m-3} (e+f x)^{m+1} \left (a \left (c^2 f (m+3) (-e h+f g (m+2)+f h (m+1) x)+c d \left (e^2 h (m+1)-2 e f \left (g \left (m^2+4 m+3\right )+h \left (m^2+4 m+5\right ) x\right )+f^2 x (2 g (m+3)+h (m+1) x)\right )+d^2 \left (e^2 (m+1) (g (m+2)+h (m+3) x)-e f x (2 g (m+1)+h (m+3) x)+2 f^2 g x^2\right )\right )+b \left (c^2 \left (2 e^2 h-e f (g (m+3)+2 h (m+1) x)+f^2 (m+1) x (g (m+3)+h (m+2) x)\right )+c d \left (e^2 (g (m+1)+2 h (m+3) x)-2 e f x \left (g \left (m^2+4 m+5\right )+h \left (m^2+4 m+3\right ) x\right )+f^2 g (m+1) x^2\right )+d^2 e (m+3) x (e g (m+1)+e h (m+2) x-f g x)\right )\right )}{(m+1) (m+2) (m+3) (c f-d e)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(c + d*x)^(-4 - m)*(e + f*x)^m*(g + h*x),x]

[Out]

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

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Maple [B]  time = 0.012, size = 906, normalized size = 2.5 \[ -{\frac{ \left ( dx+c \right ) ^{-3-m} \left ( fx+e \right ) ^{1+m} \left ( -b{c}^{2}{f}^{2}h{m}^{2}{x}^{2}+2\,bcdefh{m}^{2}{x}^{2}-b{d}^{2}{e}^{2}h{m}^{2}{x}^{2}-a{c}^{2}{f}^{2}h{m}^{2}x+2\,acdefh{m}^{2}x-acd{f}^{2}hm{x}^{2}-a{d}^{2}{e}^{2}h{m}^{2}x+a{d}^{2}efhm{x}^{2}-b{c}^{2}{f}^{2}g{m}^{2}x-3\,b{c}^{2}{f}^{2}hm{x}^{2}+2\,bcdefg{m}^{2}x+8\,bcdefhm{x}^{2}-bcd{f}^{2}gm{x}^{2}-b{d}^{2}{e}^{2}g{m}^{2}x-5\,b{d}^{2}{e}^{2}hm{x}^{2}+b{d}^{2}efgm{x}^{2}-a{c}^{2}{f}^{2}g{m}^{2}-4\,a{c}^{2}{f}^{2}hmx+2\,acdefg{m}^{2}+8\,acdefhmx-2\,acd{f}^{2}gmx-acd{f}^{2}h{x}^{2}-a{d}^{2}{e}^{2}g{m}^{2}-4\,a{d}^{2}{e}^{2}hmx+2\,a{d}^{2}efgmx+3\,a{d}^{2}efh{x}^{2}-2\,a{d}^{2}{f}^{2}g{x}^{2}+2\,b{c}^{2}efhmx-4\,b{c}^{2}{f}^{2}gmx-2\,b{c}^{2}{f}^{2}h{x}^{2}-2\,bcd{e}^{2}hmx+8\,bcdefgmx+6\,bcdefh{x}^{2}-bcd{f}^{2}g{x}^{2}-4\,b{d}^{2}{e}^{2}gmx-6\,b{d}^{2}{e}^{2}h{x}^{2}+3\,b{d}^{2}efg{x}^{2}+a{c}^{2}efhm-5\,a{c}^{2}{f}^{2}gm-3\,a{c}^{2}{f}^{2}hx-acd{e}^{2}hm+8\,acdefgm+10\,acdefhx-6\,acd{f}^{2}gx-3\,a{d}^{2}{e}^{2}gm-3\,a{d}^{2}{e}^{2}hx+2\,a{d}^{2}efgx+b{c}^{2}efgm+2\,b{c}^{2}efhx-3\,b{c}^{2}{f}^{2}gx-bcd{e}^{2}gm-6\,bcd{e}^{2}hx+10\,bcdefgx-3\,b{d}^{2}{e}^{2}gx+3\,a{c}^{2}efh-6\,a{c}^{2}{f}^{2}g-acd{e}^{2}h+6\,acdefg-2\,a{d}^{2}{e}^{2}g-2\,b{c}^{2}{e}^{2}h+3\,b{c}^{2}efg-bcd{e}^{2}g \right ) }{{c}^{3}{f}^{3}{m}^{3}-3\,{c}^{2}de{f}^{2}{m}^{3}+3\,c{d}^{2}{e}^{2}f{m}^{3}-{d}^{3}{e}^{3}{m}^{3}+6\,{c}^{3}{f}^{3}{m}^{2}-18\,{c}^{2}de{f}^{2}{m}^{2}+18\,c{d}^{2}{e}^{2}f{m}^{2}-6\,{d}^{3}{e}^{3}{m}^{2}+11\,{c}^{3}{f}^{3}m-33\,{c}^{2}de{f}^{2}m+33\,c{d}^{2}{e}^{2}fm-11\,{d}^{3}{e}^{3}m+6\,{c}^{3}{f}^{3}-18\,{c}^{2}de{f}^{2}+18\,c{d}^{2}{e}^{2}f-6\,{d}^{3}{e}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(d*x+c)^(-4-m)*(f*x+e)^m*(h*x+g),x)

