3.2359 \(\int \frac{x (a+b x+c x^2)}{(d+e x)^3} \, dx\)

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

[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 = 0.0834127, 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, Rules used = {771} \[ \frac{d \left (a e^2-b d e+c d^2\right )}{2 e^4 (d+e x)^2}-\frac{3 c d^2-e (2 b d-a e)}{e^4 (d+e x)}-\frac{(3 c d-b e) \log (d+e x)}{e^4}+\frac{c x}{e^3} \]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{x \left (a+b x+c x^2\right )}{(d+e x)^3} \, dx &=\int \left (\frac{c}{e^3}-\frac{d \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^3}+\frac{3 c d^2-e (2 b d-a e)}{e^3 (d+e x)^2}+\frac{-3 c d+b e}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{c x}{e^3}+\frac{d \left (c d^2-b d e+a e^2\right )}{2 e^4 (d+e x)^2}-\frac{3 c d^2-e (2 b d-a e)}{e^4 (d+e x)}-\frac{(3 c d-b e) \log (d+e x)}{e^4}\\ \end{align*}

Mathematica [A]  time = 0.060328, size = 80, normalized size = 0.9 \[ \frac{-\frac{2 \left (e (a e-2 b d)+3 c d^2\right )}{d+e x}+\frac{d e (a e-b d)+c d^3}{(d+e x)^2}+2 (b e-3 c d) \log (d+e x)+2 c e x}{2 e^4} \]

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 = 0.006, size = 121, normalized size = 1.4 \begin{align*}{\frac{cx}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) b}{{e}^{3}}}-3\,{\frac{\ln \left ( ex+d \right ) cd}{{e}^{4}}}+{\frac{ad}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{b{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{d}^{3}c}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\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 ) }} \end{align*}

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

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Maxima [A]  time = 1.0626, size = 130, normalized size = 1.46 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2+b*x+a)/(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 = 1.34546, size = 308, normalized size = 3.46 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2+b*x+a)/(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 = 1.0137, size = 97, normalized size = 1.09 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.08069, size = 111, normalized size = 1.25 \begin{align*} c x e^{\left (-3\right )} -{\left (3 \, c d - b e\right )} e^{\left (-4\right )} \log \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x*e^(-3) - (3*c*d - b*e)*e^(-4)*log(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