3.1502 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^4} \, dx\)

Optimal. Leaf size=119 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^4 (d+e x)^2}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac{3 c (2 c d-b e)}{e^4 (d+e x)}+\frac{2 c^2 \log (d+e x)}{e^4} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.222909, antiderivative size = 119, 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 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^4 (d+e x)^2}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac{3 c (2 c d-b e)}{e^4 (d+e x)}+\frac{2 c^2 \log (d+e x)}{e^4} \]

Antiderivative was successfully verified.

[In]  Int[((b + 2*c*x)*(a + b*x + c*x^2))/(d + e*x)^4,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 39.7675, size = 114, normalized size = 0.96 \[ \frac{2 c^{2} \log{\left (d + e x \right )}}{e^{4}} - \frac{3 c \left (b e - 2 c d\right )}{e^{4} \left (d + e x\right )} - \frac{2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}}{2 e^{4} \left (d + e x\right )^{2}} - \frac{\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{3 e^{4} \left (d + e x\right )^{3}} \]

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)**4,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0811563, size = 111, normalized size = 0.93 \[ \frac{-2 c e \left (a e (d+3 e x)+3 b \left (d^2+3 d e x+3 e^2 x^2\right )\right )-b e^2 (2 a e+b (d+3 e x))+2 c^2 d \left (11 d^2+27 d e x+18 e^2 x^2\right )+12 c^2 (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((b + 2*c*x)*(a + b*x + c*x^2))/(d + e*x)^4,x]

[Out]

(2*c^2*d*(11*d^2 + 27*d*e*x + 18*e^2*x^2) - b*e^2*(2*a*e + b*(d + 3*e*x)) - 2*c*
e*(a*e*(d + 3*e*x) + 3*b*(d^2 + 3*d*e*x + 3*e^2*x^2)) + 12*c^2*(d + e*x)^3*Log[d
+ e*x])/(6*e^4*(d + e*x)^3)

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 188, normalized size = 1.6 \[ -{\frac{ab}{3\,e \left ( ex+d \right ) ^{3}}}+{\frac{2\,acd}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}+{\frac{{b}^{2}d}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{c{d}^{2}b}{{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{2\,{c}^{2}{d}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{ac}{{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{2}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+3\,{\frac{bcd}{{e}^{3} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{c}^{2}{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{c}^{2}\ln \left ( ex+d \right ) }{{e}^{4}}}-3\,{\frac{bc}{{e}^{3} \left ( ex+d \right ) }}+6\,{\frac{{c}^{2}d}{{e}^{4} \left ( ex+d \right ) }} \]

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)^4,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.734375, size = 197, normalized size = 1.66 \[ \frac{22 \, c^{2} d^{3} - 6 \, b c d^{2} e - 2 \, a b e^{3} -{\left (b^{2} + 2 \, a c\right )} d e^{2} + 18 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} x^{2} + 3 \,{\left (18 \, c^{2} d^{2} e - 6 \, b c d e^{2} -{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac{2 \, c^{2} \log \left (e x + d\right )}{e^{4}} \]

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)^4,x, algorithm="maxima")

[Out]

1/6*(22*c^2*d^3 - 6*b*c*d^2*e - 2*a*b*e^3 - (b^2 + 2*a*c)*d*e^2 + 18*(2*c^2*d*e^
2 - b*c*e^3)*x^2 + 3*(18*c^2*d^2*e - 6*b*c*d*e^2 - (b^2 + 2*a*c)*e^3)*x)/(e^7*x^
3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4) + 2*c^2*log(e*x + d)/e^4

_______________________________________________________________________________________

Fricas [A]  time = 0.258048, size = 242, normalized size = 2.03 \[ \frac{22 \, c^{2} d^{3} - 6 \, b c d^{2} e - 2 \, a b e^{3} -{\left (b^{2} + 2 \, a c\right )} d e^{2} + 18 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} x^{2} + 3 \,{\left (18 \, c^{2} d^{2} e - 6 \, b c d e^{2} -{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x + 12 \,{\left (c^{2} e^{3} x^{3} + 3 \, c^{2} d e^{2} x^{2} + 3 \, c^{2} d^{2} e x + c^{2} d^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \]

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)^4,x, algorithm="fricas")

[Out]

1/6*(22*c^2*d^3 - 6*b*c*d^2*e - 2*a*b*e^3 - (b^2 + 2*a*c)*d*e^2 + 18*(2*c^2*d*e^
2 - b*c*e^3)*x^2 + 3*(18*c^2*d^2*e - 6*b*c*d*e^2 - (b^2 + 2*a*c)*e^3)*x + 12*(c^
2*e^3*x^3 + 3*c^2*d*e^2*x^2 + 3*c^2*d^2*e*x + c^2*d^3)*log(e*x + d))/(e^7*x^3 +
3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4)

_______________________________________________________________________________________

Sympy [A]  time = 13.5202, size = 158, normalized size = 1.33 \[ \frac{2 c^{2} \log{\left (d + e x \right )}}{e^{4}} - \frac{2 a b e^{3} + 2 a c d e^{2} + b^{2} d e^{2} + 6 b c d^{2} e - 22 c^{2} d^{3} + x^{2} \left (18 b c e^{3} - 36 c^{2} d e^{2}\right ) + x \left (6 a c e^{3} + 3 b^{2} e^{3} + 18 b c d e^{2} - 54 c^{2} d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \]

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)**4,x)

[Out]

2*c**2*log(d + e*x)/e**4 - (2*a*b*e**3 + 2*a*c*d*e**2 + b**2*d*e**2 + 6*b*c*d**2
*e - 22*c**2*d**3 + x**2*(18*b*c*e**3 - 36*c**2*d*e**2) + x*(6*a*c*e**3 + 3*b**2
*e**3 + 18*b*c*d*e**2 - 54*c**2*d**2*e))/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e*
*6*x**2 + 6*e**7*x**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270733, size = 166, normalized size = 1.39 \[ 2 \, c^{2} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (18 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} x^{2} + 3 \,{\left (18 \, c^{2} d^{2} - 6 \, b c d e - b^{2} e^{2} - 2 \, a c e^{2}\right )} x +{\left (22 \, c^{2} d^{3} - 6 \, b c d^{2} e - b^{2} d e^{2} - 2 \, a c d e^{2} - 2 \, a b e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \]

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)^4,x, algorithm="giac")

[Out]

2*c^2*e^(-4)*ln(abs(x*e + d)) + 1/6*(18*(2*c^2*d*e - b*c*e^2)*x^2 + 3*(18*c^2*d^
2 - 6*b*c*d*e - b^2*e^2 - 2*a*c*e^2)*x + (22*c^2*d^3 - 6*b*c*d^2*e - b^2*d*e^2 -
2*a*c*d*e^2 - 2*a*b*e^3)*e^(-1))*e^(-3)/(x*e + d)^3