\(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{13}} \, dx\) [1502]

Optimal result

Integrand size = 26, antiderivative size = 173 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=-\frac {(b d-a e)^6}{12 e^7 (d+e x)^{12}}+\frac {6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac {3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac {20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac {15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac {6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac {b^6}{6 e^7 (d+e x)^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 173, 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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=\frac {6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac {15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac {20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac {3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac {6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac {(b d-a e)^6}{12 e^7 (d+e x)^{12}}-\frac {b^6}{6 e^7 (d+e x)^6} \]

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^13,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=-\frac {462 a^6 e^6+252 a^5 b e^5 (d+12 e x)+126 a^4 b^2 e^4 \left (d^2+12 d e x+66 e^2 x^2\right )+56 a^3 b^3 e^3 \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+21 a^2 b^4 e^2 \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )+6 a b^5 e \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )+b^6 \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )}{5544 e^7 (d+e x)^{12}} \]

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^13,x]

[Out]

-1/5544*(462*a^6*e^6 + 252*a^5*b*e^5*(d + 12*e*x) + 126*a^4*b^2*e^4*(d^2 + 12*d*e*x + 66*e^2*x^2) + 56*a^3*b^3
*e^3*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3) + 21*a^2*b^4*e^2*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 220
*d*e^3*x^3 + 495*e^4*x^4) + 6*a*b^5*e*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 7
92*e^5*x^5) + b^6*(d^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924
*e^6*x^6))/(e^7*(d + e*x)^12)

Maple [B] (verified)

