### 3.1503 $$\int \frac{(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{14}} \, dx$$

Optimal. Leaf size=171 $\frac{3 b^5 (b d-a e)}{4 e^7 (d+e x)^8}-\frac{5 b^4 (b d-a e)^2}{3 e^7 (d+e x)^9}+\frac{2 b^3 (b d-a e)^3}{e^7 (d+e x)^{10}}-\frac{15 b^2 (b d-a e)^4}{11 e^7 (d+e x)^{11}}+\frac{b (b d-a e)^5}{2 e^7 (d+e x)^{12}}-\frac{(b d-a e)^6}{13 e^7 (d+e x)^{13}}-\frac{b^6}{7 e^7 (d+e x)^7}$

[Out]

-(b*d - a*e)^6/(13*e^7*(d + e*x)^13) + (b*(b*d - a*e)^5)/(2*e^7*(d + e*x)^12) - (15*b^2*(b*d - a*e)^4)/(11*e^7
*(d + e*x)^11) + (2*b^3*(b*d - a*e)^3)/(e^7*(d + e*x)^10) - (5*b^4*(b*d - a*e)^2)/(3*e^7*(d + e*x)^9) + (3*b^5
*(b*d - a*e))/(4*e^7*(d + e*x)^8) - b^6/(7*e^7*(d + e*x)^7)

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Rubi [A]  time = 0.13074, antiderivative size = 171, 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} $\frac{3 b^5 (b d-a e)}{4 e^7 (d+e x)^8}-\frac{5 b^4 (b d-a e)^2}{3 e^7 (d+e x)^9}+\frac{2 b^3 (b d-a e)^3}{e^7 (d+e x)^{10}}-\frac{15 b^2 (b d-a e)^4}{11 e^7 (d+e x)^{11}}+\frac{b (b d-a e)^5}{2 e^7 (d+e x)^{12}}-\frac{(b d-a e)^6}{13 e^7 (d+e x)^{13}}-\frac{b^6}{7 e^7 (d+e x)^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^14,x]

[Out]

-(b*d - a*e)^6/(13*e^7*(d + e*x)^13) + (b*(b*d - a*e)^5)/(2*e^7*(d + e*x)^12) - (15*b^2*(b*d - a*e)^4)/(11*e^7
*(d + e*x)^11) + (2*b^3*(b*d - a*e)^3)/(e^7*(d + e*x)^10) - (5*b^4*(b*d - a*e)^2)/(3*e^7*(d + e*x)^9) + (3*b^5
*(b*d - a*e))/(4*e^7*(d + e*x)^8) - b^6/(7*e^7*(d + e*x)^7)

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 )^3}{(d+e x)^{14}} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^{14}} \, dx\\ &=\int \left (\frac{(-b d+a e)^6}{e^6 (d+e x)^{14}}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^{13}}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^{12}}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^{11}}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)^{10}}-\frac{6 b^5 (b d-a e)}{e^6 (d+e x)^9}+\frac{b^6}{e^6 (d+e x)^8}\right ) \, dx\\ &=-\frac{(b d-a e)^6}{13 e^7 (d+e x)^{13}}+\frac{b (b d-a e)^5}{2 e^7 (d+e x)^{12}}-\frac{15 b^2 (b d-a e)^4}{11 e^7 (d+e x)^{11}}+\frac{2 b^3 (b d-a e)^3}{e^7 (d+e x)^{10}}-\frac{5 b^4 (b d-a e)^2}{3 e^7 (d+e x)^9}+\frac{3 b^5 (b d-a e)}{4 e^7 (d+e x)^8}-\frac{b^6}{7 e^7 (d+e x)^7}\\ \end{align*}

Mathematica [A]  time = 0.0982966, size = 277, normalized size = 1.62 $-\frac{28 a^2 b^4 e^2 \left (78 d^2 e^2 x^2+13 d^3 e x+d^4+286 d e^3 x^3+715 e^4 x^4\right )+84 a^3 b^3 e^3 \left (13 d^2 e x+d^3+78 d e^2 x^2+286 e^3 x^3\right )+210 a^4 b^2 e^4 \left (d^2+13 d e x+78 e^2 x^2\right )+462 a^5 b e^5 (d+13 e x)+924 a^6 e^6+7 a b^5 e \left (78 d^3 e^2 x^2+286 d^2 e^3 x^3+13 d^4 e x+d^5+715 d e^4 x^4+1287 e^5 x^5\right )+b^6 \left (78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+13 d^5 e x+d^6+1287 d e^5 x^5+1716 e^6 x^6\right )}{12012 e^7 (d+e x)^{13}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^14,x]

[Out]

