### 3.229 $$\int \frac{b x+c x^2}{(d+e x)^4} \, dx$$

Optimal. Leaf size=60 $\frac{2 c d-b e}{2 e^3 (d+e x)^2}-\frac{d (c d-b e)}{3 e^3 (d+e x)^3}-\frac{c}{e^3 (d+e x)}$

[Out]

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

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Rubi [A]  time = 0.0356667, antiderivative size = 60, 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} $\frac{2 c d-b e}{2 e^3 (d+e x)^2}-\frac{d (c d-b e)}{3 e^3 (d+e x)^3}-\frac{c}{e^3 (d+e x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)/(d + e*x)^4,x]

[Out]

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

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

Mathematica [A]  time = 0.0165066, size = 44, normalized size = 0.73 $-\frac{b e (d+3 e x)+2 c \left (d^2+3 d e x+3 e^2 x^2\right )}{6 e^3 (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(b*e*(d + 3*e*x) + 2*c*(d^2 + 3*d*e*x + 3*e^2*x^2))/(6*e^3*(d + e*x)^3)

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Maple [A]  time = 0.048, size = 56, normalized size = 0.9 \begin{align*} -{\frac{be-2\,cd}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{d \left ( be-cd \right ) }{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{c}{{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)^4,x)

[Out]

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

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Maxima [A]  time = 1.12797, size = 96, normalized size = 1.6 \begin{align*} -\frac{6 \, c e^{2} x^{2} + 2 \, c d^{2} + b d e + 3 \,{\left (2 \, c d e + b e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} 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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45312, size = 149, normalized size = 2.48 \begin{align*} -\frac{6 \, c e^{2} x^{2} + 2 \, c d^{2} + b d e + 3 \,{\left (2 \, c d e + b e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} 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)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.53654, size = 75, normalized size = 1.25 \begin{align*} - \frac{b d e + 2 c d^{2} + 6 c e^{2} x^{2} + x \left (3 b e^{2} + 6 c d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*d*e + 2*c*d**2 + 6*c*e**2*x**2 + x*(3*b*e**2 + 6*c*d*e))/(6*d**3*e**3 + 18*d**2*e**4*x + 18*d*e**5*x**2 +
6*e**6*x**3)

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Giac [A]  time = 1.35359, size = 61, normalized size = 1.02 \begin{align*} -\frac{{\left (6 \, c x^{2} e^{2} + 6 \, c d x e + 2 \, c d^{2} + 3 \, b x e^{2} + b d e\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((c*x^2+b*x)/(e*x+d)^4,x, algorithm="giac")

[Out]

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