3.1501 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^3} \, dx\)

Optimal. Leaf size=111 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^4 (d+e x)}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^4 (d+e x)^2}-\frac{3 c (2 c d-b e) \log (d+e x)}{e^4}+\frac{2 c^2 x}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.23844, antiderivative size = 111, 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 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^4 (d+e x)}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^4 (d+e x)^2}-\frac{3 c (2 c d-b e) \log (d+e x)}{e^4}+\frac{2 c^2 x}{e^3} \]

Antiderivative was successfully verified.

[In]  Int[((b + 2*c*x)*(a + b*x + c*x^2))/(d + e*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 40.8922, size = 107, normalized size = 0.96 \[ \frac{2 c^{2} x}{e^{3}} + \frac{3 c \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{e^{4}} - \frac{2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}}{e^{4} \left (d + e x\right )} - \frac{\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{2 e^{4} \left (d + e x\right )^{2}} \]

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)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0843757, size = 118, normalized size = 1.06 \[ \frac{c e (3 b d (3 d+4 e x)-2 a e (d+2 e x))-b e^2 (a e+b (d+2 e x))-6 c (d+e x)^2 (2 c d-b e) \log (d+e x)+c^2 \left (-10 d^3-8 d^2 e x+8 d e^2 x^2+4 e^3 x^3\right )}{2 e^4 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((b + 2*c*x)*(a + b*x + c*x^2))/(d + e*x)^3,x]

[Out]

(c^2*(-10*d^3 - 8*d^2*e*x + 8*d*e^2*x^2 + 4*e^3*x^3) - b*e^2*(a*e + b*(d + 2*e*x
)) + c*e*(-2*a*e*(d + 2*e*x) + 3*b*d*(3*d + 4*e*x)) - 6*c*(2*c*d - b*e)*(d + e*x
)^2*Log[d + e*x])/(2*e^4*(d + e*x)^2)

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Maple [A]  time = 0.01, size = 179, normalized size = 1.6 \[ 2\,{\frac{{c}^{2}x}{{e}^{3}}}-{\frac{ab}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{acd}{{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{2}d}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{3\,c{d}^{2}b}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{2}{d}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+3\,{\frac{c\ln \left ( ex+d \right ) b}{{e}^{3}}}-6\,{\frac{{c}^{2}\ln \left ( ex+d \right ) d}{{e}^{4}}}-2\,{\frac{ac}{{e}^{2} \left ( ex+d \right ) }}-{\frac{{b}^{2}}{{e}^{2} \left ( ex+d \right ) }}+6\,{\frac{bcd}{{e}^{3} \left ( ex+d \right ) }}-6\,{\frac{{c}^{2}{d}^{2}}{{e}^{4} \left ( ex+d \right ) }} \]

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)^3,x)

[Out]

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

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Maxima [A]  time = 0.717806, size = 173, normalized size = 1.56 \[ -\frac{10 \, c^{2} d^{3} - 9 \, b c d^{2} e + a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2} + 2 \,{\left (6 \, c^{2} d^{2} e - 6 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} + \frac{2 \, c^{2} x}{e^{3}} - \frac{3 \,{\left (2 \, c^{2} d - b c e\right )} \log \left (e x + d\right )}{e^{4}} \]

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)^3,x, algorithm="maxima")

[Out]

-1/2*(10*c^2*d^3 - 9*b*c*d^2*e + a*b*e^3 + (b^2 + 2*a*c)*d*e^2 + 2*(6*c^2*d^2*e
- 6*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x)/(e^6*x^2 + 2*d*e^5*x + d^2*e^4) + 2*c^2*x/
e^3 - 3*(2*c^2*d - b*c*e)*log(e*x + d)/e^4

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Fricas [A]  time = 0.266245, size = 252, normalized size = 2.27 \[ \frac{4 \, c^{2} e^{3} x^{3} + 8 \, c^{2} d e^{2} x^{2} - 10 \, c^{2} d^{3} + 9 \, b c d^{2} e - a b e^{3} -{\left (b^{2} + 2 \, a c\right )} d e^{2} - 2 \,{\left (4 \, c^{2} d^{2} e - 6 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x - 6 \,{\left (2 \, c^{2} d^{3} - b c d^{2} e +{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} x^{2} + 2 \,{\left (2 \, c^{2} d^{2} e - b c d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

1/2*(4*c^2*e^3*x^3 + 8*c^2*d*e^2*x^2 - 10*c^2*d^3 + 9*b*c*d^2*e - a*b*e^3 - (b^2
+ 2*a*c)*d*e^2 - 2*(4*c^2*d^2*e - 6*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x - 6*(2*c^2
*d^3 - b*c*d^2*e + (2*c^2*d*e^2 - b*c*e^3)*x^2 + 2*(2*c^2*d^2*e - b*c*d*e^2)*x)*
log(e*x + d))/(e^6*x^2 + 2*d*e^5*x + d^2*e^4)

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Sympy [A]  time = 7.27249, size = 139, normalized size = 1.25 \[ \frac{2 c^{2} x}{e^{3}} + \frac{3 c \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{e^{4}} - \frac{a b e^{3} + 2 a c d e^{2} + b^{2} d e^{2} - 9 b c d^{2} e + 10 c^{2} d^{3} + x \left (4 a c e^{3} + 2 b^{2} e^{3} - 12 b c d e^{2} + 12 c^{2} d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \]

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)**3,x)

[Out]

2*c**2*x/e**3 + 3*c*(b*e - 2*c*d)*log(d + e*x)/e**4 - (a*b*e**3 + 2*a*c*d*e**2 +
b**2*d*e**2 - 9*b*c*d**2*e + 10*c**2*d**3 + x*(4*a*c*e**3 + 2*b**2*e**3 - 12*b*
c*d*e**2 + 12*c**2*d**2*e))/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2)

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GIAC/XCAS [A]  time = 0.273855, size = 157, normalized size = 1.41 \[ 2 \, c^{2} x e^{\left (-3\right )} - 3 \,{\left (2 \, c^{2} d - b c e\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (10 \, c^{2} d^{3} - 9 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} + a b e^{3} + 2 \,{\left (6 \, c^{2} d^{2} e - 6 \, b c d e^{2} + b^{2} e^{3} + 2 \, a c e^{3}\right )} x\right )} e^{\left (-4\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

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)^3,x, algorithm="giac")

[Out]

2*c^2*x*e^(-3) - 3*(2*c^2*d - b*c*e)*e^(-4)*ln(abs(x*e + d)) - 1/2*(10*c^2*d^3 -
9*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 + a*b*e^3 + 2*(6*c^2*d^2*e - 6*b*c*d*e^2
+ b^2*e^3 + 2*a*c*e^3)*x)*e^(-4)/(x*e + d)^2