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

Optimal. Leaf size=188 \[ -\frac{(d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1}}{d (m+3) (d e-c f)}+\frac{(c+d x)^{-m-2} (e+f x)^{m+1} (c f h (m+1)+d (2 f g-e h (m+3)))}{d (m+2) (m+3) (d e-c f)^2}-\frac{f (c+d x)^{-m-1} (e+f x)^{m+1} (c f h (m+1)+d (2 f g-e h (m+3)))}{d (m+1) (m+2) (m+3) (d e-c f)^3} \]

[Out]

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

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Rubi [A]  time = 0.305219, antiderivative size = 186, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{(d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1}}{d (m+3) (d e-c f)}+\frac{(c+d x)^{-m-2} (e+f x)^{m+1} (c f h (m+1)-d e h (m+3)+2 d f g)}{d (m+2) (m+3) (d e-c f)^2}-\frac{f (c+d x)^{-m-1} (e+f x)^{m+1} (c f h (m+1)-d e h (m+3)+2 d f g)}{d (m+1) (m+2) (m+3) (d e-c f)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 45.3703, size = 156, normalized size = 0.83 \[ \frac{f \left (c + d x\right )^{- m - 1} \left (e + f x\right )^{m + 1} \left (2 d f g + h \left (c f \left (m + 1\right ) - d e \left (m + 3\right )\right )\right )}{d \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right ) \left (c f - d e\right )^{3}} - \frac{\left (c + d x\right )^{- m - 3} \left (e + f x\right )^{m + 1} \left (c h - d g\right )}{d \left (m + 3\right ) \left (c f - d e\right )} + \frac{\left (c + d x\right )^{- m - 2} \left (e + f x\right )^{m + 1} \left (2 d f g + h \left (c f \left (m + 1\right ) - d e \left (m + 3\right )\right )\right )}{d \left (m + 2\right ) \left (m + 3\right ) \left (c f - d e\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.655355, size = 201, normalized size = 1.07 \[ \frac{(c+d x)^{-m} (e+f x)^m \left (-\frac{f^2 (c f h (m+1)-d e h (m+3)+2 d f g)}{(m+1) (m+2) (m+3) (d e-c f)^3}+\frac{f m (c f h (m+1)-d e h (m+3)+2 d f g)}{(m+1) \left (m^2+5 m+6\right ) (c+d x) (d e-c f)^2}+\frac{c f h (2 m+3)-d (e h (m+3)+f g m)}{(m+2) (m+3) (c+d x)^2 (d e-c f)}+\frac{c h-d g}{(m+3) (c+d x)^3}\right )}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

((e + f*x)^m*(-((f^2*(2*d*f*g + c*f*h*(1 + m) - d*e*h*(3 + m)))/((d*e - c*f)^3*(
1 + m)*(2 + m)*(3 + m))) + (-(d*g) + c*h)/((3 + m)*(c + d*x)^3) + (c*f*h*(3 + 2*
m) - d*(f*g*m + e*h*(3 + m)))/((d*e - c*f)*(2 + m)*(3 + m)*(c + d*x)^2) + (f*m*(
2*d*f*g + c*f*h*(1 + m) - d*e*h*(3 + m)))/((d*e - c*f)^2*(1 + m)*(6 + 5*m + m^2)
*(c + d*x))))/(d^2*(c + d*x)^m)

