\(\int (d+e x)^7 (a^2+2 a b x+b^2 x^2)^3 \, dx\) [1482]

Optimal result

Integrand size = 26, antiderivative size = 173 \[ \int (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(b d-a e)^6 (d+e x)^8}{8 e^7}-\frac {2 b (b d-a e)^5 (d+e x)^9}{3 e^7}+\frac {3 b^2 (b d-a e)^4 (d+e x)^{10}}{2 e^7}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{11}}{11 e^7}+\frac {5 b^4 (b d-a e)^2 (d+e x)^{12}}{4 e^7}-\frac {6 b^5 (b d-a e) (d+e x)^{13}}{13 e^7}+\frac {b^6 (d+e x)^{14}}{14 e^7} \]

[Out]

1/8*(-a*e+b*d)^6*(e*x+d)^8/e^7-2/3*b*(-a*e+b*d)^5*(e*x+d)^9/e^7+3/2*b^2*(-a*e+b*d)^4*(e*x+d)^10/e^7-20/11*b^3*
(-a*e+b*d)^3*(e*x+d)^11/e^7+5/4*b^4*(-a*e+b*d)^2*(e*x+d)^12/e^7-6/13*b^5*(-a*e+b*d)*(e*x+d)^13/e^7+1/14*b^6*(e
*x+d)^14/e^7

Rubi [A] (verified)

Time = 0.28 (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} \[ \int (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=-\frac {6 b^5 (d+e x)^{13} (b d-a e)}{13 e^7}+\frac {5 b^4 (d+e x)^{12} (b d-a e)^2}{4 e^7}-\frac {20 b^3 (d+e x)^{11} (b d-a e)^3}{11 e^7}+\frac {3 b^2 (d+e x)^{10} (b d-a e)^4}{2 e^7}-\frac {2 b (d+e x)^9 (b d-a e)^5}{3 e^7}+\frac {(d+e x)^8 (b d-a e)^6}{8 e^7}+\frac {b^6 (d+e x)^{14}}{14 e^7} \]

[In]

Int[(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(684\) vs. \(2(173)=346\).

Time = 0.06 (sec) , antiderivative size = 684, normalized size of antiderivative = 3.95 \[ \int (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^6 d^7 x+\frac {1}{2} a^5 d^6 (6 b d+7 a e) x^2+a^4 d^5 \left (5 b^2 d^2+14 a b d e+7 a^2 e^2\right ) x^3+\frac {1}{4} a^3 d^4 \left (20 b^3 d^3+105 a b^2 d^2 e+126 a^2 b d e^2+35 a^3 e^3\right ) x^4+a^2 d^3 \left (3 b^4 d^4+28 a b^3 d^3 e+63 a^2 b^2 d^2 e^2+42 a^3 b d e^3+7 a^4 e^4\right ) x^5+\frac {1}{2} a d^2 \left (2 b^5 d^5+35 a b^4 d^4 e+140 a^2 b^3 d^3 e^2+175 a^3 b^2 d^2 e^3+70 a^4 b d e^4+7 a^5 e^5\right ) x^6+\frac {1}{7} d \left (b^6 d^6+42 a b^5 d^5 e+315 a^2 b^4 d^4 e^2+700 a^3 b^3 d^3 e^3+525 a^4 b^2 d^2 e^4+126 a^5 b d e^5+7 a^6 e^6\right ) x^7+\frac {1}{8} e \left (7 b^6 d^6+126 a b^5 d^5 e+525 a^2 b^4 d^4 e^2+700 a^3 b^3 d^3 e^3+315 a^4 b^2 d^2 e^4+42 a^5 b d e^5+a^6 e^6\right ) x^8+\frac {1}{3} b e^2 \left (7 b^5 d^5+70 a b^4 d^4 e+175 a^2 b^3 d^3 e^2+140 a^3 b^2 d^2 e^3+35 a^4 b d e^4+2 a^5 e^5\right ) x^9+\frac {1}{2} b^2 e^3 \left (7 b^4 d^4+42 a b^3 d^3 e+63 a^2 b^2 d^2 e^2+28 a^3 b d e^3+3 a^4 e^4\right ) x^{10}+\frac {1}{11} b^3 e^4 \left (35 b^3 d^3+126 a b^2 d^2 e+105 a^2 b d e^2+20 a^3 e^3\right ) x^{11}+\frac {1}{4} b^4 e^5 \left (7 b^2 d^2+14 a b d e+5 a^2 e^2\right ) x^{12}+\frac {1}{13} b^5 e^6 (7 b d+6 a e) x^{13}+\frac {1}{14} b^6 e^7 x^{14} \]