[Out]

-(d*x+c)^(-3-m)*(f*x+e)^(1+m)*(-b*c^2*f^2*h*m^2*x^2+2*b*c*d*e*f*h*m^2*x^2-b*d^2*
e^2*h*m^2*x^2-a*c^2*f^2*h*m^2*x+2*a*c*d*e*f*h*m^2*x-a*c*d*f^2*h*m*x^2-a*d^2*e^2*
h*m^2*x+a*d^2*e*f*h*m*x^2-b*c^2*f^2*g*m^2*x-3*b*c^2*f^2*h*m*x^2+2*b*c*d*e*f*g*m^
2*x+8*b*c*d*e*f*h*m*x^2-b*c*d*f^2*g*m*x^2-b*d^2*e^2*g*m^2*x-5*b*d^2*e^2*h*m*x^2+
b*d^2*e*f*g*m*x^2-a*c^2*f^2*g*m^2-4*a*c^2*f^2*h*m*x+2*a*c*d*e*f*g*m^2+8*a*c*d*e*
f*h*m*x-2*a*c*d*f^2*g*m*x-a*c*d*f^2*h*x^2-a*d^2*e^2*g*m^2-4*a*d^2*e^2*h*m*x+2*a*
d^2*e*f*g*m*x+3*a*d^2*e*f*h*x^2-2*a*d^2*f^2*g*x^2+2*b*c^2*e*f*h*m*x-4*b*c^2*f^2*
g*m*x-2*b*c^2*f^2*h*x^2-2*b*c*d*e^2*h*m*x+8*b*c*d*e*f*g*m*x+6*b*c*d*e*f*h*x^2-b*
c*d*f^2*g*x^2-4*b*d^2*e^2*g*m*x-6*b*d^2*e^2*h*x^2+3*b*d^2*e*f*g*x^2+a*c^2*e*f*h*
m-5*a*c^2*f^2*g*m-3*a*c^2*f^2*h*x-a*c*d*e^2*h*m+8*a*c*d*e*f*g*m+10*a*c*d*e*f*h*x
-6*a*c*d*f^2*g*x-3*a*d^2*e^2*g*m-3*a*d^2*e^2*h*x+2*a*d^2*e*f*g*x+b*c^2*e*f*g*m+2
*b*c^2*e*f*h*x-3*b*c^2*f^2*g*x-b*c*d*e^2*g*m-6*b*c*d*e^2*h*x+10*b*c*d*e*f*g*x-3*
b*d^2*e^2*g*x+3*a*c^2*e*f*h-6*a*c^2*f^2*g-a*c*d*e^2*h+6*a*c*d*e*f*g-2*a*d^2*e^2*
g-2*b*c^2*e^2*h+3*b*c^2*e*f*g-b*c*d*e^2*g)/(c^3*f^3*m^3-3*c^2*d*e*f^2*m^3+3*c*d^
2*e^2*f*m^3-d^3*e^3*m^3+6*c^3*f^3*m^2-18*c^2*d*e*f^2*m^2+18*c*d^2*e^2*f*m^2-6*d^
3*e^3*m^2+11*c^3*f^3*m-33*c^2*d*e*f^2*m+33*c*d^2*e^2*f*m-11*d^3*e^3*m+6*c^3*f^3-
18*c^2*d*e*f^2+18*c*d^2*e^2*f-6*d^3*e^3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}{\left (h x + g\right )}{\left (d x + c\right )}^{-m - 4}{\left (f x + e\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m,x, algorithm="maxima")

[Out]

integrate((b*x + a)*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m, x)

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Fricas [A]  time = 0.257777, size = 2171, normalized size = 5.98 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m,x, algorithm="fricas")

[Out]

