### 3.1479 $$\int \frac{(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{10}} \, dx$$

Optimal. Leaf size=119 $\frac{2 b^3 (b d-a e)}{3 e^5 (d+e x)^6}-\frac{6 b^2 (b d-a e)^2}{7 e^5 (d+e x)^7}+\frac{b (b d-a e)^3}{2 e^5 (d+e x)^8}-\frac{(b d-a e)^4}{9 e^5 (d+e x)^9}-\frac{b^4}{5 e^5 (d+e x)^5}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06821, antiderivative size = 119, 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} $\frac{2 b^3 (b d-a e)}{3 e^5 (d+e x)^6}-\frac{6 b^2 (b d-a e)^2}{7 e^5 (d+e x)^7}+\frac{b (b d-a e)^3}{2 e^5 (d+e x)^8}-\frac{(b d-a e)^4}{9 e^5 (d+e x)^9}-\frac{b^4}{5 e^5 (d+e x)^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^10,x]

[Out]

-(b*d - a*e)^4/(9*e^5*(d + e*x)^9) + (b*(b*d - a*e)^3)/(2*e^5*(d + e*x)^8) - (6*b^2*(b*d - a*e)^2)/(7*e^5*(d +
e*x)^7) + (2*b^3*(b*d - a*e))/(3*e^5*(d + e*x)^6) - b^4/(5*e^5*(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 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{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx &=\int \frac{(a+b x)^4}{(d+e x)^{10}} \, dx\\ &=\int \left (\frac{(-b d+a e)^4}{e^4 (d+e x)^{10}}-\frac{4 b (b d-a e)^3}{e^4 (d+e x)^9}+\frac{6 b^2 (b d-a e)^2}{e^4 (d+e x)^8}-\frac{4 b^3 (b d-a e)}{e^4 (d+e x)^7}+\frac{b^4}{e^4 (d+e x)^6}\right ) \, dx\\ &=-\frac{(b d-a e)^4}{9 e^5 (d+e x)^9}+\frac{b (b d-a e)^3}{2 e^5 (d+e x)^8}-\frac{6 b^2 (b d-a e)^2}{7 e^5 (d+e x)^7}+\frac{2 b^3 (b d-a e)}{3 e^5 (d+e x)^6}-\frac{b^4}{5 e^5 (d+e x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0512886, size = 144, normalized size = 1.21 $-\frac{15 a^2 b^2 e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+35 a^3 b e^3 (d+9 e x)+70 a^4 e^4+5 a b^3 e \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )+b^4 \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )}{630 e^5 (d+e x)^9}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^10,x]

[Out]

-(70*a^4*e^4 + 35*a^3*b*e^3*(d + 9*e*x) + 15*a^2*b^2*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + 5*a*b^3*e*(d^3 + 9*d^2
*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + b^4*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4))/(630*
e^5*(d + e*x)^9)

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 186, normalized size = 1.6 \begin{align*} -{\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}}{9\,{e}^{5} \left ( ex+d \right ) ^{9}}}-{\frac{2\,{b}^{3} \left ( ae-bd \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{b \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{2\,{e}^{5} \left ( ex+d \right ) ^{8}}}-{\frac{6\,{b}^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{{b}^{4}}{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)^10,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.14001, size = 363, normalized size = 3.05 \begin{align*} -\frac{126 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 5 \, a b^{3} d^{3} e + 15 \, a^{2} b^{2} d^{2} e^{2} + 35 \, a^{3} b d e^{3} + 70 \, a^{4} e^{4} + 84 \,{\left (b^{4} d e^{3} + 5 \, a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (b^{4} d^{2} e^{2} + 5 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 9 \,{\left (b^{4} d^{3} e + 5 \, a b^{3} d^{2} e^{2} + 15 \, a^{2} b^{2} d e^{3} + 35 \, a^{3} b e^{4}\right )} x}{630 \,{\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} 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)^10,x, algorithm="maxima")

[Out]

-1/630*(126*b^4*e^4*x^4 + b^4*d^4 + 5*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 + 35*a^3*b*d*e^3 + 70*a^4*e^4 + 84*(b^4
*d*e^3 + 5*a*b^3*e^4)*x^3 + 36*(b^4*d^2*e^2 + 5*a*b^3*d*e^3 + 15*a^2*b^2*e^4)*x^2 + 9*(b^4*d^3*e + 5*a*b^3*d^2
*e^2 + 15*a^2*b^2*d*e^3 + 35*a^3*b*e^4)*x)/(e^14*x^9 + 9*d*e^13*x^8 + 36*d^2*e^12*x^7 + 84*d^3*e^11*x^6 + 126*
d^4*e^10*x^5 + 126*d^5*e^9*x^4 + 84*d^6*e^8*x^3 + 36*d^7*e^7*x^2 + 9*d^8*e^6*x + d^9*e^5)

________________________________________________________________________________________

Fricas [B]  time = 1.69008, size = 568, normalized size = 4.77 \begin{align*} -\frac{126 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 5 \, a b^{3} d^{3} e + 15 \, a^{2} b^{2} d^{2} e^{2} + 35 \, a^{3} b d e^{3} + 70 \, a^{4} e^{4} + 84 \,{\left (b^{4} d e^{3} + 5 \, a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (b^{4} d^{2} e^{2} + 5 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 9 \,{\left (b^{4} d^{3} e + 5 \, a b^{3} d^{2} e^{2} + 15 \, a^{2} b^{2} d e^{3} + 35 \, a^{3} b e^{4}\right )} x}{630 \,{\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} 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)^10,x, algorithm="fricas")