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

Time = 2.24 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.94

method	result	size
risch	\(\frac {-\frac {b^{6} x^{6}}{6 e}-\frac {b^{5} \left (6 a e +b d \right ) x^{5}}{7 e^{2}}-\frac {5 b^{4} \left (21 a^{2} e^{2}+6 a b d e +b^{2} d^{2}\right ) x^{4}}{56 e^{3}}-\frac {5 b^{3} \left (56 a^{3} e^{3}+21 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{126 e^{4}}-\frac {b^{2} \left (126 e^{4} a^{4}+56 b \,e^{3} d \,a^{3}+21 b^{2} e^{2} d^{2} a^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{84 e^{5}}-\frac {b \left (252 a^{5} e^{5}+126 a^{4} b d \,e^{4}+56 a^{3} b^{2} d^{2} e^{3}+21 a^{2} b^{3} d^{3} e^{2}+6 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{462 e^{6}}-\frac {462 a^{6} e^{6}+252 a^{5} b d \,e^{5}+126 a^{4} b^{2} d^{2} e^{4}+56 a^{3} b^{3} d^{3} e^{3}+21 a^{2} b^{4} d^{4} e^{2}+6 a \,b^{5} d^{5} e +b^{6} d^{6}}{5544 e^{7}}}{\left (e x +d \right )^{12}}\)	\(335\)
default	\(-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{12 e^{7} \left (e x +d \right )^{12}}-\frac {6 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{11 e^{7} \left (e x +d \right )^{11}}-\frac {15 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{8 e^{7} \left (e x +d \right )^{8}}-\frac {3 b^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{2 e^{7} \left (e x +d \right )^{10}}-\frac {6 b^{5} \left (a e -b d \right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {b^{6}}{6 e^{7} \left (e x +d \right )^{6}}-\frac {20 b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{9 e^{7} \left (e x +d \right )^{9}}\)	\(357\)
norman	\(\frac {-\frac {b^{6} x^{6}}{6 e}-\frac {\left (6 a \,b^{5} e^{6}+b^{6} d \,e^{5}\right ) x^{5}}{7 e^{7}}-\frac {5 \left (21 a^{2} b^{4} e^{7}+6 a \,b^{5} d \,e^{6}+b^{6} d^{2} e^{5}\right ) x^{4}}{56 e^{8}}-\frac {5 \left (56 a^{3} b^{3} e^{8}+21 a^{2} b^{4} d \,e^{7}+6 a \,b^{5} d^{2} e^{6}+b^{6} d^{3} e^{5}\right ) x^{3}}{126 e^{9}}-\frac {\left (126 a^{4} b^{2} e^{9}+56 a^{3} b^{3} d \,e^{8}+21 a^{2} b^{4} d^{2} e^{7}+6 a \,b^{5} d^{3} e^{6}+b^{6} d^{4} e^{5}\right ) x^{2}}{84 e^{10}}-\frac {\left (252 a^{5} b \,e^{10}+126 a^{4} b^{2} d \,e^{9}+56 a^{3} b^{3} d^{2} e^{8}+21 a^{2} b^{4} d^{3} e^{7}+6 a \,b^{5} d^{4} e^{6}+b^{6} d^{5} e^{5}\right ) x}{462 e^{11}}-\frac {462 a^{6} e^{11}+252 a^{5} b d \,e^{10}+126 a^{4} b^{2} d^{2} e^{9}+56 a^{3} b^{3} d^{3} e^{8}+21 a^{2} b^{4} d^{4} e^{7}+6 a \,b^{5} d^{5} e^{6}+b^{6} d^{6} e^{5}}{5544 e^{12}}}{\left (e x +d \right )^{12}}\)	\(375\)
gosper	\(-\frac {924 x^{6} b^{6} e^{6}+4752 x^{5} a \,b^{5} e^{6}+792 x^{5} b^{6} d \,e^{5}+10395 x^{4} a^{2} b^{4} e^{6}+2970 x^{4} a \,b^{5} d \,e^{5}+495 x^{4} b^{6} d^{2} e^{4}+12320 x^{3} a^{3} b^{3} e^{6}+4620 x^{3} a^{2} b^{4} d \,e^{5}+1320 x^{3} a \,b^{5} d^{2} e^{4}+220 x^{3} b^{6} d^{3} e^{3}+8316 x^{2} a^{4} b^{2} e^{6}+3696 x^{2} a^{3} b^{3} d \,e^{5}+1386 x^{2} a^{2} b^{4} d^{2} e^{4}+396 x^{2} a \,b^{5} d^{3} e^{3}+66 x^{2} b^{6} d^{4} e^{2}+3024 x \,a^{5} b \,e^{6}+1512 x \,a^{4} b^{2} d \,e^{5}+672 x \,a^{3} b^{3} d^{2} e^{4}+252 x \,a^{2} b^{4} d^{3} e^{3}+72 x a \,b^{5} d^{4} e^{2}+12 x \,b^{6} d^{5} e +462 a^{6} e^{6}+252 a^{5} b d \,e^{5}+126 a^{4} b^{2} d^{2} e^{4}+56 a^{3} b^{3} d^{3} e^{3}+21 a^{2} b^{4} d^{4} e^{2}+6 a \,b^{5} d^{5} e +b^{6} d^{6}}{5544 e^{7} \left (e x +d \right )^{12}}\)	\(376\)
parallelrisch	\(\frac {-924 b^{6} x^{6} e^{11}-4752 a \,b^{5} e^{11} x^{5}-792 b^{6} d \,e^{10} x^{5}-10395 a^{2} b^{4} e^{11} x^{4}-2970 a \,b^{5} d \,e^{10} x^{4}-495 b^{6} d^{2} e^{9} x^{4}-12320 a^{3} b^{3} e^{11} x^{3}-4620 a^{2} b^{4} d \,e^{10} x^{3}-1320 a \,b^{5} d^{2} e^{9} x^{3}-220 b^{6} d^{3} e^{8} x^{3}-8316 a^{4} b^{2} e^{11} x^{2}-3696 a^{3} b^{3} d \,e^{10} x^{2}-1386 a^{2} b^{4} d^{2} e^{9} x^{2}-396 a \,b^{5} d^{3} e^{8} x^{2}-66 b^{6} d^{4} e^{7} x^{2}-3024 a^{5} b \,e^{11} x -1512 a^{4} b^{2} d \,e^{10} x -672 a^{3} b^{3} d^{2} e^{9} x -252 a^{2} b^{4} d^{3} e^{8} x -72 a \,b^{5} d^{4} e^{7} x -12 b^{6} d^{5} e^{6} x -462 a^{6} e^{11}-252 a^{5} b d \,e^{10}-126 a^{4} b^{2} d^{2} e^{9}-56 a^{3} b^{3} d^{3} e^{8}-21 a^{2} b^{4} d^{4} e^{7}-6 a \,b^{5} d^{5} e^{6}-b^{6} d^{6} e^{5}}{5544 e^{12} \left (e x +d \right )^{12}}\)	\(384\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^13,x,method=_RETURNVERBOSE)

[Out]