-(((b*d^3*e^2*f - 2*b*c*d^2*e*f^2 + b*c^2*d*f^3)*h*m^2 - (3*b*d^3*e*f^2 - (b*c*d
^2 + 2*a*d^3)*f^3)*g + (6*b*d^3*e^2*f - 3*(2*b*c*d^2 + a*d^3)*e*f^2 + (2*b*c^2*d
+ a*c*d^2)*f^3)*h - ((b*d^3*e*f^2 - b*c*d^2*f^3)*g - (5*b*d^3*e^2*f - (8*b*c*d^
2 + a*d^3)*e*f^2 + (3*b*c^2*d + a*c*d^2)*f^3)*h)*m)*x^4 + (a*c*d^2*e^3 - 2*a*c^2
*d*e^2*f + a*c^3*e*f^2)*g*m^2 + (((b*d^3*e^2*f - 2*b*c*d^2*e*f^2 + b*c^2*d*f^3)*
g + (b*d^3*e^3 - (b*c*d^2 - a*d^3)*e^2*f - (b*c^2*d + 2*a*c*d^2)*e*f^2 + (b*c^3
+ a*c^2*d)*f^3)*h)*m^2 - 4*(3*b*c*d^2*e*f^2 - (b*c^2*d + 2*a*c*d^2)*f^3)*g + 2*(
3*b*d^3*e^3 + 3*b*c*d^2*e^2*f - 3*(b*c^2*d + 2*a*c*d^2)*e*f^2 + (b*c^3 + 2*a*c^2
*d)*f^3)*h + ((3*b*d^3*e^2*f - 2*(4*b*c*d^2 + a*d^3)*e*f^2 + (5*b*c^2*d + 2*a*c*
d^2)*f^3)*g + (5*b*d^3*e^3 - (b*c*d^2 - 3*a*d^3)*e^2*f - (7*b*c^2*d + 8*a*c*d^2)
*e*f^2 + (3*b*c^3 + 5*a*c^2*d)*f^3)*h)*m)*x^3 + (((b*d^3*e^3 - (b*c*d^2 - a*d^3)
*e^2*f - (b*c^2*d + 2*a*c*d^2)*e*f^2 + (b*c^3 + a*c^2*d)*f^3)*g + (a*c^3*f^3 + (
b*c*d^2 + a*d^3)*e^3 - (2*b*c^2*d + a*c*d^2)*e^2*f + (b*c^3 - a*c^2*d)*e*f^2)*h)
*m^2 + 3*(b*d^3*e^3 - 3*b*c*d^2*e^2*f - 3*b*c^2*d*e*f^2 + (b*c^3 + 4*a*c^2*d)*f^
3)*g - 3*(3*a*c*d^2*e^2*f + 3*a*c^2*d*e*f^2 - a*c^3*f^3 - (4*b*c*d^2 + a*d^3)*e^
3)*h + ((4*b*d^3*e^3 - (4*b*c*d^2 - a*d^3)*e^2*f - 4*(b*c^2*d + 2*a*c*d^2)*e*f^2
+ (4*b*c^3 + 7*a*c^2*d)*f^3)*g + (4*a*c^3*f^3 + (7*b*c*d^2 + 4*a*d^3)*e^3 - 4*(
2*b*c^2*d + a*c*d^2)*e^2*f + (b*c^3 - 4*a*c^2*d)*e*f^2)*h)*m)*x^2 + (6*a*c^3*e*f
^2 + (b*c^2*d + 2*a*c*d^2)*e^3 - 3*(b*c^3 + 2*a*c^2*d)*e^2*f)*g - (3*a*c^3*e^2*f
- (2*b*c^3 + a*c^2*d)*e^3)*h + ((5*a*c^3*e*f^2 + (b*c^2*d + 3*a*c*d^2)*e^3 - (b
*c^3 + 8*a*c^2*d)*e^2*f)*g + (a*c^2*d*e^3 - a*c^3*e^2*f)*h)*m + (((a*c^3*f^3 + (
b*c*d^2 + a*d^3)*e^3 - (2*b*c^2*d + a*c*d^2)*e^2*f + (b*c^3 - a*c^2*d)*e*f^2)*g
+ (a*c*d^2*e^3 - 2*a*c^2*d*e^2*f + a*c^3*e*f^2)*h)*m^2 + 2*(3*a*c^2*d*e*f^2 + 3*
a*c^3*f^3 + (2*b*c*d^2 + a*d^3)*e^3 - 3*(2*b*c^2*d + a*c*d^2)*e^2*f)*g - 4*(3*a*
c^2*d*e^2*f - (2*b*c^2*d + a*c*d^2)*e^3)*h + ((5*a*c^3*f^3 + (5*b*c*d^2 + 3*a*d^
3)*e^3 - (8*b*c^2*d + 7*a*c*d^2)*e^2*f + (3*b*c^3 - a*c^2*d)*e*f^2)*g + (3*a*c^3
*e*f^2 + (2*b*c^2*d + 5*a*c*d^2)*e^3 - 2*(b*c^3 + 4*a*c^2*d)*e^2*f)*h)*m)*x)*(d*
x + c)^(-m - 4)*(f*x + e)^m/(6*d^3*e^3 - 18*c*d^2*e^2*f + 18*c^2*d*e*f^2 - 6*c^3
*f^3 + (d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*m^3 + 6*(d^3*e^3 - 3*
c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*m^2 + 11*(d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2
*d*e*f^2 - c^3*f^3)*m)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(d*x+c)**(-4-m)*(f*x+e)**m*(h*x+g),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}{\left (h x + g\right )}{\left (d x + c\right )}^{-m - 4}{\left (f x + e\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m,x, algorithm="giac")

[Out]

integrate((b*x + a)*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m, x)