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Maple [B]  time = 0.011, size = 509, normalized size = 2.7 \[ -{\frac{ \left ( dx+c \right ) ^{-3-m} \left ( fx+e \right ) ^{1+m} \left ( -{c}^{2}{f}^{2}h{m}^{2}x+2\,cdefh{m}^{2}x-cd{f}^{2}hm{x}^{2}-{d}^{2}{e}^{2}h{m}^{2}x+{d}^{2}efhm{x}^{2}-{c}^{2}{f}^{2}g{m}^{2}-4\,{c}^{2}{f}^{2}hmx+2\,cdefg{m}^{2}+8\,cdefhmx-2\,cd{f}^{2}gmx-cd{f}^{2}h{x}^{2}-{d}^{2}{e}^{2}g{m}^{2}-4\,{d}^{2}{e}^{2}hmx+2\,{d}^{2}efgmx+3\,{d}^{2}efh{x}^{2}-2\,{d}^{2}{f}^{2}g{x}^{2}+{c}^{2}efhm-5\,{c}^{2}{f}^{2}gm-3\,{c}^{2}{f}^{2}hx-cd{e}^{2}hm+8\,cdefgm+10\,cdefhx-6\,cd{f}^{2}gx-3\,{d}^{2}{e}^{2}gm-3\,{d}^{2}{e}^{2}hx+2\,{d}^{2}efgx+3\,{c}^{2}efh-6\,{c}^{2}{f}^{2}g-cd{e}^{2}h+6\,cdefg-2\,{d}^{2}{e}^{2}g \right ) }{{c}^{3}{f}^{3}{m}^{3}-3\,{c}^{2}de{f}^{2}{m}^{3}+3\,c{d}^{2}{e}^{2}f{m}^{3}-{d}^{3}{e}^{3}{m}^{3}+6\,{c}^{3}{f}^{3}{m}^{2}-18\,{c}^{2}de{f}^{2}{m}^{2}+18\,c{d}^{2}{e}^{2}f{m}^{2}-6\,{d}^{3}{e}^{3}{m}^{2}+11\,{c}^{3}{f}^{3}m-33\,{c}^{2}de{f}^{2}m+33\,c{d}^{2}{e}^{2}fm-11\,{d}^{3}{e}^{3}m+6\,{c}^{3}{f}^{3}-18\,{c}^{2}de{f}^{2}+18\,c{d}^{2}{e}^{2}f-6\,{d}^{3}{e}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d*x+c)^(-3-m)*(f*x+e)^(1+m)*(-c^2*f^2*h*m^2*x+2*c*d*e*f*h*m^2*x-c*d*f^2*h*m*x^
2-d^2*e^2*h*m^2*x+d^2*e*f*h*m*x^2-c^2*f^2*g*m^2-4*c^2*f^2*h*m*x+2*c*d*e*f*g*m^2+
8*c*d*e*f*h*m*x-2*c*d*f^2*g*m*x-c*d*f^2*h*x^2-d^2*e^2*g*m^2-4*d^2*e^2*h*m*x+2*d^
2*e*f*g*m*x+3*d^2*e*f*h*x^2-2*d^2*f^2*g*x^2+c^2*e*f*h*m-5*c^2*f^2*g*m-3*c^2*f^2*
h*x-c*d*e^2*h*m+8*c*d*e*f*g*m+10*c*d*e*f*h*x-6*c*d*f^2*g*x-3*d^2*e^2*g*m-3*d^2*e
^2*h*x+2*d^2*e*f*g*x+3*c^2*e*f*h-6*c^2*f^2*g-c*d*e^2*h+6*c*d*e*f*g-2*d^2*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. \[ \int{\left (h x + g\right )}{\left (d x + c\right )}^{-m - 4}{\left (f x + e\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.256331, size = 1222, normalized size = 6.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-((2*d^3*f^3*g - (d^3*e*f^2 - c*d^2*f^3)*h*m - (3*d^3*e*f^2 - c*d^2*f^3)*h)*x^4
+ (c*d^2*e^3 - 2*c^2*d*e^2*f + c^3*e*f^2)*g*m^2 + (8*c*d^2*f^3*g + (d^3*e^2*f -
2*c*d^2*e*f^2 + c^2*d*f^3)*h*m^2 - 4*(3*c*d^2*e*f^2 - c^2*d*f^3)*h - (2*(d^3*e*f
^2 - c*d^2*f^3)*g - (3*d^3*e^2*f - 8*c*d^2*e*f^2 + 5*c^2*d*f^3)*h)*m)*x^3 + (12*
c^2*d*f^3*g + ((d^3*e^2*f - 2*c*d^2*e*f^2 + c^2*d*f^3)*g + (d^3*e^3 - c*d^2*e^2*
f - c^2*d*e*f^2 + c^3*f^3)*h)*m^2 + 3*(d^3*e^3 - 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 +
c^3*f^3)*h + ((d^3*e^2*f - 8*c*d^2*e*f^2 + 7*c^2*d*f^3)*g + 4*(d^3*e^3 - c*d^2*
e^2*f - c^2*d*e*f^2 + c^3*f^3)*h)*m)*x^2 + 2*(c*d^2*e^3 - 3*c^2*d*e^2*f + 3*c^3*
e*f^2)*g + (c^2*d*e^3 - 3*c^3*e^2*f)*h + ((3*c*d^2*e^3 - 8*c^2*d*e^2*f + 5*c^3*e
*f^2)*g + (c^2*d*e^3 - c^3*e^2*f)*h)*m + (((d^3*e^3 - c*d^2*e^2*f - c^2*d*e*f^2
+ c^3*f^3)*g + (c*d^2*e^3 - 2*c^2*d*e^2*f + c^3*e*f^2)*h)*m^2 + 2*(d^3*e^3 - 3*c
*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*c^3*f^3)*g + 4*(c*d^2*e^3 - 3*c^2*d*e^2*f)*h + ((
3*d^3*e^3 - 7*c*d^2*e^2*f - c^2*d*e*f^2 + 5*c^3*f^3)*g + (5*c*d^2*e^3 - 8*c^2*d*
e^2*f + 3*c^3*e*f^2)*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((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. \[ \int{\left (h x + g\right )}{\left (d x + c\right )}^{-m - 4}{\left (f x + e\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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