(-1/6*b^6/e*x^6-1/7*b^5/e^2*(6*a*e+b*d)*x^5-5/56*b^4/e^3*(21*a^2*e^2+6*a*b*d*e+b^2*d^2)*x^4-5/126*b^3/e^4*(56*
a^3*e^3+21*a^2*b*d*e^2+6*a*b^2*d^2*e+b^3*d^3)*x^3-1/84*b^2/e^5*(126*a^4*e^4+56*a^3*b*d*e^3+21*a^2*b^2*d^2*e^2+
6*a*b^3*d^3*e+b^4*d^4)*x^2-1/462*b/e^6*(252*a^5*e^5+126*a^4*b*d*e^4+56*a^3*b^2*d^2*e^3+21*a^2*b^3*d^3*e^2+6*a*
b^4*d^4*e+b^5*d^5)*x-1/5544/e^7*(462*a^6*e^6+252*a^5*b*d*e^5+126*a^4*b^2*d^2*e^4+56*a^3*b^3*d^3*e^3+21*a^2*b^4
*d^4*e^2+6*a*b^5*d^5*e+b^6*d^6))/(e*x+d)^12

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (159) = 318\).

Time = 0.31 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.74 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=-\frac {924 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6} + 792 \, {\left (b^{6} d e^{5} + 6 \, a b^{5} e^{6}\right )} x^{5} + 495 \, {\left (b^{6} d^{2} e^{4} + 6 \, a b^{5} d e^{5} + 21 \, a^{2} b^{4} e^{6}\right )} x^{4} + 220 \, {\left (b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} + 21 \, a^{2} b^{4} d e^{5} + 56 \, a^{3} b^{3} e^{6}\right )} x^{3} + 66 \, {\left (b^{6} d^{4} e^{2} + 6 \, a b^{5} d^{3} e^{3} + 21 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 126 \, a^{4} b^{2} e^{6}\right )} x^{2} + 12 \, {\left (b^{6} d^{5} e + 6 \, a b^{5} d^{4} e^{2} + 21 \, a^{2} b^{4} d^{3} e^{3} + 56 \, a^{3} b^{3} d^{2} e^{4} + 126 \, a^{4} b^{2} d e^{5} + 252 \, a^{5} b e^{6}\right )} x}{5544 \, {\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} e^{7}\right )}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^13,x, algorithm="fricas")

[Out]

-1/5544*(924*b^6*e^6*x^6 + b^6*d^6 + 6*a*b^5*d^5*e + 21*a^2*b^4*d^4*e^2 + 56*a^3*b^3*d^3*e^3 + 126*a^4*b^2*d^2
*e^4 + 252*a^5*b*d*e^5 + 462*a^6*e^6 + 792*(b^6*d*e^5 + 6*a*b^5*e^6)*x^5 + 495*(b^6*d^2*e^4 + 6*a*b^5*d*e^5 +
21*a^2*b^4*e^6)*x^4 + 220*(b^6*d^3*e^3 + 6*a*b^5*d^2*e^4 + 21*a^2*b^4*d*e^5 + 56*a^3*b^3*e^6)*x^3 + 66*(b^6*d^
4*e^2 + 6*a*b^5*d^3*e^3 + 21*a^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 126*a^4*b^2*e^6)*x^2 + 12*(b^6*d^5*e + 6*a*b
^5*d^4*e^2 + 21*a^2*b^4*d^3*e^3 + 56*a^3*b^3*d^2*e^4 + 126*a^4*b^2*d*e^5 + 252*a^5*b*e^6)*x)/(e^19*x^12 + 12*d
*e^18*x^11 + 66*d^2*e^17*x^10 + 220*d^3*e^16*x^9 + 495*d^4*e^15*x^8 + 792*d^5*e^14*x^7 + 924*d^6*e^13*x^6 + 79
2*d^7*e^12*x^5 + 495*d^8*e^11*x^4 + 220*d^9*e^10*x^3 + 66*d^10*e^9*x^2 + 12*d^11*e^8*x + d^12*e^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=\text {Timed out} \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**13,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (159) = 318\).

