\(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{14}} \, dx\) [1503]

Optimal result

Integrand size = 26, antiderivative size = 171 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{14}} \, dx=-\frac {(b d-a e)^6}{13 e^7 (d+e x)^{13}}+\frac {b (b d-a e)^5}{2 e^7 (d+e x)^{12}}-\frac {15 b^2 (b d-a e)^4}{11 e^7 (d+e x)^{11}}+\frac {2 b^3 (b d-a e)^3}{e^7 (d+e x)^{10}}-\frac {5 b^4 (b d-a e)^2}{3 e^7 (d+e x)^9}+\frac {3 b^5 (b d-a e)}{4 e^7 (d+e x)^8}-\frac {b^6}{7 e^7 (d+e x)^7} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{14}} \, dx=\frac {3 b^5 (b d-a e)}{4 e^7 (d+e x)^8}-\frac {5 b^4 (b d-a e)^2}{3 e^7 (d+e x)^9}+\frac {2 b^3 (b d-a e)^3}{e^7 (d+e x)^{10}}-\frac {15 b^2 (b d-a e)^4}{11 e^7 (d+e x)^{11}}+\frac {b (b d-a e)^5}{2 e^7 (d+e x)^{12}}-\frac {(b d-a e)^6}{13 e^7 (d+e x)^{13}}-\frac {b^6}{7 e^7 (d+e x)^7} \]

[In]

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

[Out]

-1/13*(b*d - a*e)^6/(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 45

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*} \text {integral}& = \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] (verified)

Time = 0.07 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{14}} \, dx=-\frac {924 a^6 e^6+462 a^5 b e^5 (d+13 e x)+210 a^4 b^2 e^4 \left (d^2+13 d e x+78 e^2 x^2\right )+84 a^3 b^3 e^3 \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )+28 a^2 b^4 e^2 \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )+7 a b^5 e \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )+b^6 \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )}{12012 e^7 (d+e x)^{13}} \]

[In]

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

[Out]

-1/12012*(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 + 28
6*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))/(e^7*(d + e*x)^13)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(334\) vs. \(2(159)=318\).