-(924*a^6*e^6 + 462*a^5*b*e^5*(d + 13*e*x) + 210*a^4*b^2*e^4*(d^2 + 13*d*e*x + 78*e^2*x^2) + 84*a^3*b^3*e^3*(d
^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*e^3*x^3) + 28*a^2*b^4*e^2*(d^4 + 13*d^3*e*x + 78*d^2*e^2*x^2 + 286*d*e^3*
x^3 + 715*e^4*x^4) + 7*a*b^5*e*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 + 1287*e^5
*x^5) + b^6*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d*e^5*x^5 + 1716*e^6
*x^6))/(12012*e^7*(d + e*x)^13)

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Maple [B]  time = 0.048, size = 357, normalized size = 2.1 \begin{align*} -2\,{\frac{{b}^{3} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{{e}^{7} \left ( ex+d \right ) ^{10}}}-{\frac{{b}^{6}}{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{b \left ({a}^{5}{e}^{5}-5\,{a}^{4}bd{e}^{4}+10\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-10\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+5\,a{b}^{4}{d}^{4}e-{b}^{5}{d}^{5} \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{12}}}-{\frac{5\,{b}^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{3\,{e}^{7} \left ( ex+d \right ) ^{9}}}-{\frac{15\,{b}^{2} \left ({a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{b}^{2}{a}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) }{11\,{e}^{7} \left ( ex+d \right ) ^{11}}}-{\frac{{e}^{6}{a}^{6}-6\,{a}^{5}bd{e}^{5}+15\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-20\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+15\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-6\,a{b}^{5}{d}^{5}e+{d}^{6}{b}^{6}}{13\,{e}^{7} \left ( ex+d \right ) ^{13}}}-{\frac{3\,{b}^{5} \left ( ae-bd \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{8}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^14,x)

[Out]

-2*b^3*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/e^7/(e*x+d)^10-1/7*b^6/e^7/(e*x+d)^7-1/2*b*(a^5*e^5-5*a^4
*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)/e^7/(e*x+d)^12-5/3*b^4*(a^2*e^2-2*a*b*d*
e+b^2*d^2)/e^7/(e*x+d)^9-15/11*b^2*(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^7/(e*x+d)
^11-1/13*(a^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d^5*e+b^6*d^6
)/e^7/(e*x+d)^13-3/4*b^5*(a*e-b*d)/e^7/(e*x+d)^8

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Maxima [B]  time = 1.14424, size = 655, normalized size = 3.83 \begin{align*} -\frac{1716 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 7 \, a b^{5} d^{5} e + 28 \, a^{2} b^{4} d^{4} e^{2} + 84 \, a^{3} b^{3} d^{3} e^{3} + 210 \, a^{4} b^{2} d^{2} e^{4} + 462 \, a^{5} b d e^{5} + 924 \, a^{6} e^{6} + 1287 \,{\left (b^{6} d e^{5} + 7 \, a b^{5} e^{6}\right )} x^{5} + 715 \,{\left (b^{6} d^{2} e^{4} + 7 \, a b^{5} d e^{5} + 28 \, a^{2} b^{4} e^{6}\right )} x^{4} + 286 \,{\left (b^{6} d^{3} e^{3} + 7 \, a b^{5} d^{2} e^{4} + 28 \, a^{2} b^{4} d e^{5} + 84 \, a^{3} b^{3} e^{6}\right )} x^{3} + 78 \,{\left (b^{6} d^{4} e^{2} + 7 \, a b^{5} d^{3} e^{3} + 28 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} + 210 \, a^{4} b^{2} e^{6}\right )} x^{2} + 13 \,{\left (b^{6} d^{5} e + 7 \, a b^{5} d^{4} e^{2} + 28 \, a^{2} b^{4} d^{3} e^{3} + 84 \, a^{3} b^{3} d^{2} e^{4} + 210 \, a^{4} b^{2} d e^{5} + 462 \, a^{5} b e^{6}\right )} x}{12012 \,{\left (e^{20} x^{13} + 13 \, d e^{19} x^{12} + 78 \, d^{2} e^{18} x^{11} + 286 \, d^{3} e^{17} x^{10} + 715 \, d^{4} e^{16} x^{9} + 1287 \, d^{5} e^{15} x^{8} + 1716 \, d^{6} e^{14} x^{7} + 1716 \, d^{7} e^{13} x^{6} + 1287 \, d^{8} e^{12} x^{5} + 715 \, d^{9} e^{11} x^{4} + 286 \, d^{10} e^{10} x^{3} + 78 \, d^{11} e^{9} x^{2} + 13 \, d^{12} e^{8} x + d^{13} e^{7}\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)^3/(e*x+d)^14,x, algorithm="maxima")

