3.2311 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx\)

Optimal. Leaf size=116 \[ -\frac{\log (d+e x) \left (A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )\right )}{e^4}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}-\frac{x (-A c e-b B e+2 B c d)}{e^3}+\frac{B c x^2}{2 e^2} \]

[Out]

-(((2*B*c*d - b*B*e - A*c*e)*x)/e^3) + (B*c*x^2)/(2*e^2) + ((B*d - A*e)*(c*d^2 -
b*d*e + a*e^2))/(e^4*(d + e*x)) - ((A*e*(2*c*d - b*e) - B*(3*c*d^2 - e*(2*b*d -
a*e)))*Log[d + e*x])/e^4

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Rubi [A]  time = 0.303007, antiderivative size = 114, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{\log (d+e x) \left (-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2\right )}{e^4}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}-\frac{x (-A c e-b B e+2 B c d)}{e^3}+\frac{B c x^2}{2 e^2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2))/(d + e*x)^2,x]

[Out]

-(((2*B*c*d - b*B*e - A*c*e)*x)/e^3) + (B*c*x^2)/(2*e^2) + ((B*d - A*e)*(c*d^2 -
b*d*e + a*e^2))/(e^4*(d + e*x)) + ((3*B*c*d^2 - B*e*(2*b*d - a*e) - A*e*(2*c*d
- b*e))*Log[d + e*x])/e^4

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B c \int x\, dx}{e^{2}} + \left (A c e + B b e - 2 B c d\right ) \int \frac{1}{e^{3}}\, dx + \frac{\left (A b e^{2} - 2 A c d e + B a e^{2} - 2 B b d e + 3 B c d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} - \frac{\left (A e - B d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{e^{4} \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x+a)/(e*x+d)**2,x)

[Out]

B*c*Integral(x, x)/e**2 + (A*c*e + B*b*e - 2*B*c*d)*Integral(e**(-3), x) + (A*b*
e**2 - 2*A*c*d*e + B*a*e**2 - 2*B*b*d*e + 3*B*c*d**2)*log(d + e*x)/e**4 - (A*e -
B*d)*(a*e**2 - b*d*e + c*d**2)/(e**4*(d + e*x))

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Mathematica [A]  time = 0.160168, size = 106, normalized size = 0.91 \[ \frac{\frac{2 (B d-A e) \left (e (a e-b d)+c d^2\right )}{d+e x}+2 \log (d+e x) \left (B e (a e-2 b d)+A e (b e-2 c d)+3 B c d^2\right )+2 e x (A c e+b B e-2 B c d)+B c e^2 x^2}{2 e^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2))/(d + e*x)^2,x]

[Out]

(2*e*(-2*B*c*d + b*B*e + A*c*e)*x + B*c*e^2*x^2 + (2*(B*d - A*e)*(c*d^2 + e*(-(b
*d) + a*e)))/(d + e*x) + 2*(3*B*c*d^2 + B*e*(-2*b*d + a*e) + A*e*(-2*c*d + b*e))
*Log[d + e*x])/(2*e^4)

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Maple [A]  time = 0.011, size = 195, normalized size = 1.7 \[{\frac{Bc{x}^{2}}{2\,{e}^{2}}}+{\frac{Acx}{{e}^{2}}}+{\frac{bBx}{{e}^{2}}}-2\,{\frac{Bcdx}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) Ab}{{e}^{2}}}-2\,{\frac{\ln \left ( ex+d \right ) Acd}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) aB}{{e}^{2}}}-2\,{\frac{\ln \left ( ex+d \right ) Bbd}{{e}^{3}}}+3\,{\frac{\ln \left ( ex+d \right ) Bc{d}^{2}}{{e}^{4}}}-{\frac{aA}{e \left ( ex+d \right ) }}+{\frac{Abd}{{e}^{2} \left ( ex+d \right ) }}-{\frac{Ac{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+{\frac{aBd}{{e}^{2} \left ( ex+d \right ) }}-{\frac{bB{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+{\frac{Bc{d}^{3}}{{e}^{4} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)/(e*x+d)^2,x)

[Out]

1/2*B*c*x^2/e^2+1/e^2*A*c*x+1/e^2*b*B*x-2/e^3*B*c*d*x+1/e^2*ln(e*x+d)*A*b-2/e^3*
ln(e*x+d)*A*c*d+1/e^2*ln(e*x+d)*a*B-2/e^3*ln(e*x+d)*B*b*d+3/e^4*ln(e*x+d)*B*c*d^
2-1/e/(e*x+d)*a*A+1/e^2/(e*x+d)*A*d*b-1/e^3/(e*x+d)*A*c*d^2+1/e^2/(e*x+d)*a*B*d-
1/e^3/(e*x+d)*B*b*d^2+1/e^4/(e*x+d)*B*c*d^3