Time = 2.30 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.96

method	result	size
risch	\(\frac {-\frac {b^{6} x^{6}}{7 e}-\frac {3 b^{5} \left (7 a e +b d \right ) x^{5}}{28 e^{2}}-\frac {5 b^{4} \left (28 a^{2} e^{2}+7 a b d e +b^{2} d^{2}\right ) x^{4}}{84 e^{3}}-\frac {b^{3} \left (84 a^{3} e^{3}+28 a^{2} b d \,e^{2}+7 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{42 e^{4}}-\frac {b^{2} \left (210 e^{4} a^{4}+84 b \,e^{3} d \,a^{3}+28 b^{2} e^{2} d^{2} a^{2}+7 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{154 e^{5}}-\frac {b \left (462 a^{5} e^{5}+210 a^{4} b d \,e^{4}+84 a^{3} b^{2} d^{2} e^{3}+28 a^{2} b^{3} d^{3} e^{2}+7 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{924 e^{6}}-\frac {924 a^{6} e^{6}+462 a^{5} b d \,e^{5}+210 a^{4} b^{2} d^{2} e^{4}+84 a^{3} b^{3} d^{3} e^{3}+28 a^{2} b^{4} d^{4} e^{2}+7 a \,b^{5} d^{5} e +b^{6} d^{6}}{12012 e^{7}}}{\left (e x +d \right )^{13}}\)	\(335\)
default	\(-\frac {b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{2 e^{7} \left (e x +d \right )^{12}}-\frac {15 b^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{11 e^{7} \left (e x +d \right )^{11}}-\frac {3 b^{5} \left (a e -b d \right )}{4 e^{7} \left (e x +d \right )^{8}}-\frac {2 b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{e^{7} \left (e x +d \right )^{10}}-\frac {b^{6}}{7 e^{7} \left (e x +d \right )^{7}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{13 e^{7} \left (e x +d \right )^{13}}-\frac {5 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{3 e^{7} \left (e x +d \right )^{9}}\)	\(357\)
norman	\(\frac {-\frac {b^{6} x^{6}}{7 e}-\frac {3 \left (7 a \,b^{5} e^{7}+b^{6} d \,e^{6}\right ) x^{5}}{28 e^{8}}-\frac {5 \left (28 a^{2} b^{4} e^{8}+7 a \,b^{5} d \,e^{7}+b^{6} d^{2} e^{6}\right ) x^{4}}{84 e^{9}}-\frac {\left (84 a^{3} b^{3} e^{9}+28 a^{2} b^{4} d \,e^{8}+7 a \,b^{5} d^{2} e^{7}+b^{6} d^{3} e^{6}\right ) x^{3}}{42 e^{10}}-\frac {\left (210 a^{4} b^{2} e^{10}+84 a^{3} b^{3} d \,e^{9}+28 a^{2} b^{4} d^{2} e^{8}+7 a \,b^{5} d^{3} e^{7}+b^{6} d^{4} e^{6}\right ) x^{2}}{154 e^{11}}-\frac {\left (462 a^{5} b \,e^{11}+210 a^{4} b^{2} d \,e^{10}+84 a^{3} b^{3} d^{2} e^{9}+28 a^{2} b^{4} d^{3} e^{8}+7 a \,b^{5} d^{4} e^{7}+b^{6} d^{5} e^{6}\right ) x}{924 e^{12}}-\frac {924 a^{6} e^{12}+462 a^{5} b d \,e^{11}+210 a^{4} b^{2} d^{2} e^{10}+84 a^{3} b^{3} d^{3} e^{9}+28 a^{2} b^{4} d^{4} e^{8}+7 a \,b^{5} d^{5} e^{7}+b^{6} d^{6} e^{6}}{12012 e^{13}}}{\left (e x +d \right )^{13}}\)	\(375\)
gosper	\(-\frac {1716 x^{6} b^{6} e^{6}+9009 x^{5} a \,b^{5} e^{6}+1287 x^{5} b^{6} d \,e^{5}+20020 x^{4} a^{2} b^{4} e^{6}+5005 x^{4} a \,b^{5} d \,e^{5}+715 x^{4} b^{6} d^{2} e^{4}+24024 x^{3} a^{3} b^{3} e^{6}+8008 x^{3} a^{2} b^{4} d \,e^{5}+2002 x^{3} a \,b^{5} d^{2} e^{4}+286 x^{3} b^{6} d^{3} e^{3}+16380 x^{2} a^{4} b^{2} e^{6}+6552 x^{2} a^{3} b^{3} d \,e^{5}+2184 x^{2} a^{2} b^{4} d^{2} e^{4}+546 x^{2} a \,b^{5} d^{3} e^{3}+78 x^{2} b^{6} d^{4} e^{2}+6006 x \,a^{5} b \,e^{6}+2730 x \,a^{4} b^{2} d \,e^{5}+1092 x \,a^{3} b^{3} d^{2} e^{4}+364 x \,a^{2} b^{4} d^{3} e^{3}+91 x a \,b^{5} d^{4} e^{2}+13 x \,b^{6} d^{5} e +924 a^{6} e^{6}+462 a^{5} b d \,e^{5}+210 a^{4} b^{2} d^{2} e^{4}+84 a^{3} b^{3} d^{3} e^{3}+28 a^{2} b^{4} d^{4} e^{2}+7 a \,b^{5} d^{5} e +b^{6} d^{6}}{12012 e^{7} \left (e x +d \right )^{13}}\)	\(376\)
parallelrisch	\(\frac {-1716 b^{6} x^{6} e^{12}-9009 a \,b^{5} e^{12} x^{5}-1287 b^{6} d \,e^{11} x^{5}-20020 a^{2} b^{4} e^{12} x^{4}-5005 a \,b^{5} d \,e^{11} x^{4}-715 b^{6} d^{2} e^{10} x^{4}-24024 a^{3} b^{3} e^{12} x^{3}-8008 a^{2} b^{4} d \,e^{11} x^{3}-2002 a \,b^{5} d^{2} e^{10} x^{3}-286 b^{6} d^{3} e^{9} x^{3}-16380 a^{4} b^{2} e^{12} x^{2}-6552 a^{3} b^{3} d \,e^{11} x^{2}-2184 a^{2} b^{4} d^{2} e^{10} x^{2}-546 a \,b^{5} d^{3} e^{9} x^{2}-78 b^{6} d^{4} e^{8} x^{2}-6006 a^{5} b \,e^{12} x -2730 a^{4} b^{2} d \,e^{11} x -1092 a^{3} b^{3} d^{2} e^{10} x -364 a^{2} b^{4} d^{3} e^{9} x -91 a \,b^{5} d^{4} e^{8} x -13 b^{6} d^{5} e^{7} x -924 a^{6} e^{12}-462 a^{5} b d \,e^{11}-210 a^{4} b^{2} d^{2} e^{10}-84 a^{3} b^{3} d^{3} e^{9}-28 a^{2} b^{4} d^{4} e^{8}-7 a \,b^{5} d^{5} e^{7}-b^{6} d^{6} e^{6}}{12012 e^{13} \left (e x +d \right )^{13}}\)	\(384\)

