3.2360 \(\int \frac{x (a+b x+c x^2)}{(d+e x)^4} \, dx\)

Optimal. Leaf size=96 \[ -\frac{3 c d^2-e (2 b d-a e)}{2 e^4 (d+e x)^2}+\frac{d \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac{3 c d-b e}{e^4 (d+e x)}+\frac{c \log (d+e x)}{e^4} \]

[Out]

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

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Rubi [A]  time = 0.0778137, antiderivative size = 96, 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} \[ -\frac{3 c d^2-e (2 b d-a e)}{2 e^4 (d+e x)^2}+\frac{d \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac{3 c d-b e}{e^4 (d+e x)}+\frac{c \log (d+e x)}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0368533, size = 86, normalized size = 0.9 \[ \frac{-e \left (a e (d+3 e x)+2 b \left (d^2+3 d e x+3 e^2 x^2\right )\right )+c d \left (11 d^2+27 d e x+18 e^2 x^2\right )+6 c (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 128, normalized size = 1.3 \begin{align*}{\frac{c\ln \left ( ex+d \right ) }{{e}^{4}}}+{\frac{ad}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{b{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{{d}^{3}c}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{a}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{bd}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,c{d}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{b}{{e}^{3} \left ( ex+d \right ) }}+3\,{\frac{cd}{{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)^4,x)

[Out]

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

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Maxima [A]  time = 1.02965, size = 153, normalized size = 1.59 \begin{align*} \frac{11 \, c d^{3} - 2 \, b d^{2} e - a d e^{2} + 6 \,{\left (3 \, c d e^{2} - b e^{3}\right )} x^{2} + 3 \,{\left (9 \, c d^{2} e - 2 \, b d e^{2} - a 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{c \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)^4,x, algorithm="maxima")

[Out]

1/6*(11*c*d^3 - 2*b*d^2*e - a*d*e^2 + 6*(3*c*d*e^2 - b*e^3)*x^2 + 3*(9*c*d^2*e - 2*b*d*e^2 - a*e^3)*x)/(e^7*x^
3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4) + c*log(e*x + d)/e^4

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Fricas [A]  time = 1.28395, size = 296, normalized size = 3.08 \begin{align*} \frac{11 \, c d^{3} - 2 \, b d^{2} e - a d e^{2} + 6 \,{\left (3 \, c d e^{2} - b e^{3}\right )} x^{2} + 3 \,{\left (9 \, c d^{2} e - 2 \, b d e^{2} - a e^{3}\right )} x + 6 \,{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2+b*x+a)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/6*(11*c*d^3 - 2*b*d^2*e - a*d*e^2 + 6*(3*c*d*e^2 - b*e^3)*x^2 + 3*(9*c*d^2*e - 2*b*d*e^2 - a*e^3)*x + 6*(c*e
^3*x^3 + 3*c*d*e^2*x^2 + 3*c*d^2*e*x + c*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)

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Sympy [A]  time = 1.59645, size = 114, normalized size = 1.19 \begin{align*} \frac{c \log{\left (d + e x \right )}}{e^{4}} - \frac{a d e^{2} + 2 b d^{2} e - 11 c d^{3} + x^{2} \left (6 b e^{3} - 18 c d e^{2}\right ) + x \left (3 a e^{3} + 6 b d e^{2} - 27 c 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x**2+b*x+a)/(e*x+d)**4,x)

[Out]

c*log(d + e*x)/e**4 - (a*d*e**2 + 2*b*d**2*e - 11*c*d**3 + x**2*(6*b*e**3 - 18*c*d*e**2) + x*(3*a*e**3 + 6*b*d
*e**2 - 27*c*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)

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Giac [A]  time = 1.08241, size = 119, normalized size = 1.24 \begin{align*} c e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (6 \,{\left (3 \, c d e - b e^{2}\right )} x^{2} + 3 \,{\left (9 \, c d^{2} - 2 \, b d e - a e^{2}\right )} x +{\left (11 \, c d^{3} - 2 \, b d^{2} e - a d e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2+b*x+a)/(e*x+d)^4,x, algorithm="giac")

[Out]

c*e^(-4)*log(abs(x*e + d)) + 1/6*(6*(3*c*d*e - b*e^2)*x^2 + 3*(9*c*d^2 - 2*b*d*e - a*e^2)*x + (11*c*d^3 - 2*b*
d^2*e - a*d*e^2)*e^(-1))*e^(-3)/(x*e + d)^3