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

Optimal. Leaf size=55 \[ -\frac{d (c d-b e)}{2 e^3 (d+e x)^2}+\frac{2 c d-b e}{e^3 (d+e x)}+\frac{c \log (d+e x)}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.0373487, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {698} \[ -\frac{d (c d-b e)}{2 e^3 (d+e x)^2}+\frac{2 c d-b e}{e^3 (d+e x)}+\frac{c \log (d+e x)}{e^3} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)/(d + e*x)^3,x]

[Out]

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

Rule 698

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

Rubi steps

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

Mathematica [A]  time = 0.017849, size = 52, normalized size = 0.95 \[ \frac{-b e (d+2 e x)+c d (3 d+4 e x)+2 c (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.049, size = 70, normalized size = 1.3 \begin{align*}{\frac{bd}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{c{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{c\ln \left ( ex+d \right ) }{{e}^{3}}}-{\frac{b}{{e}^{2} \left ( ex+d \right ) }}+2\,{\frac{cd}{{e}^{3} \left ( ex+d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)/(e*x+d)^3,x)

[Out]

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

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Maxima [A]  time = 1.15968, size = 88, normalized size = 1.6 \begin{align*} \frac{3 \, c d^{2} - b d e + 2 \,{\left (2 \, c d e - b e^{2}\right )} x}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} + \frac{c \log \left (e x + d\right )}{e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.63066, size = 173, normalized size = 3.15 \begin{align*} \frac{3 \, c d^{2} - b d e + 2 \,{\left (2 \, c d e - b e^{2}\right )} x + 2 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.27856, size = 63, normalized size = 1.15 \begin{align*} \frac{c \log{\left (d + e x \right )}}{e^{3}} - \frac{b d e - 3 c d^{2} + x \left (2 b e^{2} - 4 c d e\right )}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)/(e*x+d)**3,x)

[Out]

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

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Giac [A]  time = 1.3188, size = 74, normalized size = 1.35 \begin{align*} c e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (2 \,{\left (2 \, c d - b e\right )} x +{\left (3 \, c d^{2} - b d e\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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