[In]

Integrate[(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

a^6*d^7*x + (a^5*d^6*(6*b*d + 7*a*e)*x^2)/2 + a^4*d^5*(5*b^2*d^2 + 14*a*b*d*e + 7*a^2*e^2)*x^3 + (a^3*d^4*(20*
b^3*d^3 + 105*a*b^2*d^2*e + 126*a^2*b*d*e^2 + 35*a^3*e^3)*x^4)/4 + a^2*d^3*(3*b^4*d^4 + 28*a*b^3*d^3*e + 63*a^
2*b^2*d^2*e^2 + 42*a^3*b*d*e^3 + 7*a^4*e^4)*x^5 + (a*d^2*(2*b^5*d^5 + 35*a*b^4*d^4*e + 140*a^2*b^3*d^3*e^2 + 1
75*a^3*b^2*d^2*e^3 + 70*a^4*b*d*e^4 + 7*a^5*e^5)*x^6)/2 + (d*(b^6*d^6 + 42*a*b^5*d^5*e + 315*a^2*b^4*d^4*e^2 +
700*a^3*b^3*d^3*e^3 + 525*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 + 7*a^6*e^6)*x^7)/7 + (e*(7*b^6*d^6 + 126*a*b^5*d
^5*e + 525*a^2*b^4*d^4*e^2 + 700*a^3*b^3*d^3*e^3 + 315*a^4*b^2*d^2*e^4 + 42*a^5*b*d*e^5 + a^6*e^6)*x^8)/8 + (b
*e^2*(7*b^5*d^5 + 70*a*b^4*d^4*e + 175*a^2*b^3*d^3*e^2 + 140*a^3*b^2*d^2*e^3 + 35*a^4*b*d*e^4 + 2*a^5*e^5)*x^9
)/3 + (b^2*e^3*(7*b^4*d^4 + 42*a*b^3*d^3*e + 63*a^2*b^2*d^2*e^2 + 28*a^3*b*d*e^3 + 3*a^4*e^4)*x^10)/2 + (b^3*e
^4*(35*b^3*d^3 + 126*a*b^2*d^2*e + 105*a^2*b*d*e^2 + 20*a^3*e^3)*x^11)/11 + (b^4*e^5*(7*b^2*d^2 + 14*a*b*d*e +
5*a^2*e^2)*x^12)/4 + (b^5*e^6*(7*b*d + 6*a*e)*x^13)/13 + (b^6*e^7*x^14)/14

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(696\) vs. \(2(159)=318\).

Time = 2.26 (sec) , antiderivative size = 697, normalized size of antiderivative = 4.03