[In]

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

[Out]

(-1/7*b^6/e*x^6-3/28*b^5/e^2*(7*a*e+b*d)*x^5-5/84*b^4/e^3*(28*a^2*e^2+7*a*b*d*e+b^2*d^2)*x^4-1/42*b^3/e^4*(84*
a^3*e^3+28*a^2*b*d*e^2+7*a*b^2*d^2*e+b^3*d^3)*x^3-1/154*b^2/e^5*(210*a^4*e^4+84*a^3*b*d*e^3+28*a^2*b^2*d^2*e^2
+7*a*b^3*d^3*e+b^4*d^4)*x^2-1/924*b/e^6*(462*a^5*e^5+210*a^4*b*d*e^4+84*a^3*b^2*d^2*e^3+28*a^2*b^3*d^3*e^2+7*a
*b^4*d^4*e+b^5*d^5)*x-1/12012/e^7*(924*a^6*e^6+462*a^5*b*d*e^5+210*a^4*b^2*d^2*e^4+84*a^3*b^3*d^3*e^3+28*a^2*b
^4*d^4*e^2+7*a*b^5*d^5*e+b^6*d^6))/(e*x+d)^13

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 485 vs. \(2 (159) = 318\).

Time = 0.34 (sec) , antiderivative size = 485, normalized size of antiderivative = 2.84 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{14}} \, dx=-\frac {1716 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 7 \, a b^{5} d^{5} e + 28 \, a^{2} b^{4} d^{4} e^{2} + 84 \, a^{3} b^{3} d^{3} e^{3} + 210 \, a^{4} b^{2} d^{2} e^{4} + 462 \, a^{5} b d e^{5} + 924 \, a^{6} e^{6} + 1287 \, {\left (b^{6} d e^{5} + 7 \, a b^{5} e^{6}\right )} x^{5} + 715 \, {\left (b^{6} d^{2} e^{4} + 7 \, a b^{5} d e^{5} + 28 \, a^{2} b^{4} e^{6}\right )} x^{4} + 286 \, {\left (b^{6} d^{3} e^{3} + 7 \, a b^{5} d^{2} e^{4} + 28 \, a^{2} b^{4} d e^{5} + 84 \, a^{3} b^{3} e^{6}\right )} x^{3} + 78 \, {\left (b^{6} d^{4} e^{2} + 7 \, a b^{5} d^{3} e^{3} + 28 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} + 210 \, a^{4} b^{2} e^{6}\right )} x^{2} + 13 \, {\left (b^{6} d^{5} e + 7 \, a b^{5} d^{4} e^{2} + 28 \, a^{2} b^{4} d^{3} e^{3} + 84 \, a^{3} b^{3} d^{2} e^{4} + 210 \, a^{4} b^{2} d e^{5} + 462 \, a^{5} b e^{6}\right )} x}{12012 \, {\left (e^{20} x^{13} + 13 \, d e^{19} x^{12} + 78 \, d^{2} e^{18} x^{11} + 286 \, d^{3} e^{17} x^{10} + 715 \, d^{4} e^{16} x^{9} + 1287 \, d^{5} e^{15} x^{8} + 1716 \, d^{6} e^{14} x^{7} + 1716 \, d^{7} e^{13} x^{6} + 1287 \, d^{8} e^{12} x^{5} + 715 \, d^{9} e^{11} x^{4} + 286 \, d^{10} e^{10} x^{3} + 78 \, d^{11} e^{9} x^{2} + 13 \, d^{12} e^{8} x + d^{13} e^{7}\right )}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{14}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 485 vs. \(2 (159) = 318\).

