3.1459 $$\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx$$

Optimal. Leaf size=28 $\frac{(a+b x)^3}{3 (d+e x)^3 (b d-a e)}$

[Out]

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

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Rubi [A]  time = 0.0054238, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {27, 37} $\frac{(a+b x)^3}{3 (d+e x)^3 (b d-a e)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0248631, size = 53, normalized size = 1.89 $-\frac{a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.045, size = 71, normalized size = 2.5 \begin{align*} -{\frac{{a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{ \left ( ae-bd \right ) b}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{2}}{{e}^{3} \left ( ex+d \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^4,x)

[Out]

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

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Maxima [B]  time = 1.10879, size = 113, normalized size = 4.04 \begin{align*} -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\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((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

-1/3*(3*b^2*e^2*x^2 + b^2*d^2 + a*b*d*e + a^2*e^2 + 3*(b^2*d*e + a*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 [B]  time = 1.68778, size = 170, normalized size = 6.07 \begin{align*} -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\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((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

-1/3*(3*b^2*e^2*x^2 + b^2*d^2 + a*b*d*e + a^2*e^2 + 3*(b^2*d*e + a*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 [B]  time = 0.868145, size = 88, normalized size = 3.14 \begin{align*} - \frac{a^{2} e^{2} + a b d e + b^{2} d^{2} + 3 b^{2} e^{2} x^{2} + x \left (3 a b e^{2} + 3 b^{2} d e\right )}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a**2*e**2 + a*b*d*e + b**2*d**2 + 3*b**2*e**2*x**2 + x*(3*a*b*e**2 + 3*b**2*d*e))/(3*d**3*e**3 + 9*d**2*e**4
*x + 9*d*e**5*x**2 + 3*e**6*x**3)

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Giac [B]  time = 1.17025, size = 78, normalized size = 2.79 \begin{align*} -\frac{{\left (3 \, b^{2} x^{2} e^{2} + 3 \, b^{2} d x e + b^{2} d^{2} + 3 \, a b x e^{2} + a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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