3.2168 \(\int (a c+b c x)^{-3-2 p} (f+g x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=61 \[ -\frac{(f+g x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p (a c+b c x)^{-2 p}}{2 c^3 (a+b x)^2 (b f-a g)} \]

[Out]

-((f + g*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^p)/(2*c^3*(b*f - a*g)*(a + b*x)^2*(a*c +
b*c*x)^(2*p))

_______________________________________________________________________________________

Rubi [A]  time = 0.121584, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079 \[ -\frac{(f+g x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p (a c+b c x)^{-2 p}}{2 c^3 (a+b x)^2 (b f-a g)} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + b*c*x)^(-3 - 2*p)*(f + g*x)*(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

-((f + g*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^p)/(2*c^3*(b*f - a*g)*(a + b*x)^2*(a*c +
b*c*x)^(2*p))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 41.4939, size = 88, normalized size = 1.44 \[ \frac{\left (a c + b c x\right )^{- 2 p - 2} \left (a g - b f\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{2 b^{2} c} - \frac{g \left (a c + b c x\right )^{- 2 p - 1} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{b^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*c*x+a*c)**(-3-2*p)*(g*x+f)*(b**2*x**2+2*a*b*x+a**2)**p,x)

[Out]

(a*c + b*c*x)**(-2*p - 2)*(a*g - b*f)*(a**2 + 2*a*b*x + b**2*x**2)**p/(2*b**2*c)
- g*(a*c + b*c*x)**(-2*p - 1)*(a**2 + 2*a*b*x + b**2*x**2)**p/(b**2*c**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.060496, size = 49, normalized size = 0.8 \[ -\frac{\left ((a+b x)^2\right )^p (c (a+b x))^{-2 p} (a g+b (f+2 g x))}{2 b^2 c^3 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*c + b*c*x)^(-3 - 2*p)*(f + g*x)*(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

-(((a + b*x)^2)^p*(a*g + b*(f + 2*g*x)))/(2*b^2*c^3*(a + b*x)^2*(c*(a + b*x))^(2
*p))

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 55, normalized size = 0.9 \[ -{\frac{ \left ( bx+a \right ) \left ( 2\,bgx+ag+bf \right ) \left ( bxc+ac \right ) ^{-3-2\,p} \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{2\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*c*x+a*c)^(-3-2*p)*(g*x+f)*(b^2*x^2+2*a*b*x+a^2)^p,x)

[Out]

-1/2*(b*x+a)*(2*b*g*x+a*g+b*f)*(b*c*x+a*c)^(-3-2*p)*(b^2*x^2+2*a*b*x+a^2)^p/b^2

_______________________________________________________________________________________

Maxima [A]  time = 0.725432, size = 136, normalized size = 2.23 \[ -\frac{{\left (2 \, b x + a\right )} g}{2 \,{\left (b^{4} c^{2 \, p + 3} x^{2} + 2 \, a b^{3} c^{2 \, p + 3} x + a^{2} b^{2} c^{2 \, p + 3}\right )}} - \frac{f}{2 \,{\left (b^{3} c^{2 \, p + 3} x^{2} + 2 \, a b^{2} c^{2 \, p + 3} x + a^{2} b c^{2 \, p + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)*(b^2*x^2 + 2*a*b*x + a^2)^p*(b*c*x + a*c)^(-2*p - 3),x, algorithm="maxima")

[Out]

-1/2*(2*b*x + a)*g/(b^4*c^(2*p + 3)*x^2 + 2*a*b^3*c^(2*p + 3)*x + a^2*b^2*c^(2*p
+ 3)) - 1/2*f/(b^3*c^(2*p + 3)*x^2 + 2*a*b^2*c^(2*p + 3)*x + a^2*b*c^(2*p + 3))

_______________________________________________________________________________________

Fricas [A]  time = 0.299543, size = 70, normalized size = 1.15 \[ -\frac{{\left (2 \, b g x + b f + a g\right )} \frac{1}{c^{2}}^{p}}{2 \,{\left (b^{4} c^{3} x^{2} + 2 \, a b^{3} c^{3} x + a^{2} b^{2} c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)*(b^2*x^2 + 2*a*b*x + a^2)^p*(b*c*x + a*c)^(-2*p - 3),x, algorithm="fricas")

[Out]

-1/2*(2*b*g*x + b*f + a*g)*(c^(-2))^p/(b^4*c^3*x^2 + 2*a*b^3*c^3*x + a^2*b^2*c^3
)

_______________________________________________________________________________________

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*c*x+a*c)**(-3-2*p)*(g*x+f)*(b**2*x**2+2*a*b*x+a**2)**p,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.29225, size = 177, normalized size = 2.9 \[ -\frac{2 \, b^{2} g x^{2} e^{\left (-2 \, p{\rm ln}\left (c\right ) - 3 \,{\rm ln}\left (b x + a\right ) - 3 \,{\rm ln}\left (c\right )\right )} + b^{2} f x e^{\left (-2 \, p{\rm ln}\left (c\right ) - 3 \,{\rm ln}\left (b x + a\right ) - 3 \,{\rm ln}\left (c\right )\right )} + 3 \, a b g x e^{\left (-2 \, p{\rm ln}\left (c\right ) - 3 \,{\rm ln}\left (b x + a\right ) - 3 \,{\rm ln}\left (c\right )\right )} + a b f e^{\left (-2 \, p{\rm ln}\left (c\right ) - 3 \,{\rm ln}\left (b x + a\right ) - 3 \,{\rm ln}\left (c\right )\right )} + a^{2} g e^{\left (-2 \, p{\rm ln}\left (c\right ) - 3 \,{\rm ln}\left (b x + a\right ) - 3 \,{\rm ln}\left (c\right )\right )}}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)*(b^2*x^2 + 2*a*b*x + a^2)^p*(b*c*x + a*c)^(-2*p - 3),x, algorithm="giac")

[Out]

-1/2*(2*b^2*g*x^2*e^(-2*p*ln(c) - 3*ln(b*x + a) - 3*ln(c)) + b^2*f*x*e^(-2*p*ln(
c) - 3*ln(b*x + a) - 3*ln(c)) + 3*a*b*g*x*e^(-2*p*ln(c) - 3*ln(b*x + a) - 3*ln(c
)) + a*b*f*e^(-2*p*ln(c) - 3*ln(b*x + a) - 3*ln(c)) + a^2*g*e^(-2*p*ln(c) - 3*ln
(b*x + a) - 3*ln(c)))/b^2