3.1523 $$\int \frac{1}{(d+e x)^2 (a^2+2 a b x+b^2 x^2)^2} \, dx$$

Optimal. Leaf size=132 $-\frac{e^3}{(d+e x) (b d-a e)^4}-\frac{3 b e^2}{(a+b x) (b d-a e)^4}-\frac{4 b e^3 \log (a+b x)}{(b d-a e)^5}+\frac{4 b e^3 \log (d+e x)}{(b d-a e)^5}+\frac{b e}{(a+b x)^2 (b d-a e)^3}-\frac{b}{3 (a+b x)^3 (b d-a e)^2}$

[Out]

-b/(3*(b*d - a*e)^2*(a + b*x)^3) + (b*e)/((b*d - a*e)^3*(a + b*x)^2) - (3*b*e^2)/((b*d - a*e)^4*(a + b*x)) - e
^3/((b*d - a*e)^4*(d + e*x)) - (4*b*e^3*Log[a + b*x])/(b*d - a*e)^5 + (4*b*e^3*Log[d + e*x])/(b*d - a*e)^5

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Rubi [A]  time = 0.101166, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {27, 44} $-\frac{e^3}{(d+e x) (b d-a e)^4}-\frac{3 b e^2}{(a+b x) (b d-a e)^4}-\frac{4 b e^3 \log (a+b x)}{(b d-a e)^5}+\frac{4 b e^3 \log (d+e x)}{(b d-a e)^5}+\frac{b e}{(a+b x)^2 (b d-a e)^3}-\frac{b}{3 (a+b x)^3 (b d-a e)^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

-b/(3*(b*d - a*e)^2*(a + b*x)^3) + (b*e)/((b*d - a*e)^3*(a + b*x)^2) - (3*b*e^2)/((b*d - a*e)^4*(a + b*x)) - e
^3/((b*d - a*e)^4*(d + e*x)) - (4*b*e^3*Log[a + b*x])/(b*d - a*e)^5 + (4*b*e^3*Log[d + e*x])/(b*d - a*e)^5

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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^4 (d+e x)^2} \, dx\\ &=\int \left (\frac{b^2}{(b d-a e)^2 (a+b x)^4}-\frac{2 b^2 e}{(b d-a e)^3 (a+b x)^3}+\frac{3 b^2 e^2}{(b d-a e)^4 (a+b x)^2}-\frac{4 b^2 e^3}{(b d-a e)^5 (a+b x)}+\frac{e^4}{(b d-a e)^4 (d+e x)^2}+\frac{4 b e^4}{(b d-a e)^5 (d+e x)}\right ) \, dx\\ &=-\frac{b}{3 (b d-a e)^2 (a+b x)^3}+\frac{b e}{(b d-a e)^3 (a+b x)^2}-\frac{3 b e^2}{(b d-a e)^4 (a+b x)}-\frac{e^3}{(b d-a e)^4 (d+e x)}-\frac{4 b e^3 \log (a+b x)}{(b d-a e)^5}+\frac{4 b e^3 \log (d+e x)}{(b d-a e)^5}\\ \end{align*}

Mathematica [A]  time = 0.124786, size = 121, normalized size = 0.92 $\frac{\frac{3 e^3 (a e-b d)}{d+e x}-\frac{9 b e^2 (b d-a e)}{a+b x}+\frac{3 b e (b d-a e)^2}{(a+b x)^2}-\frac{b (b d-a e)^3}{(a+b x)^3}-12 b e^3 \log (a+b x)+12 b e^3 \log (d+e x)}{3 (b d-a e)^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

(-((b*(b*d - a*e)^3)/(a + b*x)^3) + (3*b*e*(b*d - a*e)^2)/(a + b*x)^2 - (9*b*e^2*(b*d - a*e))/(a + b*x) + (3*e
^3*(-(b*d) + a*e))/(d + e*x) - 12*b*e^3*Log[a + b*x] + 12*b*e^3*Log[d + e*x])/(3*(b*d - a*e)^5)

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Maple [A]  time = 0.056, size = 132, normalized size = 1. \begin{align*} -{\frac{{e}^{3}}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}-4\,{\frac{{e}^{3}b\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{5}}}-{\frac{b}{3\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{3}}}+4\,{\frac{{e}^{3}b\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{5}}}-3\,{\frac{b{e}^{2}}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}-{\frac{be}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

