3.1499 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )}{d+e x} \, dx\)

Optimal. Leaf size=104 \[ \frac{x \left (-c e (3 b d-2 a e)+b^2 e^2+2 c^2 d^2\right )}{e^3}-\frac{(2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac{c x^2 (2 c d-3 b e)}{2 e^2}+\frac{2 c^2 x^3}{3 e} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.24334, antiderivative size = 104, 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 \[ \frac{x \left (-c e (3 b d-2 a e)+b^2 e^2+2 c^2 d^2\right )}{e^3}-\frac{(2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac{c x^2 (2 c d-3 b e)}{2 e^2}+\frac{2 c^2 x^3}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

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),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0674175, size = 95, normalized size = 0.91 \[ \frac{e x \left (3 c e (4 a e-6 b d+3 b e x)+6 b^2 e^2+2 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )}{6 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 146, normalized size = 1.4 \[{\frac{2\,{c}^{2}{x}^{3}}{3\,e}}+{\frac{3\,b{x}^{2}c}{2\,e}}-{\frac{{c}^{2}{x}^{2}d}{{e}^{2}}}+2\,{\frac{acx}{e}}+{\frac{{b}^{2}x}{e}}-3\,{\frac{bcdx}{{e}^{2}}}+2\,{\frac{{c}^{2}{d}^{2}x}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) ab}{e}}-2\,{\frac{\ln \left ( ex+d \right ) adc}{{e}^{2}}}-{\frac{\ln \left ( ex+d \right ){b}^{2}d}{{e}^{2}}}+3\,{\frac{\ln \left ( ex+d \right ) bc{d}^{2}}{{e}^{3}}}-2\,{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{3}}{{e}^{4}}} \]

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),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.702646, size = 157, normalized size = 1.51 \[ \frac{4 \, c^{2} e^{2} x^{3} - 3 \,{\left (2 \, c^{2} d e - 3 \, b c e^{2}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{2} - 3 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x}{6 \, e^{3}} - \frac{{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \]

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),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.308208, size = 158, normalized size = 1.52 \[ \frac{4 \, c^{2} e^{3} x^{3} - 3 \,{\left (2 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{2} e - 3 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x - 6 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \]

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),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 2.11421, size = 104, normalized size = 1. \[ \frac{2 c^{2} x^{3}}{3 e} + \frac{x^{2} \left (3 b c e - 2 c^{2} d\right )}{2 e^{2}} + \frac{x \left (2 a c e^{2} + b^{2} e^{2} - 3 b c d e + 2 c^{2} d^{2}\right )}{e^{3}} + \frac{\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} \]

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),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270296, size = 159, normalized size = 1.53 \[ -{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (4 \, c^{2} x^{3} e^{2} - 6 \, c^{2} d x^{2} e + 12 \, c^{2} d^{2} x + 9 \, b c x^{2} e^{2} - 18 \, b c d x e + 6 \, b^{2} x e^{2} + 12 \, a c x e^{2}\right )} e^{\left (-3\right )} \]

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),x, algorithm="giac")

[Out]

-(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3)*e^(-4)*ln(abs(x*e
+ d)) + 1/6*(4*c^2*x^3*e^2 - 6*c^2*d*x^2*e + 12*c^2*d^2*x + 9*b*c*x^2*e^2 - 18*
b*c*d*x*e + 6*b^2*x*e^2 + 12*a*c*x*e^2)*e^(-3)