### 3.1530 $$\int \frac{(d+e x)^3}{(a^2+2 a b x+b^2 x^2)^3} \, dx$$

Optimal. Leaf size=58 $\frac{e (d+e x)^4}{20 (a+b x)^4 (b d-a e)^2}-\frac{(d+e x)^4}{5 (a+b x)^5 (b d-a e)}$

[Out]

-(d + e*x)^4/(5*(b*d - a*e)*(a + b*x)^5) + (e*(d + e*x)^4)/(20*(b*d - a*e)^2*(a + b*x)^4)

________________________________________________________________________________________

Rubi [A]  time = 0.0110566, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.115, Rules used = {27, 45, 37} $\frac{e (d+e x)^4}{20 (a+b x)^4 (b d-a e)^2}-\frac{(d+e x)^4}{5 (a+b x)^5 (b d-a e)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(d + e*x)^4/(5*(b*d - a*e)*(a + b*x)^5) + (e*(d + e*x)^4)/(20*(b*d - a*e)^2*(a + b*x)^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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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{(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^3}{(a+b x)^6} \, dx\\ &=-\frac{(d+e x)^4}{5 (b d-a e) (a+b x)^5}-\frac{e \int \frac{(d+e x)^3}{(a+b x)^5} \, dx}{5 (b d-a e)}\\ &=-\frac{(d+e x)^4}{5 (b d-a e) (a+b x)^5}+\frac{e (d+e x)^4}{20 (b d-a e)^2 (a+b x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0359294, size = 97, normalized size = 1.67 $-\frac{a^2 b e^2 (2 d+5 e x)+a^3 e^3+a b^2 e \left (3 d^2+10 d e x+10 e^2 x^2\right )+b^3 \left (15 d^2 e x+4 d^3+20 d e^2 x^2+10 e^3 x^3\right )}{20 b^4 (a+b x)^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(a^3*e^3 + a^2*b*e^2*(2*d + 5*e*x) + a*b^2*e*(3*d^2 + 10*d*e*x + 10*e^2*x^2) + b^3*(4*d^3 + 15*d^2*e*x + 20*d
*e^2*x^2 + 10*e^3*x^3))/(20*b^4*(a + b*x)^5)

________________________________________________________________________________________

Maple [B]  time = 0.046, size = 121, normalized size = 2.1 \begin{align*} -{\frac{{e}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{-{a}^{3}{e}^{3}+3\,{a}^{2}bd{e}^{2}-3\,a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3}}{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}+{\frac{{e}^{2} \left ( ae-bd \right ) }{{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{3\,e \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{4\,{b}^{4} \left ( bx+a \right ) ^{4}}} \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)^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.16924, size = 216, normalized size = 3.72 \begin{align*} -\frac{10 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 2 \, a^{2} b d e^{2} + a^{3} e^{3} + 10 \,{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (3 \, b^{3} d^{2} e + 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{20 \,{\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} 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)^3,x, algorithm="maxima")

[Out]

-1/20*(10*b^3*e^3*x^3 + 4*b^3*d^3 + 3*a*b^2*d^2*e + 2*a^2*b*d*e^2 + a^3*e^3 + 10*(2*b^3*d*e^2 + a*b^2*e^3)*x^2
+ 5*(3*b^3*d^2*e + 2*a*b^2*d*e^2 + a^2*b*e^3)*x)/(b^9*x^5 + 5*a*b^8*x^4 + 10*a^2*b^7*x^3 + 10*a^3*b^6*x^2 + 5
*a^4*b^5*x + a^5*b^4)

________________________________________________________________________________________

Fricas [B]  time = 1.80828, size = 328, normalized size = 5.66 \begin{align*} -\frac{10 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 2 \, a^{2} b d e^{2} + a^{3} e^{3} + 10 \,{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (3 \, b^{3} d^{2} e + 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{20 \,{\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} 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)^3,x, algorithm="fricas")

[Out]

-1/20*(10*b^3*e^3*x^3 + 4*b^3*d^3 + 3*a*b^2*d^2*e + 2*a^2*b*d*e^2 + a^3*e^3 + 10*(2*b^3*d*e^2 + a*b^2*e^3)*x^2
+ 5*(3*b^3*d^2*e + 2*a*b^2*d*e^2 + a^2*b*e^3)*x)/(b^9*x^5 + 5*a*b^8*x^4 + 10*a^2*b^7*x^3 + 10*a^3*b^6*x^2 + 5
*a^4*b^5*x + a^5*b^4)

________________________________________________________________________________________

Sympy [B]  time = 2.68908, size = 170, normalized size = 2.93 \begin{align*} - \frac{a^{3} e^{3} + 2 a^{2} b d e^{2} + 3 a b^{2} d^{2} e + 4 b^{3} d^{3} + 10 b^{3} e^{3} x^{3} + x^{2} \left (10 a b^{2} e^{3} + 20 b^{3} d e^{2}\right ) + x \left (5 a^{2} b e^{3} + 10 a b^{2} d e^{2} + 15 b^{3} d^{2} e\right )}{20 a^{5} b^{4} + 100 a^{4} b^{5} x + 200 a^{3} b^{6} x^{2} + 200 a^{2} b^{7} x^{3} + 100 a b^{8} x^{4} + 20 b^{9} x^{5}} \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)**3,x)

[Out]

-(a**3*e**3 + 2*a**2*b*d*e**2 + 3*a*b**2*d**2*e + 4*b**3*d**3 + 10*b**3*e**3*x**3 + x**2*(10*a*b**2*e**3 + 20*
b**3*d*e**2) + x*(5*a**2*b*e**3 + 10*a*b**2*d*e**2 + 15*b**3*d**2*e))/(20*a**5*b**4 + 100*a**4*b**5*x + 200*a*
*3*b**6*x**2 + 200*a**2*b**7*x**3 + 100*a*b**8*x**4 + 20*b**9*x**5)

________________________________________________________________________________________

Giac [B]  time = 1.1305, size = 147, normalized size = 2.53 \begin{align*} -\frac{10 \, b^{3} x^{3} e^{3} + 20 \, b^{3} d x^{2} e^{2} + 15 \, b^{3} d^{2} x e + 4 \, b^{3} d^{3} + 10 \, a b^{2} x^{2} e^{3} + 10 \, a b^{2} d x e^{2} + 3 \, a b^{2} d^{2} e + 5 \, a^{2} b x e^{3} + 2 \, a^{2} b d e^{2} + a^{3} e^{3}}{20 \,{\left (b x + a\right )}^{5} 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)^3,x, algorithm="giac")

[Out]

-1/20*(10*b^3*x^3*e^3 + 20*b^3*d*x^2*e^2 + 15*b^3*d^2*x*e + 4*b^3*d^3 + 10*a*b^2*x^2*e^3 + 10*a*b^2*d*x*e^2 +
3*a*b^2*d^2*e + 5*a^2*b*x*e^3 + 2*a^2*b*d*e^2 + a^3*e^3)/((b*x + a)^5*b^4)