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

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Maxima [B]  time = 1.11037, size = 807, normalized size = 6.11 \begin{align*} -\frac{4 \, b e^{3} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac{4 \, b e^{3} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac{12 \, b^{3} e^{3} x^{3} + b^{3} d^{3} - 5 \, a b^{2} d^{2} e + 13 \, a^{2} b d e^{2} + 3 \, a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} + 5 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (b^{3} d^{2} e - 8 \, a b^{2} d e^{2} - 11 \, a^{2} b e^{3}\right )} x}{3 \,{\left (a^{3} b^{4} d^{5} - 4 \, a^{4} b^{3} d^{4} e + 6 \, a^{5} b^{2} d^{3} e^{2} - 4 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4} +{\left (b^{7} d^{4} e - 4 \, a b^{6} d^{3} e^{2} + 6 \, a^{2} b^{5} d^{2} e^{3} - 4 \, a^{3} b^{4} d e^{4} + a^{4} b^{3} e^{5}\right )} x^{4} +{\left (b^{7} d^{5} - a b^{6} d^{4} e - 6 \, a^{2} b^{5} d^{3} e^{2} + 14 \, a^{3} b^{4} d^{2} e^{3} - 11 \, a^{4} b^{3} d e^{4} + 3 \, a^{5} b^{2} e^{5}\right )} x^{3} + 3 \,{\left (a b^{6} d^{5} - 3 \, a^{2} b^{5} d^{4} e + 2 \, a^{3} b^{4} d^{3} e^{2} + 2 \, a^{4} b^{3} d^{2} e^{3} - 3 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{2} +{\left (3 \, a^{2} b^{5} d^{5} - 11 \, a^{3} b^{4} d^{4} e + 14 \, a^{4} b^{3} d^{3} e^{2} - 6 \, a^{5} b^{2} d^{2} e^{3} - a^{6} b d e^{4} + a^{7} e^{5}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

-4*b*e^3*log(b*x + a)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5
*e^5) + 4*b*e^3*log(e*x + d)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^
4 - a^5*e^5) - 1/3*(12*b^3*e^3*x^3 + b^3*d^3 - 5*a*b^2*d^2*e + 13*a^2*b*d*e^2 + 3*a^3*e^3 + 6*(b^3*d*e^2 + 5*a
*b^2*e^3)*x^2 - 2*(b^3*d^2*e - 8*a*b^2*d*e^2 - 11*a^2*b*e^3)*x)/(a^3*b^4*d^5 - 4*a^4*b^3*d^4*e + 6*a^5*b^2*d^3
*e^2 - 4*a^6*b*d^2*e^3 + a^7*d*e^4 + (b^7*d^4*e - 4*a*b^6*d^3*e^2 + 6*a^2*b^5*d^2*e^3 - 4*a^3*b^4*d*e^4 + a^4*
b^3*e^5)*x^4 + (b^7*d^5 - a*b^6*d^4*e - 6*a^2*b^5*d^3*e^2 + 14*a^3*b^4*d^2*e^3 - 11*a^4*b^3*d*e^4 + 3*a^5*b^2*
e^5)*x^3 + 3*(a*b^6*d^5 - 3*a^2*b^5*d^4*e + 2*a^3*b^4*d^3*e^2 + 2*a^4*b^3*d^2*e^3 - 3*a^5*b^2*d*e^4 + a^6*b*e^
5)*x^2 + (3*a^2*b^5*d^5 - 11*a^3*b^4*d^4*e + 14*a^4*b^3*d^3*e^2 - 6*a^5*b^2*d^2*e^3 - a^6*b*d*e^4 + a^7*e^5)*x
)