Time = 0.25 (sec) , antiderivative size = 485, normalized size of antiderivative = 2.84 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{14}} \, dx=-\frac {1716 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 7 \, a b^{5} d^{5} e + 28 \, a^{2} b^{4} d^{4} e^{2} + 84 \, a^{3} b^{3} d^{3} e^{3} + 210 \, a^{4} b^{2} d^{2} e^{4} + 462 \, a^{5} b d e^{5} + 924 \, a^{6} e^{6} + 1287 \, {\left (b^{6} d e^{5} + 7 \, a b^{5} e^{6}\right )} x^{5} + 715 \, {\left (b^{6} d^{2} e^{4} + 7 \, a b^{5} d e^{5} + 28 \, a^{2} b^{4} e^{6}\right )} x^{4} + 286 \, {\left (b^{6} d^{3} e^{3} + 7 \, a b^{5} d^{2} e^{4} + 28 \, a^{2} b^{4} d e^{5} + 84 \, a^{3} b^{3} e^{6}\right )} x^{3} + 78 \, {\left (b^{6} d^{4} e^{2} + 7 \, a b^{5} d^{3} e^{3} + 28 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} + 210 \, a^{4} b^{2} e^{6}\right )} x^{2} + 13 \, {\left (b^{6} d^{5} e + 7 \, a b^{5} d^{4} e^{2} + 28 \, a^{2} b^{4} d^{3} e^{3} + 84 \, a^{3} b^{3} d^{2} e^{4} + 210 \, a^{4} b^{2} d e^{5} + 462 \, a^{5} b e^{6}\right )} x}{12012 \, {\left (e^{20} x^{13} + 13 \, d e^{19} x^{12} + 78 \, d^{2} e^{18} x^{11} + 286 \, d^{3} e^{17} x^{10} + 715 \, d^{4} e^{16} x^{9} + 1287 \, d^{5} e^{15} x^{8} + 1716 \, d^{6} e^{14} x^{7} + 1716 \, d^{7} e^{13} x^{6} + 1287 \, d^{8} e^{12} x^{5} + 715 \, d^{9} e^{11} x^{4} + 286 \, d^{10} e^{10} x^{3} + 78 \, d^{11} e^{9} x^{2} + 13 \, d^{12} e^{8} x + d^{13} e^{7}\right )}} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (159) = 318\).

Time = 0.26 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.19 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{14}} \, dx=-\frac {1716 \, b^{6} e^{6} x^{6} + 1287 \, b^{6} d e^{5} x^{5} + 9009 \, a b^{5} e^{6} x^{5} + 715 \, b^{6} d^{2} e^{4} x^{4} + 5005 \, a b^{5} d e^{5} x^{4} + 20020 \, a^{2} b^{4} e^{6} x^{4} + 286 \, b^{6} d^{3} e^{3} x^{3} + 2002 \, a b^{5} d^{2} e^{4} x^{3} + 8008 \, a^{2} b^{4} d e^{5} x^{3} + 24024 \, a^{3} b^{3} e^{6} x^{3} + 78 \, b^{6} d^{4} e^{2} x^{2} + 546 \, a b^{5} d^{3} e^{3} x^{2} + 2184 \, a^{2} b^{4} d^{2} e^{4} x^{2} + 6552 \, a^{3} b^{3} d e^{5} x^{2} + 16380 \, a^{4} b^{2} e^{6} x^{2} + 13 \, b^{6} d^{5} e x + 91 \, a b^{5} d^{4} e^{2} x + 364 \, a^{2} b^{4} d^{3} e^{3} x + 1092 \, a^{3} b^{3} d^{2} e^{4} x + 2730 \, a^{4} b^{2} d e^{5} x + 6006 \, a^{5} b e^{6} x + b^{6} d^{6} + 7 \, a b^{5} d^{5} e + 28 \, a^{2} b^{4} d^{4} e^{2} + 84 \, a^{3} b^{3} d^{3} e^{3} + 210 \, a^{4} b^{2} d^{2} e^{4} + 462 \, a^{5} b d e^{5} + 924 \, a^{6} e^{6}}{12012 \, {\left (e x + d\right )}^{13} e^{7}} \]

