### 3.1477 $$\int \frac{(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^8} \, dx$$

Optimal. Leaf size=89 $\frac{b^2 (a+b x)^5}{105 (d+e x)^5 (b d-a e)^3}+\frac{b (a+b x)^5}{21 (d+e x)^6 (b d-a e)^2}+\frac{(a+b x)^5}{7 (d+e x)^7 (b d-a e)}$

[Out]

(a + b*x)^5/(7*(b*d - a*e)*(d + e*x)^7) + (b*(a + b*x)^5)/(21*(b*d - a*e)^2*(d + e*x)^6) + (b^2*(a + b*x)^5)/(
105*(b*d - a*e)^3*(d + e*x)^5)

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Rubi [A]  time = 0.0214929, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.115, Rules used = {27, 45, 37} $\frac{b^2 (a+b x)^5}{105 (d+e x)^5 (b d-a e)^3}+\frac{b (a+b x)^5}{21 (d+e x)^6 (b d-a e)^2}+\frac{(a+b x)^5}{7 (d+e x)^7 (b d-a e)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^8,x]

[Out]

(a + b*x)^5/(7*(b*d - a*e)*(d + e*x)^7) + (b*(a + b*x)^5)/(21*(b*d - a*e)^2*(d + e*x)^6) + (b^2*(a + b*x)^5)/(
105*(b*d - a*e)^3*(d + e*x)^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 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{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx &=\int \frac{(a+b x)^4}{(d+e x)^8} \, dx\\ &=\frac{(a+b x)^5}{7 (b d-a e) (d+e x)^7}+\frac{(2 b) \int \frac{(a+b x)^4}{(d+e x)^7} \, dx}{7 (b d-a e)}\\ &=\frac{(a+b x)^5}{7 (b d-a e) (d+e x)^7}+\frac{b (a+b x)^5}{21 (b d-a e)^2 (d+e x)^6}+\frac{b^2 \int \frac{(a+b x)^4}{(d+e x)^6} \, dx}{21 (b d-a e)^2}\\ &=\frac{(a+b x)^5}{7 (b d-a e) (d+e x)^7}+\frac{b (a+b x)^5}{21 (b d-a e)^2 (d+e x)^6}+\frac{b^2 (a+b x)^5}{105 (b d-a e)^3 (d+e x)^5}\\ \end{align*}

Mathematica [A]  time = 0.050105, size = 144, normalized size = 1.62 $-\frac{6 a^2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+10 a^3 b e^3 (d+7 e x)+15 a^4 e^4+3 a b^3 e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )}{105 e^5 (d+e x)^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^8,x]

[Out]

-(15*a^4*e^4 + 10*a^3*b*e^3*(d + 7*e*x) + 6*a^2*b^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*a*b^3*e*(d^3 + 7*d^2*
e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + b^4*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4))/(105*e^
5*(d + e*x)^7)

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Maple [B]  time = 0.046, size = 186, normalized size = 2.1 \begin{align*} -{\frac{{b}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{2\,b \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{3} \left ( ae-bd \right ) }{{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{{a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{b}^{2}{a}^{2}-4\,{d}^{3}ea{b}^{3}+{b}^{4}{d}^{4}}{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{6\,{b}^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^8,x)

[Out]

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

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Maxima [B]  time = 1.1831, size = 333, normalized size = 3.74 \begin{align*} -\frac{35 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 3 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4} + 35 \,{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 21 \,{\left (b^{4} d^{2} e^{2} + 3 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} + 7 \,{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 10 \, a^{3} b e^{4}\right )} x}{105 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\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)^2/(e*x+d)^8,x, algorithm="maxima")

[Out]

-1/105*(35*b^4*e^4*x^4 + b^4*d^4 + 3*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + 15*a^4*e^4 + 35*(b^4*d
*e^3 + 3*a*b^3*e^4)*x^3 + 21*(b^4*d^2*e^2 + 3*a*b^3*d*e^3 + 6*a^2*b^2*e^4)*x^2 + 7*(b^4*d^3*e + 3*a*b^3*d^2*e^
2 + 6*a^2*b^2*d*e^3 + 10*a^3*b*e^4)*x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^
8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