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Fricas [B]  time = 1.81873, size = 1501, normalized size = 11.37 \begin{align*} -\frac{b^{4} d^{4} - 6 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 10 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} - 5 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \,{\left (b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} - 3 \, a^{2} b^{2} d e^{3} + 11 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} e^{4} x^{4} + a^{3} b d e^{3} +{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} +{\left (3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right ) - 12 \,{\left (b^{4} e^{4} x^{4} + a^{3} b d e^{3} +{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} +{\left (3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (a^{3} b^{5} d^{6} - 5 \, a^{4} b^{4} d^{5} e + 10 \, a^{5} b^{3} d^{4} e^{2} - 10 \, a^{6} b^{2} d^{3} e^{3} + 5 \, a^{7} b d^{2} e^{4} - a^{8} d e^{5} +{\left (b^{8} d^{5} e - 5 \, a b^{7} d^{4} e^{2} + 10 \, a^{2} b^{6} d^{3} e^{3} - 10 \, a^{3} b^{5} d^{2} e^{4} + 5 \, a^{4} b^{4} d e^{5} - a^{5} b^{3} e^{6}\right )} x^{4} +{\left (b^{8} d^{6} - 2 \, a b^{7} d^{5} e - 5 \, a^{2} b^{6} d^{4} e^{2} + 20 \, a^{3} b^{5} d^{3} e^{3} - 25 \, a^{4} b^{4} d^{2} e^{4} + 14 \, a^{5} b^{3} d e^{5} - 3 \, a^{6} b^{2} e^{6}\right )} x^{3} + 3 \,{\left (a b^{7} d^{6} - 4 \, a^{2} b^{6} d^{5} e + 5 \, a^{3} b^{5} d^{4} e^{2} - 5 \, a^{5} b^{3} d^{2} e^{4} + 4 \, a^{6} b^{2} d e^{5} - a^{7} b e^{6}\right )} x^{2} +{\left (3 \, a^{2} b^{6} d^{6} - 14 \, a^{3} b^{5} d^{5} e + 25 \, a^{4} b^{4} d^{4} e^{2} - 20 \, a^{5} b^{3} d^{3} e^{3} + 5 \, a^{6} b^{2} d^{2} e^{4} + 2 \, a^{7} b d e^{5} - a^{8} e^{6}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

-1/3*(b^4*d^4 - 6*a*b^3*d^3*e + 18*a^2*b^2*d^2*e^2 - 10*a^3*b*d*e^3 - 3*a^4*e^4 + 12*(b^4*d*e^3 - a*b^3*e^4)*x
^3 + 6*(b^4*d^2*e^2 + 4*a*b^3*d*e^3 - 5*a^2*b^2*e^4)*x^2 - 2*(b^4*d^3*e - 9*a*b^3*d^2*e^2 - 3*a^2*b^2*d*e^3 +
11*a^3*b*e^4)*x + 12*(b^4*e^4*x^4 + a^3*b*d*e^3 + (b^4*d*e^3 + 3*a*b^3*e^4)*x^3 + 3*(a*b^3*d*e^3 + a^2*b^2*e^4
)*x^2 + (3*a^2*b^2*d*e^3 + a^3*b*e^4)*x)*log(b*x + a) - 12*(b^4*e^4*x^4 + a^3*b*d*e^3 + (b^4*d*e^3 + 3*a*b^3*e
^4)*x^3 + 3*(a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + (3*a^2*b^2*d*e^3 + a^3*b*e^4)*x)*log(e*x + d))/(a^3*b^5*d^6 - 5*
a^4*b^4*d^5*e + 10*a^5*b^3*d^4*e^2 - 10*a^6*b^2*d^3*e^3 + 5*a^7*b*d^2*e^4 - a^8*d*e^5 + (b^8*d^5*e - 5*a*b^7*d
^4*e^2 + 10*a^2*b^6*d^3*e^3 - 10*a^3*b^5*d^2*e^4 + 5*a^4*b^4*d*e^5 - a^5*b^3*e^6)*x^4 + (b^8*d^6 - 2*a*b^7*d^5
*e - 5*a^2*b^6*d^4*e^2 + 20*a^3*b^5*d^3*e^3 - 25*a^4*b^4*d^2*e^4 + 14*a^5*b^3*d*e^5 - 3*a^6*b^2*e^6)*x^3 + 3*(
a*b^7*d^6 - 4*a^2*b^6*d^5*e + 5*a^3*b^5*d^4*e^2 - 5*a^5*b^3*d^2*e^4 + 4*a^6*b^2*d*e^5 - a^7*b*e^6)*x^2 + (3*a^
2*b^6*d^6 - 14*a^3*b^5*d^5*e + 25*a^4*b^4*d^4*e^2 - 20*a^5*b^3*d^3*e^3 + 5*a^6*b^2*d^2*e^4 + 2*a^7*b*d*e^5 - a
^8*e^6)*x)

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Sympy [B]  time = 3.59816, size = 881, normalized size = 6.67 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