method	result	size
norman	\(a^{6} d^{7} x +\left (\frac {7}{2} a^{6} d^{6} e +3 a^{5} b \,d^{7}\right ) x^{2}+\left (7 a^{6} d^{5} e^{2}+14 a^{5} b \,d^{6} e +5 a^{4} b^{2} d^{7}\right ) x^{3}+\left (\frac {35}{4} a^{6} d^{4} e^{3}+\frac {63}{2} a^{5} b \,d^{5} e^{2}+\frac {105}{4} a^{4} b^{2} d^{6} e +5 a^{3} b^{3} d^{7}\right ) x^{4}+\left (7 a^{6} d^{3} e^{4}+42 a^{5} b \,d^{4} e^{3}+63 a^{4} b^{2} d^{5} e^{2}+28 a^{3} b^{3} d^{6} e +3 a^{2} b^{4} d^{7}\right ) x^{5}+\left (\frac {7}{2} a^{6} d^{2} e^{5}+35 a^{5} b \,d^{3} e^{4}+\frac {175}{2} a^{4} b^{2} d^{4} e^{3}+70 a^{3} b^{3} d^{5} e^{2}+\frac {35}{2} a^{2} b^{4} d^{6} e +a \,b^{5} d^{7}\right ) x^{6}+\left (a^{6} d \,e^{6}+18 a^{5} b \,d^{2} e^{5}+75 a^{4} b^{2} d^{3} e^{4}+100 a^{3} b^{3} d^{4} e^{3}+45 a^{2} b^{4} d^{5} e^{2}+6 a \,b^{5} d^{6} e +\frac {1}{7} b^{6} d^{7}\right ) x^{7}+\left (\frac {1}{8} a^{6} e^{7}+\frac {21}{4} a^{5} b d \,e^{6}+\frac {315}{8} a^{4} b^{2} d^{2} e^{5}+\frac {175}{2} a^{3} b^{3} d^{3} e^{4}+\frac {525}{8} a^{2} b^{4} d^{4} e^{3}+\frac {63}{4} a \,b^{5} d^{5} e^{2}+\frac {7}{8} b^{6} d^{6} e \right ) x^{8}+\left (\frac {2}{3} a^{5} b \,e^{7}+\frac {35}{3} a^{4} b^{2} d \,e^{6}+\frac {140}{3} a^{3} b^{3} d^{2} e^{5}+\frac {175}{3} a^{2} b^{4} d^{3} e^{4}+\frac {70}{3} a \,b^{5} d^{4} e^{3}+\frac {7}{3} b^{6} d^{5} e^{2}\right ) x^{9}+\left (\frac {3}{2} a^{4} b^{2} e^{7}+14 a^{3} b^{3} d \,e^{6}+\frac {63}{2} a^{2} b^{4} d^{2} e^{5}+21 a \,b^{5} d^{3} e^{4}+\frac {7}{2} b^{6} d^{4} e^{3}\right ) x^{10}+\left (\frac {20}{11} a^{3} b^{3} e^{7}+\frac {105}{11} a^{2} b^{4} d \,e^{6}+\frac {126}{11} a \,b^{5} d^{2} e^{5}+\frac {35}{11} b^{6} d^{3} e^{4}\right ) x^{11}+\left (\frac {5}{4} a^{2} b^{4} e^{7}+\frac {7}{2} a \,b^{5} d \,e^{6}+\frac {7}{4} b^{6} d^{2} e^{5}\right ) x^{12}+\left (\frac {6}{13} a \,b^{5} e^{7}+\frac {7}{13} b^{6} d \,e^{6}\right ) x^{13}+\frac {b^{6} e^{7} x^{14}}{14}\)	\(697\)
