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

Optimal. Leaf size=362 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f h \left (m^2+5 m+6\right )-a b d (m+3) (2 c f h (m+1)+d (e h+f g))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )+c d (m+1) (e h+f g)+2 d^2 e g\right )\right )}{b d^2 (m+2) (m+3) (b c-a d)^2}+\frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f h \left (m^2+5 m+6\right )-a b d (m+3) (2 c f h (m+1)+d (e h+f g))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )+c d (m+1) (e h+f g)+2 d^2 e g\right )\right )}{d^2 (m+1) (m+2) (m+3) (b c-a d)^3}+\frac{(a+b x)^{m+1} (c+d x)^{-m-3} \left (-d f h (m+3) x (b c-a d)+a c d f h (m+3)+b \left (c^2 (-f) h (m+2)-c d (e h+f g)+d^2 e g\right )\right )}{b d^2 (m+3) (b c-a d)} \]

[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))

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Rubi [A]  time = 0.988157, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f h \left (m^2+5 m+6\right )-a b d (m+3) (2 c f h (m+1)+d (e h+f g))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )+c d (m+1) (e h+f g)+2 d^2 e g\right )\right )}{b d^2 (m+2) (m+3) (b c-a d)^2}+\frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f h \left (m^2+5 m+6\right )-a b d (m+3) (2 c f h (m+1)+d (e h+f g))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )+c d (m+1) (e h+f g)+2 d^2 e g\right )\right )}{d^2 (m+1) (m+2) (m+3) (b c-a d)^3}+\frac{(a+b x)^{m+1} (c+d x)^{-m-3} \left (-d f h (m+3) x (b c-a d)+a c d f h (m+3)+b \left (c^2 (-f) h (m+2)-c d (e h+f g)+d^2 e g\right )\right )}{b d^2 (m+3) (b c-a d)} \]

Antiderivative was successfully verified.

[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))

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Rubi in Sympy [A]  time = 127.151, size = 337, normalized size = 0.93 \[ - \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (b^{2} c^{2} f h \left (m + 1\right ) \left (m + 2\right ) + b c d \left (m + 1\right ) \left (- 2 a f h \left (m + 3\right ) + b \left (e h + f g\right )\right ) + d^{2} \left (a^{2} f h \left (m + 2\right ) \left (m + 3\right ) - a b \left (m + 3\right ) \left (e h + f g\right ) + 2 b^{2} e g\right )\right )}{d^{2} \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right ) \left (a d - b c\right )^{3}} + \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 3} \left (b c^{2} f h \left (m + 2\right ) - b d^{2} e g + c d \left (- a f h \left (m + 3\right ) + b \left (e h + f g\right )\right ) - d f h x \left (m + 3\right ) \left (a d - b c\right )\right )}{b d^{2} \left (m + 3\right ) \left (a d - b c\right )} + \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 2} \left (b^{2} c^{2} f h \left (m + 1\right ) \left (m + 2\right ) + b c d \left (m + 1\right ) \left (- 2 a f h \left (m + 3\right ) + b \left (e h + f g\right )\right ) + d^{2} \left (a^{2} f h \left (m + 2\right ) \left (m + 3\right ) - a b \left (m + 3\right ) \left (e h + f g\right ) + 2 b^{2} e g\right )\right )}{b d^{2} \left (m + 2\right ) \left (m + 3\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.42076, size = 315, normalized size = 0.87 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-3} \left (a^2 \left (2 c^2 f h+c d (e h (m+1)+f g (m+1)+2 f h (m+3) x)+d^2 (e (m+1) (g (m+2)+h (m+3) x)+f (m+3) x (g (m+1)+h (m+2) x))\right )-a b \left (c^2 (e h (m+3)+f g (m+3)+2 f h (m+1) x)+2 c d \left (e \left (g \left (m^2+4 m+3\right )+h \left (m^2+4 m+5\right ) x\right )+f x \left (g \left (m^2+4 m+5\right )+h \left (m^2+4 m+3\right ) x\right )\right )+d^2 x (2 e g (m+1)+e h (m+3) x+f g (m+3) x)\right )+b^2 \left (c^2 (e (m+3) (g (m+2)+h (m+1) x)+f (m+1) x (g (m+3)+h (m+2) x))+c d x (2 e g (m+3)+e h (m+1) x+f g (m+1) x)+2 d^2 e g x^2\right )\right )}{(m+1) (m+2) (m+3) (b c-a d)^3} \]

Antiderivative was successfully verified.

