3.2358 \(\int \frac{x \left (a+b x+c x^2\right )}{(d+e x)^3} \, dx\)

Optimal. Leaf size=89 \[ -\frac{3 c d^2-e (2 b d-a e)}{e^4 (d+e x)}+\frac{d \left (a e^2-b d e+c d^2\right )}{2 e^4 (d+e x)^2}-\frac{(3 c d-b e) \log (d+e x)}{e^4}+\frac{c x}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.177188, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{3 c d^2-e (2 b d-a e)}{e^4 (d+e x)}+\frac{d \left (a e^2-b d e+c d^2\right )}{2 e^4 (d+e x)^2}-\frac{(3 c d-b e) \log (d+e x)}{e^4}+\frac{c x}{e^3} \]

Antiderivative was successfully verified.

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

[Out]

(c*x)/e^3 + (d*(c*d^2 - b*d*e + a*e^2))/(2*e^4*(d + e*x)^2) - (3*c*d^2 - e*(2*b*
d - a*e))/(e^4*(d + e*x)) - ((3*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. \[ \frac{d \left (a e^{2} - b d e + c d^{2}\right )}{2 e^{4} \left (d + e x\right )^{2}} + \frac{\int c\, dx}{e^{3}} + \frac{\left (b e - 3 c d\right ) \log{\left (d + e x \right )}}{e^{4}} - \frac{a e^{2} - 2 b d e + 3 c d^{2}}{e^{4} \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.112069, size = 80, normalized size = 0.9 \[ \frac{\frac{d e (a e-b d)+c d^3}{(d+e x)^2}-\frac{2 \left (e (a e-2 b d)+3 c d^2\right )}{d+e x}+2 (b e-3 c d) \log (d+e x)+2 c e x}{2 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 121, normalized size = 1.4 \[{\frac{cx}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) b}{{e}^{3}}}-3\,{\frac{\ln \left ( ex+d \right ) cd}{{e}^{4}}}-{\frac{a}{{e}^{2} \left ( ex+d \right ) }}+2\,{\frac{bd}{{e}^{3} \left ( ex+d \right ) }}-3\,{\frac{c{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}+{\frac{da}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{d}^{2}b}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{d}^{3}c}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.705979, size = 130, normalized size = 1.46 \[ -\frac{5 \, c d^{3} - 3 \, b d^{2} e + a d e^{2} + 2 \,{\left (3 \, c d^{2} e - 2 \, b d e^{2} + a e^{3}\right )} x}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} + \frac{c x}{e^{3}} - \frac{{\left (3 \, c d - b e\right )} \log \left (e x + d\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*x/(e*x + d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.260436, size = 198, normalized size = 2.22 \[ \frac{2 \, c e^{3} x^{3} + 4 \, c d e^{2} x^{2} - 5 \, c d^{3} + 3 \, b d^{2} e - a d e^{2} - 2 \,{\left (2 \, c d^{2} e - 2 \, b d e^{2} + a e^{3}\right )} x - 2 \,{\left (3 \, c d^{3} - b d^{2} e +{\left (3 \, c d e^{2} - b e^{3}\right )} x^{2} + 2 \,{\left (3 \, c d^{2} e - b d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*x/(e*x + d)^3,x, algorithm="fricas")

[Out]

1/2*(2*c*e^3*x^3 + 4*c*d*e^2*x^2 - 5*c*d^3 + 3*b*d^2*e - a*d*e^2 - 2*(2*c*d^2*e
- 2*b*d*e^2 + a*e^3)*x - 2*(3*c*d^3 - b*d^2*e + (3*c*d*e^2 - b*e^3)*x^2 + 2*(3*c
*d^2*e - b*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 = 3.96618, size = 97, normalized size = 1.09 \[ \frac{c x}{e^{3}} - \frac{a d e^{2} - 3 b d^{2} e + 5 c d^{3} + x \left (2 a e^{3} - 4 b d e^{2} + 6 c d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} + \frac{\left (b e - 3 c d\right ) \log{\left (d + e x \right )}}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.258705, size = 111, normalized size = 1.25 \[ c x e^{\left (-3\right )} -{\left (3 \, c d - b e\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (5 \, c d^{3} - 3 \, b d^{2} e + a d e^{2} + 2 \,{\left (3 \, c d^{2} e - 2 \, b d e^{2} + a e^{3}\right )} x\right )} e^{\left (-4\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*x/(e*x + d)^3,x, algorithm="giac")

[Out]

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