[Out]

-1/12012*(1716*b^6*e^6*x^6 + b^6*d^6 + 7*a*b^5*d^5*e + 28*a^2*b^4*d^4*e^2 + 84*a^3*b^3*d^3*e^3 + 210*a^4*b^2*d
^2*e^4 + 462*a^5*b*d*e^5 + 924*a^6*e^6 + 1287*(b^6*d*e^5 + 7*a*b^5*e^6)*x^5 + 715*(b^6*d^2*e^4 + 7*a*b^5*d*e^5
+ 28*a^2*b^4*e^6)*x^4 + 286*(b^6*d^3*e^3 + 7*a*b^5*d^2*e^4 + 28*a^2*b^4*d*e^5 + 84*a^3*b^3*e^6)*x^3 + 78*(b^6
*d^4*e^2 + 7*a*b^5*d^3*e^3 + 28*a^2*b^4*d^2*e^4 + 84*a^3*b^3*d*e^5 + 210*a^4*b^2*e^6)*x^2 + 13*(b^6*d^5*e + 7*
a*b^5*d^4*e^2 + 28*a^2*b^4*d^3*e^3 + 84*a^3*b^3*d^2*e^4 + 210*a^4*b^2*d*e^5 + 462*a^5*b*e^6)*x)/(e^20*x^13 + 1
3*d*e^19*x^12 + 78*d^2*e^18*x^11 + 286*d^3*e^17*x^10 + 715*d^4*e^16*x^9 + 1287*d^5*e^15*x^8 + 1716*d^6*e^14*x^
7 + 1716*d^7*e^13*x^6 + 1287*d^8*e^12*x^5 + 715*d^9*e^11*x^4 + 286*d^10*e^10*x^3 + 78*d^11*e^9*x^2 + 13*d^12*e
^8*x + d^13*e^7)

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Fricas [B]  time = 1.75834, size = 1064, normalized size = 6.22 \begin{align*} -\frac{1716 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 7 \, a b^{5} d^{5} e + 28 \, a^{2} b^{4} d^{4} e^{2} + 84 \, a^{3} b^{3} d^{3} e^{3} + 210 \, a^{4} b^{2} d^{2} e^{4} + 462 \, a^{5} b d e^{5} + 924 \, a^{6} e^{6} + 1287 \,{\left (b^{6} d e^{5} + 7 \, a b^{5} e^{6}\right )} x^{5} + 715 \,{\left (b^{6} d^{2} e^{4} + 7 \, a b^{5} d e^{5} + 28 \, a^{2} b^{4} e^{6}\right )} x^{4} + 286 \,{\left (b^{6} d^{3} e^{3} + 7 \, a b^{5} d^{2} e^{4} + 28 \, a^{2} b^{4} d e^{5} + 84 \, a^{3} b^{3} e^{6}\right )} x^{3} + 78 \,{\left (b^{6} d^{4} e^{2} + 7 \, a b^{5} d^{3} e^{3} + 28 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} + 210 \, a^{4} b^{2} e^{6}\right )} x^{2} + 13 \,{\left (b^{6} d^{5} e + 7 \, a b^{5} d^{4} e^{2} + 28 \, a^{2} b^{4} d^{3} e^{3} + 84 \, a^{3} b^{3} d^{2} e^{4} + 210 \, a^{4} b^{2} d e^{5} + 462 \, a^{5} b e^{6}\right )} x}{12012 \,{\left (e^{20} x^{13} + 13 \, d e^{19} x^{12} + 78 \, d^{2} e^{18} x^{11} + 286 \, d^{3} e^{17} x^{10} + 715 \, d^{4} e^{16} x^{9} + 1287 \, d^{5} e^{15} x^{8} + 1716 \, d^{6} e^{14} x^{7} + 1716 \, d^{7} e^{13} x^{6} + 1287 \, d^{8} e^{12} x^{5} + 715 \, d^{9} e^{11} x^{4} + 286 \, d^{10} e^{10} x^{3} + 78 \, d^{11} e^{9} x^{2} + 13 \, d^{12} e^{8} x + d^{13} e^{7}\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)^3/(e*x+d)^14,x, algorithm="fricas")

[Out]