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

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Maple [B]  time = 0.012, size = 894, normalized size = 2.5 \[ -{\frac{ \left ( bx+a \right ) ^{1+m} \left ( dx+c \right ) ^{-3-m} \left ({a}^{2}{d}^{2}fh{m}^{2}{x}^{2}-2\,abcdfh{m}^{2}{x}^{2}+{b}^{2}{c}^{2}fh{m}^{2}{x}^{2}+{a}^{2}{d}^{2}eh{m}^{2}x+{a}^{2}{d}^{2}fg{m}^{2}x+5\,{a}^{2}{d}^{2}fhm{x}^{2}-2\,abcdeh{m}^{2}x-2\,abcdfg{m}^{2}x-8\,abcdfhm{x}^{2}-ab{d}^{2}ehm{x}^{2}-ab{d}^{2}fgm{x}^{2}+{b}^{2}{c}^{2}eh{m}^{2}x+{b}^{2}{c}^{2}fg{m}^{2}x+3\,{b}^{2}{c}^{2}fhm{x}^{2}+{b}^{2}cdehm{x}^{2}+{b}^{2}cdfgm{x}^{2}+2\,{a}^{2}cdfhmx+{a}^{2}{d}^{2}eg{m}^{2}+4\,{a}^{2}{d}^{2}ehmx+4\,{a}^{2}{d}^{2}fgmx+6\,{a}^{2}{d}^{2}fh{x}^{2}-2\,ab{c}^{2}fhmx-2\,abcdeg{m}^{2}-8\,abcdehmx-8\,abcdfgmx-6\,abcdfh{x}^{2}-2\,ab{d}^{2}egmx-3\,ab{d}^{2}eh{x}^{2}-3\,ab{d}^{2}fg{x}^{2}+{b}^{2}{c}^{2}eg{m}^{2}+4\,{b}^{2}{c}^{2}ehmx+4\,{b}^{2}{c}^{2}fgmx+2\,{b}^{2}{c}^{2}fh{x}^{2}+2\,{b}^{2}cdegmx+{b}^{2}cdeh{x}^{2}+{b}^{2}cdfg{x}^{2}+2\,{b}^{2}{d}^{2}eg{x}^{2}+{a}^{2}cdehm+{a}^{2}cdfgm+6\,{a}^{2}cdfhx+3\,{a}^{2}{d}^{2}egm+3\,{a}^{2}{d}^{2}ehx+3\,{a}^{2}{d}^{2}fgx-ab{c}^{2}ehm-ab{c}^{2}fgm-2\,ab{c}^{2}fhx-8\,abcdegm-10\,abcdehx-10\,abcdfgx-2\,ab{d}^{2}egx+5\,{b}^{2}{c}^{2}egm+3\,{b}^{2}{c}^{2}ehx+3\,{b}^{2}{c}^{2}fgx+6\,{b}^{2}cdegx+2\,{a}^{2}{c}^{2}fh+{a}^{2}cdeh+{a}^{2}cdfg+2\,{a}^{2}{d}^{2}eg-3\,ab{c}^{2}eh-3\,ab{c}^{2}fg-6\,abcdeg+6\,{b}^{2}{c}^{2}eg \right ) }{{a}^{3}{d}^{3}{m}^{3}-3\,{a}^{2}bc{d}^{2}{m}^{3}+3\,a{b}^{2}{c}^{2}d{m}^{3}-{b}^{3}{c}^{3}{m}^{3}+6\,{a}^{3}{d}^{3}{m}^{2}-18\,{a}^{2}bc{d}^{2}{m}^{2}+18\,a{b}^{2}{c}^{2}d{m}^{2}-6\,{b}^{3}{c}^{3}{m}^{2}+11\,{a}^{3}{d}^{3}m-33\,{a}^{2}bc{d}^{2}m+33\,a{b}^{2}{c}^{2}dm-11\,{b}^{3}{c}^{3}m+6\,{a}^{3}{d}^{3}-18\,{a}^{2}bc{d}^{2}+18\,a{b}^{2}{c}^{2}d-6\,{b}^{3}{c}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*x+a)^(1+m)*(d*x+c)^(-3-m)*(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)/(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^3-1
8*a^2*b*c*d^2+18*a*b^2*c^2*d-6*b^3*c^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.2607, size = 2240, normalized size = 6.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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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)**m*(d*x+c)**(-4-m)*(f*x+e)*(h*x+g),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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