Time = 0.22 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.74 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=-\frac {924 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6} + 792 \, {\left (b^{6} d e^{5} + 6 \, a b^{5} e^{6}\right )} x^{5} + 495 \, {\left (b^{6} d^{2} e^{4} + 6 \, a b^{5} d e^{5} + 21 \, a^{2} b^{4} e^{6}\right )} x^{4} + 220 \, {\left (b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} + 21 \, a^{2} b^{4} d e^{5} + 56 \, a^{3} b^{3} e^{6}\right )} x^{3} + 66 \, {\left (b^{6} d^{4} e^{2} + 6 \, a b^{5} d^{3} e^{3} + 21 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 126 \, a^{4} b^{2} e^{6}\right )} x^{2} + 12 \, {\left (b^{6} d^{5} e + 6 \, a b^{5} d^{4} e^{2} + 21 \, a^{2} b^{4} d^{3} e^{3} + 56 \, a^{3} b^{3} d^{2} e^{4} + 126 \, a^{4} b^{2} d e^{5} + 252 \, a^{5} b e^{6}\right )} x}{5544 \, {\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} e^{7}\right )}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^13,x, algorithm="maxima")

[Out]

-1/5544*(924*b^6*e^6*x^6 + b^6*d^6 + 6*a*b^5*d^5*e + 21*a^2*b^4*d^4*e^2 + 56*a^3*b^3*d^3*e^3 + 126*a^4*b^2*d^2
*e^4 + 252*a^5*b*d*e^5 + 462*a^6*e^6 + 792*(b^6*d*e^5 + 6*a*b^5*e^6)*x^5 + 495*(b^6*d^2*e^4 + 6*a*b^5*d*e^5 +
21*a^2*b^4*e^6)*x^4 + 220*(b^6*d^3*e^3 + 6*a*b^5*d^2*e^4 + 21*a^2*b^4*d*e^5 + 56*a^3*b^3*e^6)*x^3 + 66*(b^6*d^
4*e^2 + 6*a*b^5*d^3*e^3 + 21*a^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 126*a^4*b^2*e^6)*x^2 + 12*(b^6*d^5*e + 6*a*b
^5*d^4*e^2 + 21*a^2*b^4*d^3*e^3 + 56*a^3*b^3*d^2*e^4 + 126*a^4*b^2*d*e^5 + 252*a^5*b*e^6)*x)/(e^19*x^12 + 12*d
*e^18*x^11 + 66*d^2*e^17*x^10 + 220*d^3*e^16*x^9 + 495*d^4*e^15*x^8 + 792*d^5*e^14*x^7 + 924*d^6*e^13*x^6 + 79
2*d^7*e^12*x^5 + 495*d^8*e^11*x^4 + 220*d^9*e^10*x^3 + 66*d^10*e^9*x^2 + 12*d^11*e^8*x + d^12*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 = 0.25 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.17 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=-\frac {924 \, b^{6} e^{6} x^{6} + 792 \, b^{6} d e^{5} x^{5} + 4752 \, a b^{5} e^{6} x^{5} + 495 \, b^{6} d^{2} e^{4} x^{4} + 2970 \, a b^{5} d e^{5} x^{4} + 10395 \, a^{2} b^{4} e^{6} x^{4} + 220 \, b^{6} d^{3} e^{3} x^{3} + 1320 \, a b^{5} d^{2} e^{4} x^{3} + 4620 \, a^{2} b^{4} d e^{5} x^{3} + 12320 \, a^{3} b^{3} e^{6} x^{3} + 66 \, b^{6} d^{4} e^{2} x^{2} + 396 \, a b^{5} d^{3} e^{3} x^{2} + 1386 \, a^{2} b^{4} d^{2} e^{4} x^{2} + 3696 \, a^{3} b^{3} d e^{5} x^{2} + 8316 \, a^{4} b^{2} e^{6} x^{2} + 12 \, b^{6} d^{5} e x + 72 \, a b^{5} d^{4} e^{2} x + 252 \, a^{2} b^{4} d^{3} e^{3} x + 672 \, a^{3} b^{3} d^{2} e^{4} x + 1512 \, a^{4} b^{2} d e^{5} x + 3024 \, a^{5} b e^{6} x + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6}}{5544 \, {\left (e x + d\right )}^{12} e^{7}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^13,x, algorithm="giac")

[Out]

