3.2357 \(\int \frac{x (a+b x+c x^2)}{d+e x} \, dx\)

Optimal. Leaf size=79 \[ \frac{x \left (a e^2-b d e+c d^2\right )}{e^3}-\frac{d \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac{x^2 (c d-b e)}{2 e^2}+\frac{c x^3}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.073496, antiderivative size = 79, 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} \[ \frac{x \left (a e^2-b d e+c d^2\right )}{e^3}-\frac{d \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac{x^2 (c d-b e)}{2 e^2}+\frac{c x^3}{3 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0288964, size = 74, normalized size = 0.94 \[ \frac{e x \left (3 e (2 a e-2 b d+b e x)+c \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 \log (d+e x) \left (d e (a e-b d)+c d^3\right )}{6 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 95, normalized size = 1.2 \begin{align*}{\frac{c{x}^{3}}{3\,e}}+{\frac{b{x}^{2}}{2\,e}}-{\frac{c{x}^{2}d}{2\,{e}^{2}}}+{\frac{ax}{e}}-{\frac{bdx}{{e}^{2}}}+{\frac{c{d}^{2}x}{{e}^{3}}}-{\frac{d\ln \left ( ex+d \right ) a}{{e}^{2}}}+{\frac{{d}^{2}\ln \left ( ex+d \right ) b}{{e}^{3}}}-{\frac{{d}^{3}\ln \left ( ex+d \right ) c}{{e}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c*x^2+b*x+a)/(e*x+d),x)

[Out]

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

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Maxima [A]  time = 1.00997, size = 109, normalized size = 1.38 \begin{align*} \frac{2 \, c e^{2} x^{3} - 3 \,{\left (c d e - b e^{2}\right )} x^{2} + 6 \,{\left (c d^{2} - b d e + a e^{2}\right )} x}{6 \, e^{3}} - \frac{{\left (c d^{3} - b d^{2} e + a d e^{2}\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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.30979, size = 173, normalized size = 2.19 \begin{align*} \frac{2 \, c e^{3} x^{3} - 3 \,{\left (c d e^{2} - b e^{3}\right )} x^{2} + 6 \,{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} x - 6 \,{\left (c d^{3} - b d^{2} e + a d e^{2}\right )} \log \left (e x + d\right )}{6 \, 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.427311, size = 70, normalized size = 0.89 \begin{align*} \frac{c x^{3}}{3 e} - \frac{d \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} + \frac{x^{2} \left (b e - c d\right )}{2 e^{2}} + \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{e^{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),x)

[Out]

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

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Giac [A]  time = 1.11624, size = 111, normalized size = 1.41 \begin{align*} -{\left (c d^{3} - b d^{2} e + a d e^{2}\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, c x^{3} e^{2} - 3 \, c d x^{2} e + 6 \, c d^{2} x + 3 \, b x^{2} e^{2} - 6 \, b d x e + 6 \, a x e^{2}\right )} e^{\left (-3\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),x, algorithm="giac")

[Out]

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