-4*b*e**3*log(x + (-4*a**6*b*e**9/(a*e - b*d)**5 + 24*a**5*b**2*d*e**8/(a*e - b*d)**5 - 60*a**4*b**3*d**2*e**7
/(a*e - b*d)**5 + 80*a**3*b**4*d**3*e**6/(a*e - b*d)**5 - 60*a**2*b**5*d**4*e**5/(a*e - b*d)**5 + 24*a*b**6*d*
*5*e**4/(a*e - b*d)**5 + 4*a*b*e**4 - 4*b**7*d**6*e**3/(a*e - b*d)**5 + 4*b**2*d*e**3)/(8*b**2*e**4))/(a*e - b
*d)**5 + 4*b*e**3*log(x + (4*a**6*b*e**9/(a*e - b*d)**5 - 24*a**5*b**2*d*e**8/(a*e - b*d)**5 + 60*a**4*b**3*d*
*2*e**7/(a*e - b*d)**5 - 80*a**3*b**4*d**3*e**6/(a*e - b*d)**5 + 60*a**2*b**5*d**4*e**5/(a*e - b*d)**5 - 24*a*
b**6*d**5*e**4/(a*e - b*d)**5 + 4*a*b*e**4 + 4*b**7*d**6*e**3/(a*e - b*d)**5 + 4*b**2*d*e**3)/(8*b**2*e**4))/(
a*e - b*d)**5 - (3*a**3*e**3 + 13*a**2*b*d*e**2 - 5*a*b**2*d**2*e + b**3*d**3 + 12*b**3*e**3*x**3 + x**2*(30*a
*b**2*e**3 + 6*b**3*d*e**2) + x*(22*a**2*b*e**3 + 16*a*b**2*d*e**2 - 2*b**3*d**2*e))/(3*a**7*d*e**4 - 12*a**6*
b*d**2*e**3 + 18*a**5*b**2*d**3*e**2 - 12*a**4*b**3*d**4*e + 3*a**3*b**4*d**5 + x**4*(3*a**4*b**3*e**5 - 12*a*
*3*b**4*d*e**4 + 18*a**2*b**5*d**2*e**3 - 12*a*b**6*d**3*e**2 + 3*b**7*d**4*e) + x**3*(9*a**5*b**2*e**5 - 33*a
**4*b**3*d*e**4 + 42*a**3*b**4*d**2*e**3 - 18*a**2*b**5*d**3*e**2 - 3*a*b**6*d**4*e + 3*b**7*d**5) + x**2*(9*a
**6*b*e**5 - 27*a**5*b**2*d*e**4 + 18*a**4*b**3*d**2*e**3 + 18*a**3*b**4*d**3*e**2 - 27*a**2*b**5*d**4*e + 9*a
*b**6*d**5) + x*(3*a**7*e**5 - 3*a**6*b*d*e**4 - 18*a**5*b**2*d**2*e**3 + 42*a**4*b**3*d**3*e**2 - 33*a**3*b**
4*d**4*e + 9*a**2*b**5*d**5))

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Giac [B]  time = 1.23866, size = 377, normalized size = 2.86 \begin{align*} -\frac{4 \, b e^{4} \log \left ({\left | b - \frac{b d}{x e + d} + \frac{a e}{x e + d} \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac{e^{7}}{{\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )}{\left (x e + d\right )}} - \frac{13 \, b^{4} e^{3} - \frac{30 \,{\left (b^{4} d e^{4} - a b^{3} e^{5}\right )} e^{\left (-1\right )}}{x e + d} + \frac{18 \,{\left (b^{4} d^{2} e^{5} - 2 \, a b^{3} d e^{6} + a^{2} b^{2} e^{7}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}}{3 \,{\left (b d - a e\right )}^{5}{\left (b - \frac{b d}{x e + d} + \frac{a e}{x e + d}\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

-4*b*e^4*log(abs(b - b*d/(x*e + d) + a*e/(x*e + d)))/(b^5*d^5*e - 5*a*b^4*d^4*e^2 + 10*a^2*b^3*d^3*e^3 - 10*a^
3*b^2*d^2*e^4 + 5*a^4*b*d*e^5 - a^5*e^6) - e^7/((b^4*d^4*e^4 - 4*a*b^3*d^3*e^5 + 6*a^2*b^2*d^2*e^6 - 4*a^3*b*d
*e^7 + a^4*e^8)*(x*e + d)) - 1/3*(13*b^4*e^3 - 30*(b^4*d*e^4 - a*b^3*e^5)*e^(-1)/(x*e + d) + 18*(b^4*d^2*e^5 -
2*a*b^3*d*e^6 + a^2*b^2*e^7)*e^(-2)/(x*e + d)^2)/((b*d - a*e)^5*(b - b*d/(x*e + d) + a*e/(x*e + d))^3)