3.1500 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx\)

Optimal. Leaf size=102 \[ \frac{\log (d+e x) \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}-\frac{c x (4 c d-3 b e)}{e^3}+\frac{c^2 x^2}{e^2} \]

[Out]

-((c*(4*c*d - 3*b*e)*x)/e^3) + (c^2*x^2)/e^2 + ((2*c*d - b*e)*(c*d^2 - b*d*e + a
*e^2))/(e^4*(d + e*x)) + ((6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*Log[d + e*
x])/e^4

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Rubi [A]  time = 0.245144, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{\log (d+e x) \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}-\frac{c x (4 c d-3 b e)}{e^3}+\frac{c^2 x^2}{e^2} \]

Antiderivative was successfully verified.

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

[Out]

-((c*(4*c*d - 3*b*e)*x)/e^3) + (c^2*x^2)/e^2 + ((2*c*d - b*e)*(c*d^2 - b*d*e + a
*e^2))/(e^4*(d + e*x)) + ((6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*Log[d + e*
x])/e^4

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 c^{2} \int x\, dx}{e^{2}} + \frac{\left (3 b e - 4 c d\right ) \int c\, dx}{e^{3}} + \frac{\left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} - \frac{\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{e^{4} \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c**2*Integral(x, x)/e**2 + (3*b*e - 4*c*d)*Integral(c, x)/e**3 + (2*a*c*e**2 +
b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)*log(d + e*x)/e**4 - (b*e - 2*c*d)*(a*e**2
- b*d*e + c*d**2)/(e**4*(d + e*x))

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Mathematica [A]  time = 0.123282, size = 97, normalized size = 0.95 \[ \frac{\log (d+e x) \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+\frac{(2 c d-b e) \left (e (a e-b d)+c d^2\right )}{d+e x}-c e x (4 c d-3 b e)+c^2 e^2 x^2}{e^4} \]

Antiderivative was successfully verified.

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

[Out]

(-(c*e*(4*c*d - 3*b*e)*x) + c^2*e^2*x^2 + ((2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*
e)))/(d + e*x) + (6*c^2*d^2 + b^2*e^2 + 2*c*e*(-3*b*d + a*e))*Log[d + e*x])/e^4

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Maple [A]  time = 0.011, size = 166, normalized size = 1.6 \[{\frac{{c}^{2}{x}^{2}}{{e}^{2}}}+3\,{\frac{bxc}{{e}^{2}}}-4\,{\frac{{c}^{2}dx}{{e}^{3}}}+2\,{\frac{\ln \left ( ex+d \right ) ac}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){b}^{2}}{{e}^{2}}}-6\,{\frac{\ln \left ( ex+d \right ) bcd}{{e}^{3}}}+6\,{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{2}}{{e}^{4}}}-{\frac{ab}{e \left ( ex+d \right ) }}+2\,{\frac{acd}{{e}^{2} \left ( ex+d \right ) }}+{\frac{{b}^{2}d}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{c{d}^{2}b}{{e}^{3} \left ( ex+d \right ) }}+2\,{\frac{{c}^{2}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.705886, size = 158, normalized size = 1.55 \[ \frac{2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}}{e^{5} x + d e^{4}} + \frac{c^{2} e x^{2} -{\left (4 \, c^{2} d - 3 \, b c e\right )} x}{e^{3}} + \frac{{\left (6 \, c^{2} d^{2} - 6 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)/(e^5*x + d*e^4) + (c^2
*e*x^2 - (4*c^2*d - 3*b*c*e)*x)/e^3 + (6*c^2*d^2 - 6*b*c*d*e + (b^2 + 2*a*c)*e^2
)*log(e*x + d)/e^4

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Fricas [A]  time = 0.276787, size = 232, normalized size = 2.27 \[ \frac{c^{2} e^{3} x^{3} + 2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2} - 3 \,{\left (c^{2} d e^{2} - b c e^{3}\right )} x^{2} -{\left (4 \, c^{2} d^{2} e - 3 \, b c d e^{2}\right )} x +{\left (6 \, c^{2} d^{3} - 6 \, b c d^{2} e +{\left (b^{2} + 2 \, a c\right )} d e^{2} +{\left (6 \, c^{2} d^{2} e - 6 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x\right )} \log \left (e x + d\right )}{e^{5} x + d e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c^2*e^3*x^3 + 2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2 - 3*(c^2*
d*e^2 - b*c*e^3)*x^2 - (4*c^2*d^2*e - 3*b*c*d*e^2)*x + (6*c^2*d^3 - 6*b*c*d^2*e
+ (b^2 + 2*a*c)*d*e^2 + (6*c^2*d^2*e - 6*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x)*log(e
*x + d))/(e^5*x + d*e^4)

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Sympy [A]  time = 3.43553, size = 124, normalized size = 1.22 \[ \frac{c^{2} x^{2}}{e^{2}} - \frac{a b e^{3} - 2 a c d e^{2} - b^{2} d e^{2} + 3 b c d^{2} e - 2 c^{2} d^{3}}{d e^{4} + e^{5} x} + \frac{x \left (3 b c e - 4 c^{2} d\right )}{e^{3}} + \frac{\left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*x**2/e**2 - (a*b*e**3 - 2*a*c*d*e**2 - b**2*d*e**2 + 3*b*c*d**2*e - 2*c**2*
d**3)/(d*e**4 + e**5*x) + x*(3*b*c*e - 4*c**2*d)/e**3 + (2*a*c*e**2 + b**2*e**2
- 6*b*c*d*e + 6*c**2*d**2)*log(d + e*x)/e**4

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GIAC/XCAS [A]  time = 0.273621, size = 239, normalized size = 2.34 \[{\left (c^{2} - \frac{3 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-4\right )} -{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2} + 2 \, a c e^{2}\right )} e^{\left (-4\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{2 \, c^{2} d^{3} e^{2}}{x e + d} - \frac{3 \, b c d^{2} e^{3}}{x e + d} + \frac{b^{2} d e^{4}}{x e + d} + \frac{2 \, a c d e^{4}}{x e + d} - \frac{a b e^{5}}{x e + d}\right )} e^{\left (-6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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