-1/12012*(1716*b^6*e^6*x^6 + b^6*d^6 + 7*a*b^5*d^5*e + 28*a^2*b^4*d^4*e^2 + 84*a^3*b^3*d^3*e^3 + 210*a^4*b^2*d
^2*e^4 + 462*a^5*b*d*e^5 + 924*a^6*e^6 + 1287*(b^6*d*e^5 + 7*a*b^5*e^6)*x^5 + 715*(b^6*d^2*e^4 + 7*a*b^5*d*e^5
+ 28*a^2*b^4*e^6)*x^4 + 286*(b^6*d^3*e^3 + 7*a*b^5*d^2*e^4 + 28*a^2*b^4*d*e^5 + 84*a^3*b^3*e^6)*x^3 + 78*(b^6
*d^4*e^2 + 7*a*b^5*d^3*e^3 + 28*a^2*b^4*d^2*e^4 + 84*a^3*b^3*d*e^5 + 210*a^4*b^2*e^6)*x^2 + 13*(b^6*d^5*e + 7*
a*b^5*d^4*e^2 + 28*a^2*b^4*d^3*e^3 + 84*a^3*b^3*d^2*e^4 + 210*a^4*b^2*d*e^5 + 462*a^5*b*e^6)*x)/(e^20*x^13 + 1
3*d*e^19*x^12 + 78*d^2*e^18*x^11 + 286*d^3*e^17*x^10 + 715*d^4*e^16*x^9 + 1287*d^5*e^15*x^8 + 1716*d^6*e^14*x^
7 + 1716*d^7*e^13*x^6 + 1287*d^8*e^12*x^5 + 715*d^9*e^11*x^4 + 286*d^10*e^10*x^3 + 78*d^11*e^9*x^2 + 13*d^12*e
^8*x + d^13*e^7)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**14,x)

[Out]

Timed out

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Giac [B]  time = 1.13461, size = 475, normalized size = 2.78 \begin{align*} -\frac{{\left (1716 \, b^{6} x^{6} e^{6} + 1287 \, b^{6} d x^{5} e^{5} + 715 \, b^{6} d^{2} x^{4} e^{4} + 286 \, b^{6} d^{3} x^{3} e^{3} + 78 \, b^{6} d^{4} x^{2} e^{2} + 13 \, b^{6} d^{5} x e + b^{6} d^{6} + 9009 \, a b^{5} x^{5} e^{6} + 5005 \, a b^{5} d x^{4} e^{5} + 2002 \, a b^{5} d^{2} x^{3} e^{4} + 546 \, a b^{5} d^{3} x^{2} e^{3} + 91 \, a b^{5} d^{4} x e^{2} + 7 \, a b^{5} d^{5} e + 20020 \, a^{2} b^{4} x^{4} e^{6} + 8008 \, a^{2} b^{4} d x^{3} e^{5} + 2184 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 364 \, a^{2} b^{4} d^{3} x e^{3} + 28 \, a^{2} b^{4} d^{4} e^{2} + 24024 \, a^{3} b^{3} x^{3} e^{6} + 6552 \, a^{3} b^{3} d x^{2} e^{5} + 1092 \, a^{3} b^{3} d^{2} x e^{4} + 84 \, a^{3} b^{3} d^{3} e^{3} + 16380 \, a^{4} b^{2} x^{2} e^{6} + 2730 \, a^{4} b^{2} d x e^{5} + 210 \, a^{4} b^{2} d^{2} e^{4} + 6006 \, a^{5} b x e^{6} + 462 \, a^{5} b d e^{5} + 924 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{12012 \,{\left (x e + d\right )}^{13}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^14,x, algorithm="giac")

[Out]

-1/12012*(1716*b^6*x^6*e^6 + 1287*b^6*d*x^5*e^5 + 715*b^6*d^2*x^4*e^4 + 286*b^6*d^3*x^3*e^3 + 78*b^6*d^4*x^2*e
^2 + 13*b^6*d^5*x*e + b^6*d^6 + 9009*a*b^5*x^5*e^6 + 5005*a*b^5*d*x^4*e^5 + 2002*a*b^5*d^2*x^3*e^4 + 546*a*b^5
*d^3*x^2*e^3 + 91*a*b^5*d^4*x*e^2 + 7*a*b^5*d^5*e + 20020*a^2*b^4*x^4*e^6 + 8008*a^2*b^4*d*x^3*e^5 + 2184*a^2*
b^4*d^2*x^2*e^4 + 364*a^2*b^4*d^3*x*e^3 + 28*a^2*b^4*d^4*e^2 + 24024*a^3*b^3*x^3*e^6 + 6552*a^3*b^3*d*x^2*e^5
+ 1092*a^3*b^3*d^2*x*e^4 + 84*a^3*b^3*d^3*e^3 + 16380*a^4*b^2*x^2*e^6 + 2730*a^4*b^2*d*x*e^5 + 210*a^4*b^2*d^2
*e^4 + 6006*a^5*b*x*e^6 + 462*a^5*b*d*e^5 + 924*a^6*e^6)*e^(-7)/(x*e + d)^13