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Fricas [B]  time = 1.76871, size = 512, normalized size = 5.75 \begin{align*} -\frac{35 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 3 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4} + 35 \,{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 21 \,{\left (b^{4} d^{2} e^{2} + 3 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} + 7 \,{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 10 \, a^{3} b e^{4}\right )} x}{105 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\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)^2/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/105*(35*b^4*e^4*x^4 + b^4*d^4 + 3*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + 15*a^4*e^4 + 35*(b^4*d
*e^3 + 3*a*b^3*e^4)*x^3 + 21*(b^4*d^2*e^2 + 3*a*b^3*d*e^3 + 6*a^2*b^2*e^4)*x^2 + 7*(b^4*d^3*e + 3*a*b^3*d^2*e^
2 + 6*a^2*b^2*d*e^3 + 10*a^3*b*e^4)*x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^
8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

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Sympy [B]  time = 16.6001, size = 264, normalized size = 2.97 \begin{align*} - \frac{15 a^{4} e^{4} + 10 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} + 3 a b^{3} d^{3} e + b^{4} d^{4} + 35 b^{4} e^{4} x^{4} + x^{3} \left (105 a b^{3} e^{4} + 35 b^{4} d e^{3}\right ) + x^{2} \left (126 a^{2} b^{2} e^{4} + 63 a b^{3} d e^{3} + 21 b^{4} d^{2} e^{2}\right ) + x \left (70 a^{3} b e^{4} + 42 a^{2} b^{2} d e^{3} + 21 a b^{3} d^{2} e^{2} + 7 b^{4} d^{3} e\right )}{105 d^{7} e^{5} + 735 d^{6} e^{6} x + 2205 d^{5} e^{7} x^{2} + 3675 d^{4} e^{8} x^{3} + 3675 d^{3} e^{9} x^{4} + 2205 d^{2} e^{10} x^{5} + 735 d e^{11} x^{6} + 105 e^{12} x^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**8,x)

[Out]

-(15*a**4*e**4 + 10*a**3*b*d*e**3 + 6*a**2*b**2*d**2*e**2 + 3*a*b**3*d**3*e + b**4*d**4 + 35*b**4*e**4*x**4 +
x**3*(105*a*b**3*e**4 + 35*b**4*d*e**3) + x**2*(126*a**2*b**2*e**4 + 63*a*b**3*d*e**3 + 21*b**4*d**2*e**2) + x
*(70*a**3*b*e**4 + 42*a**2*b**2*d*e**3 + 21*a*b**3*d**2*e**2 + 7*b**4*d**3*e))/(105*d**7*e**5 + 735*d**6*e**6*
x + 2205*d**5*e**7*x**2 + 3675*d**4*e**8*x**3 + 3675*d**3*e**9*x**4 + 2205*d**2*e**10*x**5 + 735*d*e**11*x**6
+ 105*e**12*x**7)

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Giac [B]  time = 1.10957, size = 235, normalized size = 2.64 \begin{align*} -\frac{{\left (35 \, b^{4} x^{4} e^{4} + 35 \, b^{4} d x^{3} e^{3} + 21 \, b^{4} d^{2} x^{2} e^{2} + 7 \, b^{4} d^{3} x e + b^{4} d^{4} + 105 \, a b^{3} x^{3} e^{4} + 63 \, a b^{3} d x^{2} e^{3} + 21 \, a b^{3} d^{2} x e^{2} + 3 \, a b^{3} d^{3} e + 126 \, a^{2} b^{2} x^{2} e^{4} + 42 \, a^{2} b^{2} d x e^{3} + 6 \, a^{2} b^{2} d^{2} e^{2} + 70 \, a^{3} b x e^{4} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{105 \,{\left (x e + d\right )}^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/105*(35*b^4*x^4*e^4 + 35*b^4*d*x^3*e^3 + 21*b^4*d^2*x^2*e^2 + 7*b^4*d^3*x*e + b^4*d^4 + 105*a*b^3*x^3*e^4 +
63*a*b^3*d*x^2*e^3 + 21*a*b^3*d^2*x*e^2 + 3*a*b^3*d^3*e + 126*a^2*b^2*x^2*e^4 + 42*a^2*b^2*d*x*e^3 + 6*a^2*b^
2*d^2*e^2 + 70*a^3*b*x*e^4 + 10*a^3*b*d*e^3 + 15*a^4*e^4)*e^(-5)/(x*e + d)^7