-1/5544*(924*b^6*e^6*x^6 + 792*b^6*d*e^5*x^5 + 4752*a*b^5*e^6*x^5 + 495*b^6*d^2*e^4*x^4 + 2970*a*b^5*d*e^5*x^4
+ 10395*a^2*b^4*e^6*x^4 + 220*b^6*d^3*e^3*x^3 + 1320*a*b^5*d^2*e^4*x^3 + 4620*a^2*b^4*d*e^5*x^3 + 12320*a^3*b
^3*e^6*x^3 + 66*b^6*d^4*e^2*x^2 + 396*a*b^5*d^3*e^3*x^2 + 1386*a^2*b^4*d^2*e^4*x^2 + 3696*a^3*b^3*d*e^5*x^2 +
8316*a^4*b^2*e^6*x^2 + 12*b^6*d^5*e*x + 72*a*b^5*d^4*e^2*x + 252*a^2*b^4*d^3*e^3*x + 672*a^3*b^3*d^2*e^4*x + 1
512*a^4*b^2*d*e^5*x + 3024*a^5*b*e^6*x + b^6*d^6 + 6*a*b^5*d^5*e + 21*a^2*b^4*d^4*e^2 + 56*a^3*b^3*d^3*e^3 + 1
26*a^4*b^2*d^2*e^4 + 252*a^5*b*d*e^5 + 462*a^6*e^6)/((e*x + d)^12*e^7)

Mupad [B] (verification not implemented)

Time = 10.10 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.64 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=-\frac {\frac {462\,a^6\,e^6+252\,a^5\,b\,d\,e^5+126\,a^4\,b^2\,d^2\,e^4+56\,a^3\,b^3\,d^3\,e^3+21\,a^2\,b^4\,d^4\,e^2+6\,a\,b^5\,d^5\,e+b^6\,d^6}{5544\,e^7}+\frac {b^6\,x^6}{6\,e}+\frac {5\,b^3\,x^3\,\left (56\,a^3\,e^3+21\,a^2\,b\,d\,e^2+6\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{126\,e^4}+\frac {b\,x\,\left (252\,a^5\,e^5+126\,a^4\,b\,d\,e^4+56\,a^3\,b^2\,d^2\,e^3+21\,a^2\,b^3\,d^3\,e^2+6\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{462\,e^6}+\frac {b^5\,x^5\,\left (6\,a\,e+b\,d\right )}{7\,e^2}+\frac {b^2\,x^2\,\left (126\,a^4\,e^4+56\,a^3\,b\,d\,e^3+21\,a^2\,b^2\,d^2\,e^2+6\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{84\,e^5}+\frac {5\,b^4\,x^4\,\left (21\,a^2\,e^2+6\,a\,b\,d\,e+b^2\,d^2\right )}{56\,e^3}}{d^{12}+12\,d^{11}\,e\,x+66\,d^{10}\,e^2\,x^2+220\,d^9\,e^3\,x^3+495\,d^8\,e^4\,x^4+792\,d^7\,e^5\,x^5+924\,d^6\,e^6\,x^6+792\,d^5\,e^7\,x^7+495\,d^4\,e^8\,x^8+220\,d^3\,e^9\,x^9+66\,d^2\,e^{10}\,x^{10}+12\,d\,e^{11}\,x^{11}+e^{12}\,x^{12}} \]

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^13,x)

[Out]

-((462*a^6*e^6 + b^6*d^6 + 21*a^2*b^4*d^4*e^2 + 56*a^3*b^3*d^3*e^3 + 126*a^4*b^2*d^2*e^4 + 6*a*b^5*d^5*e + 252
*a^5*b*d*e^5)/(5544*e^7) + (b^6*x^6)/(6*e) + (5*b^3*x^3*(56*a^3*e^3 + b^3*d^3 + 6*a*b^2*d^2*e + 21*a^2*b*d*e^2
))/(126*e^4) + (b*x*(252*a^5*e^5 + b^5*d^5 + 21*a^2*b^3*d^3*e^2 + 56*a^3*b^2*d^2*e^3 + 6*a*b^4*d^4*e + 126*a^4
*b*d*e^4))/(462*e^6) + (b^5*x^5*(6*a*e + b*d))/(7*e^2) + (b^2*x^2*(126*a^4*e^4 + b^4*d^4 + 21*a^2*b^2*d^2*e^2
+ 6*a*b^3*d^3*e + 56*a^3*b*d*e^3))/(84*e^5) + (5*b^4*x^4*(21*a^2*e^2 + b^2*d^2 + 6*a*b*d*e))/(56*e^3))/(d^12 +
e^12*x^12 + 12*d*e^11*x^11 + 66*d^10*e^2*x^2 + 220*d^9*e^3*x^3 + 495*d^8*e^4*x^4 + 792*d^7*e^5*x^5 + 924*d^6*
e^6*x^6 + 792*d^5*e^7*x^7 + 495*d^4*e^8*x^8 + 220*d^3*e^9*x^9 + 66*d^2*e^10*x^10 + 12*d^11*e*x)