3.133 \(\int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx\)

Optimal. Leaf size=572 \[ -\frac{(d g-c h) (c+d x)^{-m-2} (e+f x)^{m+1} \left (b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))-2 d f \left (a^2 d f+a b (c f (m+1)-d e (m+3))+b^2 c e\right )\right )}{d^3 f (m+2) (m+3) (d e-c f)^2}+\frac{(d g-c h) (c+d x)^{-m-1} (e+f x)^{m+1} \left (b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))-2 d f \left (a^2 d f+a b (c f (m+1)-d e (m+3))+b^2 c e\right )\right )}{d^3 (m+1) (m+2) (m+3) (d e-c f)^3}+\frac{(b c-a d) (d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1} (a d f+b (c f (m+2)-d e (m+3)))}{d^3 f (m+3) (d e-c f)}-\frac{h (b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^3 (m+2) (d e-c f)}-\frac{h (b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f-b (2 d e (m+2)-c f (2 m+3)))}{d^3 (m+1) (m+2) (d e-c f)^2}-\frac{b (a+b x) (d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1}}{d^2 f}-\frac{b^2 h (c+d x)^{-m} (e+f x)^m \left (\frac{d (e+f x)}{d e-c f}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{f (c+d x)}{d e-c f}\right )}{d^4 m} \]

[Out]

((b*c - a*d)*(d*g - c*h)*(a*d*f + b*(c*f*(2 + m) - d*e*(3 + m)))*(c + d*x)^(-3 -
m)*(e + f*x)^(1 + m))/(d^3*f*(d*e - c*f)*(3 + m)) - (b*(d*g - c*h)*(a + b*x)*(c
+ d*x)^(-3 - m)*(e + f*x)^(1 + m))/(d^2*f) - ((b*c - a*d)^2*h*(c + d*x)^(-2 - m
)*(e + f*x)^(1 + m))/(d^3*(d*e - c*f)*(2 + m)) - ((d*g - c*h)*(b^2*(d*e - c*f)*(
2 + m)*(c*f*(1 + m) - d*e*(3 + m)) - 2*d*f*(b^2*c*e + a^2*d*f + a*b*(c*f*(1 + m)
- d*e*(3 + m))))*(c + d*x)^(-2 - m)*(e + f*x)^(1 + m))/(d^3*f*(d*e - c*f)^2*(2
+ m)*(3 + m)) - ((b*c - a*d)*h*(a*d*f - b*(2*d*e*(2 + m) - c*f*(3 + 2*m)))*(c +
d*x)^(-1 - m)*(e + f*x)^(1 + m))/(d^3*(d*e - c*f)^2*(1 + m)*(2 + m)) + ((d*g - c
*h)*(b^2*(d*e - c*f)*(2 + m)*(c*f*(1 + m) - d*e*(3 + m)) - 2*d*f*(b^2*c*e + a^2*
d*f + a*b*(c*f*(1 + m) - d*e*(3 + m))))*(c + d*x)^(-1 - m)*(e + f*x)^(1 + m))/(d
^3*(d*e - c*f)^3*(1 + m)*(2 + m)*(3 + m)) - (b^2*h*(e + f*x)^m*Hypergeometric2F1
[-m, -m, 1 - m, -((f*(c + d*x))/(d*e - c*f))])/(d^4*m*(c + d*x)^m*((d*(e + f*x))
/(d*e - c*f))^m)

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Rubi [A]  time = 2.12932, antiderivative size = 566, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258 \[ -\frac{(d g-c h) (c+d x)^{-m-2} (e+f x)^{m+1} \left (b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))-2 d f \left (a^2 d f+b (a c f (m+1)-a d e (m+3)+b c e)\right )\right )}{d^3 f (m+2) (m+3) (d e-c f)^2}+\frac{(d g-c h) (c+d x)^{-m-1} (e+f x)^{m+1} \left (b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))-2 d f \left (a^2 d f+b (a c f (m+1)-a d e (m+3)+b c e)\right )\right )}{d^3 (m+1) (m+2) (m+3) (d e-c f)^3}+\frac{(b c-a d) (d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1} (a d f+b c f (m+2)-b d e (m+3))}{d^3 f (m+3) (d e-c f)}-\frac{h (b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^3 (m+2) (d e-c f)}-\frac{h (b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{d^3 (m+1) (m+2) (d e-c f)^2}-\frac{b (a+b x) (d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1}}{d^2 f}-\frac{b^2 h (c+d x)^{-m} (e+f x)^m \left (\frac{d (e+f x)}{d e-c f}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{f (c+d x)}{d e-c f}\right )}{d^4 m} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2*(c + d*x)^(-4 - m)*(e + f*x)^m*(g + h*x),x]

