### 3.1518 $$\int \frac{(d+e x)^3}{(a^2+2 a b x+b^2 x^2)^2} \, dx$$

Optimal. Leaf size=86 $-\frac{3 e^2 (b d-a e)}{b^4 (a+b x)}-\frac{3 e (b d-a e)^2}{2 b^4 (a+b x)^2}-\frac{(b d-a e)^3}{3 b^4 (a+b x)^3}+\frac{e^3 \log (a+b x)}{b^4}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0595502, antiderivative size = 86, 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, 43} $-\frac{3 e^2 (b d-a e)}{b^4 (a+b x)}-\frac{3 e (b d-a e)^2}{2 b^4 (a+b x)^2}-\frac{(b d-a e)^3}{3 b^4 (a+b x)^3}+\frac{e^3 \log (a+b x)}{b^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0414583, size = 80, normalized size = 0.93 $\frac{6 e^3 \log (a+b x)-\frac{(b d-a e) \left (11 a^2 e^2+a b e (5 d+27 e x)+b^2 \left (2 d^2+9 d e x+18 e^2 x^2\right )\right )}{(a+b x)^3}}{6 b^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-(((b*d - a*e)*(11*a^2*e^2 + a*b*e*(5*d + 27*e*x) + b^2*(2*d^2 + 9*d*e*x + 18*e^2*x^2)))/(a + b*x)^3) + 6*e^3
*Log[a + b*x])/(6*b^4)

________________________________________________________________________________________

Maple [B]  time = 0.046, size = 166, normalized size = 1.9 \begin{align*} -{\frac{3\,{a}^{2}{e}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}+3\,{\frac{ad{e}^{2}}{{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{d}^{2}e}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}+{\frac{{a}^{3}{e}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{d{e}^{2}{a}^{2}}{{b}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{{d}^{2}ea}{{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{{d}^{3}}{3\,b \left ( bx+a \right ) ^{3}}}+{\frac{{e}^{3}\ln \left ( bx+a \right ) }{{b}^{4}}}+3\,{\frac{a{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{d{e}^{2}}{{b}^{3} \left ( bx+a \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.20806, size = 192, normalized size = 2.23 \begin{align*} -\frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 18 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac{e^{3} \log \left (b x + a\right )}{b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*b^3*d^3 + 3*a*b^2*d^2*e + 6*a^2*b*d*e^2 - 11*a^3*e^3 + 18*(b^3*d*e^2 - a*b^2*e^3)*x^2 + 9*(b^3*d^2*e +
2*a*b^2*d*e^2 - 3*a^2*b*e^3)*x)/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4) + e^3*log(b*x + a)/b^4

________________________________________________________________________________________

Fricas [B]  time = 1.74703, size = 360, normalized size = 4.19 \begin{align*} -\frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 18 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x - 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (b x + a\right )}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*b^3*d^3 + 3*a*b^2*d^2*e + 6*a^2*b*d*e^2 - 11*a^3*e^3 + 18*(b^3*d*e^2 - a*b^2*e^3)*x^2 + 9*(b^3*d^2*e +
2*a*b^2*d*e^2 - 3*a^2*b*e^3)*x - 6*(b^3*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*log(b*x + a))/(b
^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4)

________________________________________________________________________________________

Sympy [A]  time = 1.52887, size = 148, normalized size = 1.72 \begin{align*} \frac{11 a^{3} e^{3} - 6 a^{2} b d e^{2} - 3 a b^{2} d^{2} e - 2 b^{3} d^{3} + x^{2} \left (18 a b^{2} e^{3} - 18 b^{3} d e^{2}\right ) + x \left (27 a^{2} b e^{3} - 18 a b^{2} d e^{2} - 9 b^{3} d^{2} e\right )}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac{e^{3} \log{\left (a + b x \right )}}{b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11*a**3*e**3 - 6*a**2*b*d*e**2 - 3*a*b**2*d**2*e - 2*b**3*d**3 + x**2*(18*a*b**2*e**3 - 18*b**3*d*e**2) + x*(
27*a**2*b*e**3 - 18*a*b**2*d*e**2 - 9*b**3*d**2*e))/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x*
*3) + e**3*log(a + b*x)/b**4

________________________________________________________________________________________

Giac [A]  time = 1.16156, size = 153, normalized size = 1.78 \begin{align*} \frac{e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac{18 \,{\left (b^{2} d e^{2} - a b e^{3}\right )} x^{2} + 9 \,{\left (b^{2} d^{2} e + 2 \, a b d e^{2} - 3 \, a^{2} e^{3}\right )} x + \frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3}}{b}}{6 \,{\left (b x + a\right )}^{3} b^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^3*log(abs(b*x + a))/b^4 - 1/6*(18*(b^2*d*e^2 - a*b*e^3)*x^2 + 9*(b^2*d^2*e + 2*a*b*d*e^2 - 3*a^2*e^3)*x + (2
*b^3*d^3 + 3*a*b^2*d^2*e + 6*a^2*b*d*e^2 - 11*a^3*e^3)/b)/((b*x + a)^3*b^3)