\(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{11}} \, dx\) [1500]

Optimal result

Integrand size = 26, antiderivative size = 120 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{11}} \, dx=\frac {(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^7}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 (a+b x)^7}{120 (b d-a e)^3 (d+e x)^8}+\frac {b^3 (a+b x)^7}{840 (b d-a e)^4 (d+e x)^7} \]

[Out]

1/10*(b*x+a)^7/(-a*e+b*d)/(e*x+d)^10+1/30*b*(b*x+a)^7/(-a*e+b*d)^2/(e*x+d)^9+1/120*b^2*(b*x+a)^7/(-a*e+b*d)^3/
(e*x+d)^8+1/840*b^3*(b*x+a)^7/(-a*e+b*d)^4/(e*x+d)^7

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {27, 47, 37} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{11}} \, dx=\frac {b^3 (a+b x)^7}{840 (d+e x)^7 (b d-a e)^4}+\frac {b^2 (a+b x)^7}{120 (d+e x)^8 (b d-a e)^3}+\frac {b (a+b x)^7}{30 (d+e x)^9 (b d-a e)^2}+\frac {(a+b x)^7}{10 (d+e x)^{10} (b d-a e)} \]

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^11,x]

[Out]

(a + b*x)^7/(10*(b*d - a*e)*(d + e*x)^10) + (b*(a + b*x)^7)/(30*(b*d - a*e)^2*(d + e*x)^9) + (b^2*(a + b*x)^7)
/(120*(b*d - a*e)^3*(d + e*x)^8) + (b^3*(a + b*x)^7)/(840*(b*d - a*e)^4*(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 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]

Rule 47

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])

Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^6}{(d+e x)^{11}} \, dx \\ & = \frac {(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac {(3 b) \int \frac {(a+b x)^6}{(d+e x)^{10}} \, dx}{10 (b d-a e)} \\ & = \frac {(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^7}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{15 (b d-a e)^2} \\ & = \frac {(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^7}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 (a+b x)^7}{120 (b d-a e)^3 (d+e x)^8}+\frac {b^3 \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{120 (b d-a e)^3} \\ & = \frac {(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac {b (a+b x)^7}{30 (b d-a e)^2 (d+e x)^9}+\frac {b^2 (a+b x)^7}{120 (b d-a e)^3 (d+e x)^8}+\frac {b^3 (a+b x)^7}{840 (b d-a e)^4 (d+e x)^7} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(277\) vs. \(2(120)=240\).

Time = 0.06 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.31 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {84 a^6 e^6+56 a^5 b e^5 (d+10 e x)+35 a^4 b^2 e^4 \left (d^2+10 d e x+45 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+10 a^2 b^4 e^2 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+4 a b^5 e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+b^6 \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )}{840 e^7 (d+e x)^{10}} \]

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^11,x]

[Out]

-1/840*(84*a^6*e^6 + 56*a^5*b*e^5*(d + 10*e*x) + 35*a^4*b^2*e^4*(d^2 + 10*d*e*x + 45*e^2*x^2) + 20*a^3*b^3*e^3
*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 10*a^2*b^4*e^2*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e
^3*x^3 + 210*e^4*x^4) + 4*a*b^5*e*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e
^5*x^5) + b^6*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6
*x^6))/(e^7*(d + e*x)^10)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(334\) vs. \(2(112)=224\).