[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*e^6*x^6 + 1287*b^6*d*e^5*x^5 + 9009*a*b^5*e^6*x^5 + 715*b^6*d^2*e^4*x^4 + 5005*a*b^5*d*e^5*
x^4 + 20020*a^2*b^4*e^6*x^4 + 286*b^6*d^3*e^3*x^3 + 2002*a*b^5*d^2*e^4*x^3 + 8008*a^2*b^4*d*e^5*x^3 + 24024*a^
3*b^3*e^6*x^3 + 78*b^6*d^4*e^2*x^2 + 546*a*b^5*d^3*e^3*x^2 + 2184*a^2*b^4*d^2*e^4*x^2 + 6552*a^3*b^3*d*e^5*x^2
+ 16380*a^4*b^2*e^6*x^2 + 13*b^6*d^5*e*x + 91*a*b^5*d^4*e^2*x + 364*a^2*b^4*d^3*e^3*x + 1092*a^3*b^3*d^2*e^4*
x + 2730*a^4*b^2*d*e^5*x + 6006*a^5*b*e^6*x + 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)/((e*x + d)^13*e^7)

Mupad [B] (verification not implemented)

Time = 10.29 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.73 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{14}} \, dx=-\frac {\frac {924\,a^6\,e^6+462\,a^5\,b\,d\,e^5+210\,a^4\,b^2\,d^2\,e^4+84\,a^3\,b^3\,d^3\,e^3+28\,a^2\,b^4\,d^4\,e^2+7\,a\,b^5\,d^5\,e+b^6\,d^6}{12012\,e^7}+\frac {b^6\,x^6}{7\,e}+\frac {b^3\,x^3\,\left (84\,a^3\,e^3+28\,a^2\,b\,d\,e^2+7\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{42\,e^4}+\frac {b\,x\,\left (462\,a^5\,e^5+210\,a^4\,b\,d\,e^4+84\,a^3\,b^2\,d^2\,e^3+28\,a^2\,b^3\,d^3\,e^2+7\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{924\,e^6}+\frac {3\,b^5\,x^5\,\left (7\,a\,e+b\,d\right )}{28\,e^2}+\frac {b^2\,x^2\,\left (210\,a^4\,e^4+84\,a^3\,b\,d\,e^3+28\,a^2\,b^2\,d^2\,e^2+7\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{154\,e^5}+\frac {5\,b^4\,x^4\,\left (28\,a^2\,e^2+7\,a\,b\,d\,e+b^2\,d^2\right )}{84\,e^3}}{d^{13}+13\,d^{12}\,e\,x+78\,d^{11}\,e^2\,x^2+286\,d^{10}\,e^3\,x^3+715\,d^9\,e^4\,x^4+1287\,d^8\,e^5\,x^5+1716\,d^7\,e^6\,x^6+1716\,d^6\,e^7\,x^7+1287\,d^5\,e^8\,x^8+715\,d^4\,e^9\,x^9+286\,d^3\,e^{10}\,x^{10}+78\,d^2\,e^{11}\,x^{11}+13\,d\,e^{12}\,x^{12}+e^{13}\,x^{13}} \]

[In]

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

[Out]

-((924*a^6*e^6 + b^6*d^6 + 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 + 7*a*b^5*d^5*e + 462
*a^5*b*d*e^5)/(12012*e^7) + (b^6*x^6)/(7*e) + (b^3*x^3*(84*a^3*e^3 + b^3*d^3 + 7*a*b^2*d^2*e + 28*a^2*b*d*e^2)
)/(42*e^4) + (b*x*(462*a^5*e^5 + b^5*d^5 + 28*a^2*b^3*d^3*e^2 + 84*a^3*b^2*d^2*e^3 + 7*a*b^4*d^4*e + 210*a^4*b
*d*e^4))/(924*e^6) + (3*b^5*x^5*(7*a*e + b*d))/(28*e^2) + (b^2*x^2*(210*a^4*e^4 + b^4*d^4 + 28*a^2*b^2*d^2*e^2
+ 7*a*b^3*d^3*e + 84*a^3*b*d*e^3))/(154*e^5) + (5*b^4*x^4*(28*a^2*e^2 + b^2*d^2 + 7*a*b*d*e))/(84*e^3))/(d^13
+ e^13*x^13 + 13*d*e^12*x^12 + 78*d^11*e^2*x^2 + 286*d^10*e^3*x^3 + 715*d^9*e^4*x^4 + 1287*d^8*e^5*x^5 + 1716
*d^7*e^6*x^6 + 1716*d^6*e^7*x^7 + 1287*d^5*e^8*x^8 + 715*d^4*e^9*x^9 + 286*d^3*e^10*x^10 + 78*d^2*e^11*x^11 +
13*d^12*e*x)