[Out]

-1/630*(126*b^4*e^4*x^4 + b^4*d^4 + 5*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 + 35*a^3*b*d*e^3 + 70*a^4*e^4 + 84*(b^4
*d*e^3 + 5*a*b^3*e^4)*x^3 + 36*(b^4*d^2*e^2 + 5*a*b^3*d*e^3 + 15*a^2*b^2*e^4)*x^2 + 9*(b^4*d^3*e + 5*a*b^3*d^2
*e^2 + 15*a^2*b^2*d*e^3 + 35*a^3*b*e^4)*x)/(e^14*x^9 + 9*d*e^13*x^8 + 36*d^2*e^12*x^7 + 84*d^3*e^11*x^6 + 126*
d^4*e^10*x^5 + 126*d^5*e^9*x^4 + 84*d^6*e^8*x^3 + 36*d^7*e^7*x^2 + 9*d^8*e^6*x + d^9*e^5)

________________________________________________________________________________________

Sympy [B]  time = 44.3185, size = 287, normalized size = 2.41 \begin{align*} - \frac{70 a^{4} e^{4} + 35 a^{3} b d e^{3} + 15 a^{2} b^{2} d^{2} e^{2} + 5 a b^{3} d^{3} e + b^{4} d^{4} + 126 b^{4} e^{4} x^{4} + x^{3} \left (420 a b^{3} e^{4} + 84 b^{4} d e^{3}\right ) + x^{2} \left (540 a^{2} b^{2} e^{4} + 180 a b^{3} d e^{3} + 36 b^{4} d^{2} e^{2}\right ) + x \left (315 a^{3} b e^{4} + 135 a^{2} b^{2} d e^{3} + 45 a b^{3} d^{2} e^{2} + 9 b^{4} d^{3} e\right )}{630 d^{9} e^{5} + 5670 d^{8} e^{6} x + 22680 d^{7} e^{7} x^{2} + 52920 d^{6} e^{8} x^{3} + 79380 d^{5} e^{9} x^{4} + 79380 d^{4} e^{10} x^{5} + 52920 d^{3} e^{11} x^{6} + 22680 d^{2} e^{12} x^{7} + 5670 d e^{13} x^{8} + 630 e^{14} x^{9}} \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)**10,x)

[Out]

-(70*a**4*e**4 + 35*a**3*b*d*e**3 + 15*a**2*b**2*d**2*e**2 + 5*a*b**3*d**3*e + b**4*d**4 + 126*b**4*e**4*x**4
+ x**3*(420*a*b**3*e**4 + 84*b**4*d*e**3) + x**2*(540*a**2*b**2*e**4 + 180*a*b**3*d*e**3 + 36*b**4*d**2*e**2)
+ x*(315*a**3*b*e**4 + 135*a**2*b**2*d*e**3 + 45*a*b**3*d**2*e**2 + 9*b**4*d**3*e))/(630*d**9*e**5 + 5670*d**8
*e**6*x + 22680*d**7*e**7*x**2 + 52920*d**6*e**8*x**3 + 79380*d**5*e**9*x**4 + 79380*d**4*e**10*x**5 + 52920*d
**3*e**11*x**6 + 22680*d**2*e**12*x**7 + 5670*d*e**13*x**8 + 630*e**14*x**9)

________________________________________________________________________________________

Giac [A]  time = 1.19851, size = 235, normalized size = 1.97 \begin{align*} -\frac{{\left (126 \, b^{4} x^{4} e^{4} + 84 \, b^{4} d x^{3} e^{3} + 36 \, b^{4} d^{2} x^{2} e^{2} + 9 \, b^{4} d^{3} x e + b^{4} d^{4} + 420 \, a b^{3} x^{3} e^{4} + 180 \, a b^{3} d x^{2} e^{3} + 45 \, a b^{3} d^{2} x e^{2} + 5 \, a b^{3} d^{3} e + 540 \, a^{2} b^{2} x^{2} e^{4} + 135 \, a^{2} b^{2} d x e^{3} + 15 \, a^{2} b^{2} d^{2} e^{2} + 315 \, a^{3} b x e^{4} + 35 \, a^{3} b d e^{3} + 70 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{630 \,{\left (x e + d\right )}^{9}} \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)^10,x, algorithm="giac")

[Out]

-1/630*(126*b^4*x^4*e^4 + 84*b^4*d*x^3*e^3 + 36*b^4*d^2*x^2*e^2 + 9*b^4*d^3*x*e + b^4*d^4 + 420*a*b^3*x^3*e^4
+ 180*a*b^3*d*x^2*e^3 + 45*a*b^3*d^2*x*e^2 + 5*a*b^3*d^3*e + 540*a^2*b^2*x^2*e^4 + 135*a^2*b^2*d*x*e^3 + 15*a^
2*b^2*d^2*e^2 + 315*a^3*b*x*e^4 + 35*a^3*b*d*e^3 + 70*a^4*e^4)*e^(-5)/(x*e + d)^9