Time = 2.21 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.79

method	result	size
risch	\(\frac {-\frac {b^{6} x^{6}}{4 e}-\frac {3 b^{5} \left (4 a e +b d \right ) x^{5}}{10 e^{2}}-\frac {b^{4} \left (10 a^{2} e^{2}+4 a b d e +b^{2} d^{2}\right ) x^{4}}{4 e^{3}}-\frac {b^{3} \left (20 a^{3} e^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{7 e^{4}}-\frac {3 b^{2} \left (35 e^{4} a^{4}+20 b \,e^{3} d \,a^{3}+10 b^{2} e^{2} d^{2} a^{2}+4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{56 e^{5}}-\frac {b \left (56 a^{5} e^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{84 e^{6}}-\frac {84 a^{6} e^{6}+56 a^{5} b d \,e^{5}+35 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}+10 a^{2} b^{4} d^{4} e^{2}+4 a \,b^{5} d^{5} e +b^{6} d^{6}}{840 e^{7}}}{\left (e x +d \right )^{10}}\)	\(335\)
default	\(-\frac {6 b^{5} \left (a e -b d \right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {15 b^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{8 e^{7} \left (e x +d \right )^{8}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{10 e^{7} \left (e x +d \right )^{10}}-\frac {20 b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {b^{6}}{4 e^{7} \left (e x +d \right )^{4}}-\frac {5 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{2 e^{7} \left (e x +d \right )^{6}}-\frac {2 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{3 e^{7} \left (e x +d \right )^{9}}\)	\(357\)
norman	\(\frac {-\frac {b^{6} x^{6}}{4 e}-\frac {3 \left (4 e^{4} a \,b^{5}+d \,e^{3} b^{6}\right ) x^{5}}{10 e^{5}}-\frac {\left (10 e^{5} a^{2} b^{4}+4 d \,e^{4} a \,b^{5}+d^{2} e^{3} b^{6}\right ) x^{4}}{4 e^{6}}-\frac {\left (20 a^{3} b^{3} e^{6}+10 a^{2} b^{4} d \,e^{5}+4 a \,b^{5} d^{2} e^{4}+b^{6} d^{3} e^{3}\right ) x^{3}}{7 e^{7}}-\frac {3 \left (35 a^{4} b^{2} e^{7}+20 a^{3} b^{3} d \,e^{6}+10 a^{2} b^{4} d^{2} e^{5}+4 a \,b^{5} d^{3} e^{4}+b^{6} d^{4} e^{3}\right ) x^{2}}{56 e^{8}}-\frac {\left (56 a^{5} b \,e^{8}+35 a^{4} b^{2} d \,e^{7}+20 a^{3} b^{3} d^{2} e^{6}+10 a^{2} b^{4} d^{3} e^{5}+4 a \,b^{5} d^{4} e^{4}+b^{6} d^{5} e^{3}\right ) x}{84 e^{9}}-\frac {84 a^{6} e^{9}+56 a^{5} b d \,e^{8}+35 a^{4} b^{2} d^{2} e^{7}+20 a^{3} b^{3} d^{3} e^{6}+10 a^{2} b^{4} d^{4} e^{5}+4 a \,b^{5} d^{5} e^{4}+b^{6} d^{6} e^{3}}{840 e^{10}}}{\left (e x +d \right )^{10}}\)	\(375\)
gosper	\(-\frac {210 x^{6} b^{6} e^{6}+1008 x^{5} a \,b^{5} e^{6}+252 x^{5} b^{6} d \,e^{5}+2100 x^{4} a^{2} b^{4} e^{6}+840 x^{4} a \,b^{5} d \,e^{5}+210 x^{4} b^{6} d^{2} e^{4}+2400 x^{3} a^{3} b^{3} e^{6}+1200 x^{3} a^{2} b^{4} d \,e^{5}+480 x^{3} a \,b^{5} d^{2} e^{4}+120 x^{3} b^{6} d^{3} e^{3}+1575 x^{2} a^{4} b^{2} e^{6}+900 x^{2} a^{3} b^{3} d \,e^{5}+450 x^{2} a^{2} b^{4} d^{2} e^{4}+180 x^{2} a \,b^{5} d^{3} e^{3}+45 x^{2} b^{6} d^{4} e^{2}+560 x \,a^{5} b \,e^{6}+350 x \,a^{4} b^{2} d \,e^{5}+200 x \,a^{3} b^{3} d^{2} e^{4}+100 x \,a^{2} b^{4} d^{3} e^{3}+40 x a \,b^{5} d^{4} e^{2}+10 x \,b^{6} d^{5} e +84 a^{6} e^{6}+56 a^{5} b d \,e^{5}+35 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}+10 a^{2} b^{4} d^{4} e^{2}+4 a \,b^{5} d^{5} e +b^{6} d^{6}}{840 e^{7} \left (e x +d \right )^{10}}\)	\(376\)
parallelrisch	\(\frac {-210 b^{6} x^{6} e^{9}-1008 a \,b^{5} e^{9} x^{5}-252 b^{6} d \,e^{8} x^{5}-2100 a^{2} b^{4} e^{9} x^{4}-840 a \,b^{5} d \,e^{8} x^{4}-210 b^{6} d^{2} e^{7} x^{4}-2400 a^{3} b^{3} e^{9} x^{3}-1200 a^{2} b^{4} d \,e^{8} x^{3}-480 a \,b^{5} d^{2} e^{7} x^{3}-120 b^{6} d^{3} e^{6} x^{3}-1575 a^{4} b^{2} e^{9} x^{2}-900 a^{3} b^{3} d \,e^{8} x^{2}-450 a^{2} b^{4} d^{2} e^{7} x^{2}-180 a \,b^{5} d^{3} e^{6} x^{2}-45 b^{6} d^{4} e^{5} x^{2}-560 a^{5} b \,e^{9} x -350 a^{4} b^{2} d \,e^{8} x -200 a^{3} b^{3} d^{2} e^{7} x -100 a^{2} b^{4} d^{3} e^{6} x -40 a \,b^{5} d^{4} e^{5} x -10 b^{6} d^{5} e^{4} x -84 a^{6} e^{9}-56 a^{5} b d \,e^{8}-35 a^{4} b^{2} d^{2} e^{7}-20 a^{3} b^{3} d^{3} e^{6}-10 a^{2} b^{4} d^{4} e^{5}-4 a \,b^{5} d^{5} e^{4}-b^{6} d^{6} e^{3}}{840 e^{10} \left (e x +d \right )^{10}}\)	\(384\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^11,x,method=_RETURNVERBOSE)