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Maxima [A]  time = 0.691669, size = 170, normalized size = 1.47 \[ \frac{B c d^{3} - A a e^{3} -{\left (B b + A c\right )} d^{2} e +{\left (B a + A b\right )} d e^{2}}{e^{5} x + d e^{4}} + \frac{B c e x^{2} - 2 \,{\left (2 \, B c d -{\left (B b + A c\right )} e\right )} x}{2 \, e^{3}} + \frac{{\left (3 \, B c d^{2} - 2 \,{\left (B b + A c\right )} d e +{\left (B a + A b\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)/(e*x + d)^2,x, algorithm="maxima")

[Out]

(B*c*d^3 - A*a*e^3 - (B*b + A*c)*d^2*e + (B*a + A*b)*d*e^2)/(e^5*x + d*e^4) + 1/
2*(B*c*e*x^2 - 2*(2*B*c*d - (B*b + A*c)*e)*x)/e^3 + (3*B*c*d^2 - 2*(B*b + A*c)*d
*e + (B*a + A*b)*e^2)*log(e*x + d)/e^4

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Fricas [A]  time = 0.257845, size = 259, normalized size = 2.23 \[ \frac{B c e^{3} x^{3} + 2 \, B c d^{3} - 2 \, A a e^{3} - 2 \,{\left (B b + A c\right )} d^{2} e + 2 \,{\left (B a + A b\right )} d e^{2} -{\left (3 \, B c d e^{2} - 2 \,{\left (B b + A c\right )} e^{3}\right )} x^{2} - 2 \,{\left (2 \, B c d^{2} e -{\left (B b + A c\right )} d e^{2}\right )} x + 2 \,{\left (3 \, B c d^{3} - 2 \,{\left (B b + A c\right )} d^{2} e +{\left (B a + A b\right )} d e^{2} +{\left (3 \, B c d^{2} e - 2 \,{\left (B b + A c\right )} d e^{2} +{\left (B a + A b\right )} e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x + d e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)/(e*x + d)^2,x, algorithm="fricas")

[Out]

1/2*(B*c*e^3*x^3 + 2*B*c*d^3 - 2*A*a*e^3 - 2*(B*b + A*c)*d^2*e + 2*(B*a + A*b)*d
*e^2 - (3*B*c*d*e^2 - 2*(B*b + A*c)*e^3)*x^2 - 2*(2*B*c*d^2*e - (B*b + A*c)*d*e^
2)*x + 2*(3*B*c*d^3 - 2*(B*b + A*c)*d^2*e + (B*a + A*b)*d*e^2 + (3*B*c*d^2*e - 2
*(B*b + A*c)*d*e^2 + (B*a + A*b)*e^3)*x)*log(e*x + d))/(e^5*x + d*e^4)

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Sympy [A]  time = 5.22561, size = 139, normalized size = 1.2 \[ \frac{B c x^{2}}{2 e^{2}} + \frac{- A a e^{3} + A b d e^{2} - A c d^{2} e + B a d e^{2} - B b d^{2} e + B c d^{3}}{d e^{4} + e^{5} x} + \frac{x \left (A c e + B b e - 2 B c d\right )}{e^{3}} + \frac{\left (A b e^{2} - 2 A c d e + B a e^{2} - 2 B b d e + 3 B c d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)/(e*x+d)**2,x)

[Out]

B*c*x**2/(2*e**2) + (-A*a*e**3 + A*b*d*e**2 - A*c*d**2*e + B*a*d*e**2 - B*b*d**2
*e + B*c*d**3)/(d*e**4 + e**5*x) + x*(A*c*e + B*b*e - 2*B*c*d)/e**3 + (A*b*e**2
- 2*A*c*d*e + B*a*e**2 - 2*B*b*d*e + 3*B*c*d**2)*log(d + e*x)/e**4

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GIAC/XCAS [A]  time = 0.283342, size = 270, normalized size = 2.33 \[ \frac{1}{2} \,{\left (B c - \frac{2 \,{\left (3 \, B c d e - B b e^{2} - A c e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-4\right )} -{\left (3 \, B c d^{2} - 2 \, B b d e - 2 \, A c d e + B a e^{2} + A b e^{2}\right )} e^{\left (-4\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{B c d^{3} e^{2}}{x e + d} - \frac{B b d^{2} e^{3}}{x e + d} - \frac{A c d^{2} e^{3}}{x e + d} + \frac{B a d e^{4}}{x e + d} + \frac{A b d e^{4}}{x e + d} - \frac{A a e^{5}}{x e + d}\right )} e^{\left (-6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)/(e*x + d)^2,x, algorithm="giac")

[Out]

1/2*(B*c - 2*(3*B*c*d*e - B*b*e^2 - A*c*e^2)*e^(-1)/(x*e + d))*(x*e + d)^2*e^(-4
) - (3*B*c*d^2 - 2*B*b*d*e - 2*A*c*d*e + B*a*e^2 + A*b*e^2)*e^(-4)*ln(abs(x*e +
d)*e^(-1)/(x*e + d)^2) + (B*c*d^3*e^2/(x*e + d) - B*b*d^2*e^3/(x*e + d) - A*c*d^
2*e^3/(x*e + d) + B*a*d*e^4/(x*e + d) + A*b*d*e^4/(x*e + d) - A*a*e^5/(x*e + d))
*e^(-6)