default	\(\frac {b^{6} e^{7} x^{14}}{14}+\frac {\left (6 a \,b^{5} e^{7}+7 b^{6} d \,e^{6}\right ) x^{13}}{13}+\frac {\left (15 a^{2} b^{4} e^{7}+42 a \,b^{5} d \,e^{6}+21 b^{6} d^{2} e^{5}\right ) x^{12}}{12}+\frac {\left (20 a^{3} b^{3} e^{7}+105 a^{2} b^{4} d \,e^{6}+126 a \,b^{5} d^{2} e^{5}+35 b^{6} d^{3} e^{4}\right ) x^{11}}{11}+\frac {\left (15 a^{4} b^{2} e^{7}+140 a^{3} b^{3} d \,e^{6}+315 a^{2} b^{4} d^{2} e^{5}+210 a \,b^{5} d^{3} e^{4}+35 b^{6} d^{4} e^{3}\right ) x^{10}}{10}+\frac {\left (6 a^{5} b \,e^{7}+105 a^{4} b^{2} d \,e^{6}+420 a^{3} b^{3} d^{2} e^{5}+525 a^{2} b^{4} d^{3} e^{4}+210 a \,b^{5} d^{4} e^{3}+21 b^{6} d^{5} e^{2}\right ) x^{9}}{9}+\frac {\left (a^{6} e^{7}+42 a^{5} b d \,e^{6}+315 a^{4} b^{2} d^{2} e^{5}+700 a^{3} b^{3} d^{3} e^{4}+525 a^{2} b^{4} d^{4} e^{3}+126 a \,b^{5} d^{5} e^{2}+7 b^{6} d^{6} e \right ) x^{8}}{8}+\frac {\left (7 a^{6} d \,e^{6}+126 a^{5} b \,d^{2} e^{5}+525 a^{4} b^{2} d^{3} e^{4}+700 a^{3} b^{3} d^{4} e^{3}+315 a^{2} b^{4} d^{5} e^{2}+42 a \,b^{5} d^{6} e +b^{6} d^{7}\right ) x^{7}}{7}+\frac {\left (21 a^{6} d^{2} e^{5}+210 a^{5} b \,d^{3} e^{4}+525 a^{4} b^{2} d^{4} e^{3}+420 a^{3} b^{3} d^{5} e^{2}+105 a^{2} b^{4} d^{6} e +6 a \,b^{5} d^{7}\right ) x^{6}}{6}+\frac {\left (35 a^{6} d^{3} e^{4}+210 a^{5} b \,d^{4} e^{3}+315 a^{4} b^{2} d^{5} e^{2}+140 a^{3} b^{3} d^{6} e +15 a^{2} b^{4} d^{7}\right ) x^{5}}{5}+\frac {\left (35 a^{6} d^{4} e^{3}+126 a^{5} b \,d^{5} e^{2}+105 a^{4} b^{2} d^{6} e +20 a^{3} b^{3} d^{7}\right ) x^{4}}{4}+\frac {\left (21 a^{6} d^{5} e^{2}+42 a^{5} b \,d^{6} e +15 a^{4} b^{2} d^{7}\right ) x^{3}}{3}+\frac {\left (7 a^{6} d^{6} e +6 a^{5} b \,d^{7}\right ) x^{2}}{2}+a^{6} d^{7} x\)	\(709\)
risch	\(\frac {7}{2} x^{2} a^{6} d^{6} e +3 x^{2} a^{5} b \,d^{7}+7 a^{6} d^{3} e^{4} x^{5}+3 a^{2} b^{4} d^{7} x^{5}+7 a^{6} d^{5} e^{2} x^{3}+5 a^{4} b^{2} d^{7} x^{3}+\frac {7}{2} x^{10} b^{6} d^{4} e^{3}+\frac {2}{3} x^{9} a^{5} b \,e^{7}+\frac {7}{4} x^{12} b^{6} d^{2} e^{5}+\frac {20}{11} x^{11} a^{3} b^{3} e^{7}+\frac {35}{11} x^{11} b^{6} d^{3} e^{4}+x^{7} a^{6} d \,e^{6}+\frac {5}{4} x^{12} a^{2} b^{4} e^{7}+\frac {1}{8} x^{8} a^{6} e^{7}+\frac {1}{7} x^{7} b^{6} d^{7}+\frac {70}{3} x^{9} a \,b^{5} d^{4} e^{3}+\frac {21}{4} x^{8} a^{5} b d \,e^{6}+\frac {315}{8} x^{8} a^{4} b^{2} d^{2} e^{5}+\frac {175}{2} x^{8} a^{3} b^{3} d^{3} e^{4}+\frac {525}{8} x^{8} a^{2} b^{4} d^{4} e^{3}+\frac {63}{4} x^{8} a \,b^{5} d^{5} e^{2}+18 x^{7} a^{5} b \,d^{2} e^{5}+75 x^{7} a^{4} b^{2} d^{3} e^{4}+21 x^{10} a \,b^{5} d^{3} e^{4}+\frac {7}{8} x^{8} b^{6} d^{6} e +100 