[Out]

(-1/4*b^6/e*x^6-3/10*b^5/e^2*(4*a*e+b*d)*x^5-1/4/e^3*b^4*(10*a^2*e^2+4*a*b*d*e+b^2*d^2)*x^4-1/7*b^3/e^4*(20*a^
3*e^3+10*a^2*b*d*e^2+4*a*b^2*d^2*e+b^3*d^3)*x^3-3/56*b^2/e^5*(35*a^4*e^4+20*a^3*b*d*e^3+10*a^2*b^2*d^2*e^2+4*a
*b^3*d^3*e+b^4*d^4)*x^2-1/84*b/e^6*(56*a^5*e^5+35*a^4*b*d*e^4+20*a^3*b^2*d^2*e^3+10*a^2*b^3*d^3*e^2+4*a*b^4*d^
4*e+b^5*d^5)*x-1/840/e^7*(84*a^6*e^6+56*a^5*b*d*e^5+35*a^4*b^2*d^2*e^4+20*a^3*b^3*d^3*e^3+10*a^2*b^4*d^4*e^2+4
*a*b^5*d^5*e+b^6*d^6))/(e*x+d)^10

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (112) = 224\).

Time = 0.34 (sec) , antiderivative size = 452, normalized size of antiderivative = 3.77 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {210 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 4 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6} + 252 \, {\left (b^{6} d e^{5} + 4 \, a b^{5} e^{6}\right )} x^{5} + 210 \, {\left (b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} + 120 \, {\left (b^{6} d^{3} e^{3} + 4 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} + 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 45 \, {\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \, {\left (b^{6} d^{5} e + 4 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 20 \, a^{3} b^{3} d^{2} e^{4} + 35 \, a^{4} b^{2} d e^{5} + 56 \, a^{5} b e^{6}\right )} x}{840 \, {\left (e^{17} x^{10} + 10 \, d e^{16} x^{9} + 45 \, d^{2} e^{15} x^{8} + 120 \, d^{3} e^{14} x^{7} + 210 \, d^{4} e^{13} x^{6} + 252 \, d^{5} e^{12} x^{5} + 210 \, d^{6} e^{11} x^{4} + 120 \, d^{7} e^{10} x^{3} + 45 \, d^{8} e^{9} x^{2} + 10 \, d^{9} e^{8} x + d^{10} e^{7}\right )}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^11,x, algorithm="fricas")

[Out]

-1/840*(210*b^6*e^6*x^6 + b^6*d^6 + 4*a*b^5*d^5*e + 10*a^2*b^4*d^4*e^2 + 20*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e
^4 + 56*a^5*b*d*e^5 + 84*a^6*e^6 + 252*(b^6*d*e^5 + 4*a*b^5*e^6)*x^5 + 210*(b^6*d^2*e^4 + 4*a*b^5*d*e^5 + 10*a
^2*b^4*e^6)*x^4 + 120*(b^6*d^3*e^3 + 4*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5 + 20*a^3*b^3*e^6)*x^3 + 45*(b^6*d^4*e^
2 + 4*a*b^5*d^3*e^3 + 10*a^2*b^4*d^2*e^4 + 20*a^3*b^3*d*e^5 + 35*a^4*b^2*e^6)*x^2 + 10*(b^6*d^5*e + 4*a*b^5*d^
4*e^2 + 10*a^2*b^4*d^3*e^3 + 20*a^3*b^3*d^2*e^4 + 35*a^4*b^2*d*e^5 + 56*a^5*b*e^6)*x)/(e^17*x^10 + 10*d*e^16*x
^9 + 45*d^2*e^15*x^8 + 120*d^3*e^14*x^7 + 210*d^4*e^13*x^6 + 252*d^5*e^12*x^5 + 210*d^6*e^11*x^4 + 120*d^7*e^1
0*x^3 + 45*d^8*e^9*x^2 + 10*d^9*e^8*x + d^10*e^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{11}} \, dx=\text {Timed out} \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**11,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (112) = 224\).