[Out]

((b*c - a*d)*(d*g - c*h)*(a*d*f + b*c*f*(2 + m) - b*d*e*(3 + m))*(c + d*x)^(-3 -
m)*(e + f*x)^(1 + m))/(d^3*f*(d*e - c*f)*(3 + m)) - (b*(d*g - c*h)*(a + b*x)*(c
+ d*x)^(-3 - m)*(e + f*x)^(1 + m))/(d^2*f) - ((b*c - a*d)^2*h*(c + d*x)^(-2 - m
)*(e + f*x)^(1 + m))/(d^3*(d*e - c*f)*(2 + m)) - ((d*g - c*h)*(b^2*(d*e - c*f)*(
2 + m)*(c*f*(1 + m) - d*e*(3 + m)) - 2*d*f*(a^2*d*f + b*(b*c*e + a*c*f*(1 + m) -
a*d*e*(3 + m))))*(c + d*x)^(-2 - m)*(e + f*x)^(1 + m))/(d^3*f*(d*e - c*f)^2*(2
+ m)*(3 + m)) - ((b*c - a*d)*h*(a*d*f - 2*b*d*e*(2 + m) + b*c*f*(3 + 2*m))*(c +
d*x)^(-1 - m)*(e + f*x)^(1 + m))/(d^3*(d*e - c*f)^2*(1 + m)*(2 + m)) + ((d*g - c
*h)*(b^2*(d*e - c*f)*(2 + m)*(c*f*(1 + m) - d*e*(3 + m)) - 2*d*f*(a^2*d*f + b*(b
*c*e + a*c*f*(1 + m) - a*d*e*(3 + m))))*(c + d*x)^(-1 - m)*(e + f*x)^(1 + m))/(d
^3*(d*e - c*f)^3*(1 + m)*(2 + m)*(3 + m)) - (b^2*h*(e + f*x)^m*Hypergeometric2F1
[-m, -m, 1 - m, -((f*(c + d*x))/(d*e - c*f))])/(d^4*m*(c + d*x)^m*((d*(e + f*x))
/(d*e - c*f))^m)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(d*x+c)**(-4-m)*(f*x+e)**m*(h*x+g),x)

[Out]

Timed out

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Mathematica [C]  time = 8.02083, size = 10700, normalized size = 18.71 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^2*(c + d*x)^(-4 - m)*(e + f*x)^m*(g + h*x),x]

[Out]

Result too large to show

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{2} \left ( dx+c \right ) ^{-4-m} \left ( fx+e \right ) ^{m} \left ( hx+g \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(d*x+c)^(-4-m)*(f*x+e)^m*(h*x+g),x)

[Out]

int((b*x+a)^2*(d*x+c)^(-4-m)*(f*x+e)^m*(h*x+g),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{2}{\left (h x + g\right )}{\left (d x + c\right )}^{-m - 4}{\left (f x + e\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m,x, algorithm="maxima")

[Out]

integrate((b*x + a)^2*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b^{2} h x^{3} + a^{2} g +{\left (b^{2} g + 2 \, a b h\right )} x^{2} +{\left (2 \, a b g + a^{2} h\right )} x\right )}{\left (d x + c\right )}^{-m - 4}{\left (f x + e\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m,x, algorithm="fricas")

[Out]

integral((b^2*h*x^3 + a^2*g + (b^2*g + 2*a*b*h)*x^2 + (2*a*b*g + a^2*h)*x)*(d*x
+ c)^(-m - 4)*(f*x + e)^m, x)

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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)**2*(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. \[ \int{\left (b x + a\right )}^{2}{\left (h x + g\right )}{\left (d x + c\right )}^{-m - 4}{\left (f x + e\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m,x, algorithm="giac")

[Out]

integrate((b*x + a)^2*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m, x)