Time = 0.23 (sec) , antiderivative size = 452, normalized size of antiderivative = 3.77 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {210 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 4 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6} + 252 \, {\left (b^{6} d e^{5} + 4 \, a b^{5} e^{6}\right )} x^{5} + 210 \, {\left (b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} + 120 \, {\left (b^{6} d^{3} e^{3} + 4 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} + 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 45 \, {\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \, {\left (b^{6} d^{5} e + 4 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 20 \, a^{3} b^{3} d^{2} e^{4} + 35 \, a^{4} b^{2} d e^{5} + 56 \, a^{5} b e^{6}\right )} x}{840 \, {\left (e^{17} x^{10} + 10 \, d e^{16} x^{9} + 45 \, d^{2} e^{15} x^{8} + 120 \, d^{3} e^{14} x^{7} + 210 \, d^{4} e^{13} x^{6} + 252 \, d^{5} e^{12} x^{5} + 210 \, d^{6} e^{11} x^{4} + 120 \, d^{7} e^{10} x^{3} + 45 \, d^{8} e^{9} x^{2} + 10 \, d^{9} e^{8} x + d^{10} e^{7}\right )}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^11,x, algorithm="maxima")

[Out]

-1/840*(210*b^6*e^6*x^6 + b^6*d^6 + 4*a*b^5*d^5*e + 10*a^2*b^4*d^4*e^2 + 20*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e
^4 + 56*a^5*b*d*e^5 + 84*a^6*e^6 + 252*(b^6*d*e^5 + 4*a*b^5*e^6)*x^5 + 210*(b^6*d^2*e^4 + 4*a*b^5*d*e^5 + 10*a
^2*b^4*e^6)*x^4 + 120*(b^6*d^3*e^3 + 4*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5 + 20*a^3*b^3*e^6)*x^3 + 45*(b^6*d^4*e^
2 + 4*a*b^5*d^3*e^3 + 10*a^2*b^4*d^2*e^4 + 20*a^3*b^3*d*e^5 + 35*a^4*b^2*e^6)*x^2 + 10*(b^6*d^5*e + 4*a*b^5*d^
4*e^2 + 10*a^2*b^4*d^3*e^3 + 20*a^3*b^3*d^2*e^4 + 35*a^4*b^2*d*e^5 + 56*a^5*b*e^6)*x)/(e^17*x^10 + 10*d*e^16*x
^9 + 45*d^2*e^15*x^8 + 120*d^3*e^14*x^7 + 210*d^4*e^13*x^6 + 252*d^5*e^12*x^5 + 210*d^6*e^11*x^4 + 120*d^7*e^1
0*x^3 + 45*d^8*e^9*x^2 + 10*d^9*e^8*x + d^10*e^7)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (112) = 224\).

Time = 0.26 (sec) , antiderivative size = 375, normalized size of antiderivative = 3.12 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {210 \, b^{6} e^{6} x^{6} + 252 \, b^{6} d e^{5} x^{5} + 1008 \, a b^{5} e^{6} x^{5} + 210 \, b^{6} d^{2} e^{4} x^{4} + 840 \, a b^{5} d e^{5} x^{4} + 2100 \, a^{2} b^{4} e^{6} x^{4} + 120 \, b^{6} d^{3} e^{3} x^{3} + 480 \, a b^{5} d^{2} e^{4} x^{3} + 1200 \, a^{2} b^{4} d e^{5} x^{3} + 2400 \, a^{3} b^{3} e^{6} x^{3} + 45 \, b^{6} d^{4} e^{2} x^{2} + 180 \, a b^{5} d^{3} e^{3} x^{2} + 450 \, a^{2} b^{4} d^{2} e^{4} x^{2} + 900 \, a^{3} b^{3} d e^{5} x^{2} + 1575 \, a^{4} b^{2} e^{6} x^{2} + 10 \, b^{6} d^{5} e x + 40 \, a b^{5} d^{4} e^{2} x + 100 \, a^{2} b^{4} d^{3} e^{3} x + 200 \, a^{3} b^{3} d^{2} e^{4} x + 350 \, a^{4} b^{2} d e^{5} x + 560 \, a^{5} b e^{6} x + b^{6} d^{6} + 4 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6}}{840 \, {\left (e x + d\right )}^{10} e^{7}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^11,x, algorithm="giac")

[Out]

-1/840*(210*b^6*e^6*x^6 + 252*b^6*d*e^5*x^5 + 1008*a*b^5*e^6*x^5 + 210*b^6*d^2*e^4*x^4 + 840*a*b^5*d*e^5*x^4 +
2100*a^2*b^4*e^6*x^4 + 120*b^6*d^3*e^3*x^3 + 480*a*b^5*d^2*e^4*x^3 + 1200*a^2*b^4*d*e^5*x^3 + 2400*a^3*b^3*e^
6*x^3 + 45*b^6*d^4*e^2*x^2 + 180*a*b^5*d^3*e^3*x^2 + 450*a^2*b^4*d^2*e^4*x^2 + 900*a^3*b^3*d*e^5*x^2 + 1575*a^
4*b^2*e^6*x^2 + 10*b^6*d^5*e*x + 40*a*b^5*d^4*e^2*x + 100*a^2*b^4*d^3*e^3*x + 200*a^3*b^3*d^2*e^4*x + 350*a^4*
b^2*d*e^5*x + 560*a^5*b*e^6*x + b^6*d^6 + 4*a*b^5*d^5*e + 10*a^2*b^4*d^4*e^2 + 20*a^3*b^3*d^3*e^3 + 35*a^4*b^2
*d^2*e^4 + 56*a^5*b*d*e^5 + 84*a^6*e^6)/((e*x + d)^10*e^7)

Mupad [B] (verification not implemented)

Time = 9.85 (sec) , antiderivative size = 434, normalized size of antiderivative = 3.62 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {\frac {84\,a^6\,e^6+56\,a^5\,b\,d\,e^5+35\,a^4\,b^2\,d^2\,e^4+20\,a^3\,b^3\,d^3\,e^3+10\,a^2\,b^4\,d^4\,e^2+4\,a\,b^5\,d^5\,e+b^6\,d^6}{840\,e^7}+\frac {b^6\,x^6}{4\,e}+\frac {b^3\,x^3\,\left (20\,a^3\,e^3+10\,a^2\,b\,d\,e^2+4\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{7\,e^4}+\frac {b\,x\,\left (56\,a^5\,e^5+35\,a^4\,b\,d\,e^4+20\,a^3\,b^2\,d^2\,e^3+10\,a^2\,b^3\,d^3\,e^2+4\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{84\,e^6}+\frac {3\,b^5\,x^5\,\left (4\,a\,e+b\,d\right )}{10\,e^2}+\frac {3\,b^2\,x^2\,\left (35\,a^4\,e^4+20\,a^3\,b\,d\,e^3+10\,a^2\,b^2\,d^2\,e^2+4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{56\,e^5}+\frac {b^4\,x^4\,\left (10\,a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )}{4\,e^3}}{d^{10}+10\,d^9\,e\,x+45\,d^8\,e^2\,x^2+120\,d^7\,e^3\,x^3+210\,d^6\,e^4\,x^4+252\,d^5\,e^5\,x^5+210\,d^4\,e^6\,x^6+120\,d^3\,e^7\,x^7+45\,d^2\,e^8\,x^8+10\,d\,e^9\,x^9+e^{10}\,x^{10}} \]

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^11,x)

[Out]

-((84*a^6*e^6 + b^6*d^6 + 10*a^2*b^4*d^4*e^2 + 20*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 4*a*b^5*d^5*e + 56*a^
5*b*d*e^5)/(840*e^7) + (b^6*x^6)/(4*e) + (b^3*x^3*(20*a^3*e^3 + b^3*d^3 + 4*a*b^2*d^2*e + 10*a^2*b*d*e^2))/(7*
e^4) + (b*x*(56*a^5*e^5 + b^5*d^5 + 10*a^2*b^3*d^3*e^2 + 20*a^3*b^2*d^2*e^3 + 4*a*b^4*d^4*e + 35*a^4*b*d*e^4))
/(84*e^6) + (3*b^5*x^5*(4*a*e + b*d))/(10*e^2) + (3*b^2*x^2*(35*a^4*e^4 + b^4*d^4 + 10*a^2*b^2*d^2*e^2 + 4*a*b
^3*d^3*e + 20*a^3*b*d*e^3))/(56*e^5) + (b^4*x^4*(10*a^2*e^2 + b^2*d^2 + 4*a*b*d*e))/(4*e^3))/(d^10 + e^10*x^10
+ 10*d*e^9*x^9 + 45*d^8*e^2*x^2 + 120*d^7*e^3*x^3 + 210*d^6*e^4*x^4 + 252*d^5*e^5*x^5 + 210*d^4*e^6*x^6 + 120
*d^3*e^7*x^7 + 45*d^2*e^8*x^8 + 10*d^9*e*x)