x^{7} a^{3} b^{3} d^{4} e^{3}+45 x^{7} a^{2} b^{4} d^{5} e^{2}+6 x^{7} a \,b^{5} d^{6} e +35 x^{6} a^{5} b \,d^{3} e^{4}+\frac {175}{2} x^{6} a^{4} b^{2} d^{4} e^{3}+70 x^{6} a^{3} b^{3} d^{5} e^{2}+\frac {35}{2} x^{6} a^{2} b^{4} d^{6} e +\frac {63}{2} x^{4} a^{5} b \,d^{5} e^{2}+\frac {105}{4} x^{4} a^{4} b^{2} d^{6} e +42 a^{5} b \,d^{4} e^{3} x^{5}+63 a^{4} b^{2} d^{5} e^{2} x^{5}+28 a^{3} b^{3} d^{6} e \,x^{5}+14 a^{5} b \,d^{6} e \,x^{3}+\frac {35}{3} x^{9} a^{4} b^{2} d \,e^{6}+\frac {140}{3} x^{9} a^{3} b^{3} d^{2} e^{5}+\frac {1}{14} b^{6} e^{7} x^{14}+14 x^{10} a^{3} b^{3} d \,e^{6}+\frac {63}{2} x^{10} a^{2} b^{4} d^{2} e^{5}+\frac {7}{3} x^{9} b^{6} d^{5} e^{2}+a^{6} d^{7} x +\frac {6}{13} x^{13} a \,b^{5} e^{7}+\frac {7}{13} x^{13} b^{6} d \,e^{6}+\frac {7}{2} x^{12} a \,b^{5} d \,e^{6}+\frac {105}{11} x^{11} a^{2} b^{4} d \,e^{6}+\frac {126}{11} x^{11} a \,b^{5} d^{2} e^{5}+\frac {3}{2} x^{10} a^{4} b^{2} e^{7}+\frac {7}{2} x^{6} a^{6} d^{2} e^{5}+x^{6} a \,b^{5} d^{7}+\frac {35}{4} x^{4} a^{6} d^{4} e^{3}+5 x^{4} a^{3} b^{3} d^{7}+\frac {175}{3} x^{9} a^{2} b^{4} d^{3} e^{4}\)	\(799\)
parallelrisch	\(\frac {7}{2} x^{2} a^{6} d^{6} e +3 x^{2} a^{5} b \,d^{7}+7 a^{6} d^{3} e^{4} x^{5}+3 a^{2} b^{4} d^{7} x^{5}+7 a^{6} d^{5} e^{2} x^{3}+5 a^{4} b^{2} d^{7} x^{3}+\frac {7}{2} x^{10} b^{6} d^{4} e^{3}+\frac {2}{3} x^{9} a^{5} b \,e^{7}+\frac {7}{4} x^{12} b^{6} d^{2} e^{5}+\frac {20}{11} x^{11} a^{3} b^{3} e^{7}+\frac {35}{11} x^{11} b^{6} d^{3} e^{4}+x^{7} a^{6} d \,e^{6}+\frac {5}{4} x^{12} a^{2} b^{4} e^{7}+\frac {1}{8} x^{8} a^{6} e^{7}+\frac {1}{7} x^{7} b^{6} d^{7}+\frac {70}{3} x^{9} a \,b^{5} d^{4} e^{3}+\frac {21}{4} x^{8} a^{5} b d \,e^{6}+\frac {315}{8} x^{8} a^{4} b^{2} d^{2} e^{5}+\frac {175}{2} x^{8} a^{3} b^{3} d^{3} e^{4}+\frac {525}{8} x^{8} a^{2} b^{4} d^{4} e^{3}+\frac {63}{4} x^{8} a \,b^{5} d^{5} e^{2}+18 x^{7} a^{5} b \,d^{2} e^{5}+75 x^{7} a^{4} b^{2} d^{3} e^{4}+21 x^{10} a \,b^{5} d^{3} e^{4}+\frac {7}{8} x^{8} b^{6} d^{6} e +100 x^{7} a^{3} b^{3} d^{4} e^{3}+45 x^{7} a^{2} b^{4} d^{5} e^{2}+6 x^{7} a \,b^{5} d^{6} e +35 x^{6} a^{5} b \,d^{3} e^{4}+\frac {175}{2} x^{6} a^{4} b^{2} d^{4} e^{3}+70 x^{6} a^{3} b^{3} d^{5} e^{2}+\frac {35}{2} x^{6} a^{2} b^{4} d^{6} e +\frac {63}{2} x^{4} a^{5} b \,d^{5} e^{2}+\frac {105}{4} x^{4} a^{4} b^{2} d^{6} e +42 a^{5} b \,d^{4} e^{3} x^{5}+63 a^{4} b^{2} d^{5} e^{2} x^{5}+28 a^{3} b^{3} d^{6} e \,x^{5}+14 a^{5} b \,d^{6} e \,x^{3}+\frac {35}{3} x^{9} a^{4} b^{2} d \,e^{6}+\frac {140}{3} x^{9} a^{3} b^{3} d^{2} e^{5}+\frac {1}{14} b^{6} e^{7} x^{14}+14 x^{10} a^{3} b^{3} d \,e^{6}+\frac {63}{2} x^{10} a^{2} b^{4} d^{2} e^{5}+\frac {7}{3} x^{9} b^{6} d^{5} e^{2}+a^{6} d^{7} x +\frac {6}{13} x^{13} a \,b^{5} e^{7}+\frac {7}{13} x^{13} b^{6} d \,e^{6}+\frac {7}{2} x^{12} a \,b^{5} d \,e^{6}+\frac {105}{11} x^{11} a^{2} b^{4} d \,e^{6}+\frac {126}{11} x^{11} a \,b^{5} d^{2} e^{5}+\frac {3}{2} x^{10} a^{4} b^{2} e^{7}+\frac {7}{2} x^{6} a^{6} d^{2} e^{5}+x^{6} a \,b^{5} d^{7}+\frac {35}{4} x^{4} a^{6} d^{4} e^{3}+5 x^{4} a^{3} b^{3} d^{7}+\frac {175}{3} x^{9} a^{2} b^{4} d^{3} e^{4}\)	\(799\)
gosper	\(\frac {x \left (1716 b^{6} e^{7} x^{13}+11088 x^{12} a \,b^{5} e^{7}+12936 x^{12} b^{6} d \,e^{6}+30030 x^{11} a^{2} b^{4} e^{7}+84084 x^{11} a \,b^{5} d \,e^{6}+42042 x^{11} b^{6} d^{2} e^{5}+43680 x^{10} a^{3} b^{3} e^{7}+229320 x^{10} a^{2} b^{4} d \,e^{6}+275184 x^{10} a \,b^{5} d^{2} e^{5}+76440 x^{10} b^{6} d^{3} e^{4}+36036 x^{9} a^{4} b^{2} e^{7}+336336 x^{9} a^{3} b^{3} d \,e^{6}+756756 x^{9} a^{2} b^{4} d^{2} e^{5}+504504 x^{9} a \,b^{5} d^{3} e^{4}+84084 x^{9} b^{6} d^{4} e^{3}+16016 x^{8} a^{5} b \,e^{7}+280280 x^{8} a^{4} b^{2} d \,e^{6}+1121120 x^{8} a^{3} b^{3} d^{2} e^{5}+1401400 x^{8} a^{2} b^{4} d^{3} e^{4}+560560 x^{8} a \,b^{5} d^{4} e^{3}+56056 x^{8} b^{6} d^{5} e^{2}+3003 x^{7} a^{6} e^{7}+126126 x^{7} a^{5} b d \,e^{6}+945945 x^{7} a^{4} b^{2} d^{2} e^{5}+2102100 x^{7} a^{3} b^{3} d^{3} e^{4}+1576575 x^{7} a^{2} b^{4} d^{4} e^{3}+378378 x^{7} a \,b^{5} d^{5} e^{2}+21021 x^{7} b^{6} d^{6} e +24024 x^{6} a^{6} d \,e^{6}+432432 x^{6} a^{5} b \,d^{2} e^{5}+1801800 x^{6} a^{4} b^{2} d^{3} e^{4}+2402400 x^{6} a^{3} b^{3} d^{4} e^{3}+1081080 x^{6} a^{2} b^{4} d^{5} e^{2}+144144 x^{6} a \,b^{5} d^{6} e +3432 x^{6} b^{6} d^{7}+84084 x^{5} a^{6} d^{2} e^{5}+840840 x^{5} a^{5} b \,d^{3} e^{4}+2102100 x^{5} a^{4} b^{2} d^{4} e^{3}+1681680 x^{5} a^{3} b^{3} d^{5} e^{2}+420420 x^{5} a^{2} b^{4} d^{6} e +24024 x^{5} a \,b^{5} d^{7}+168168 a^{6} d^{3} e^{4} x^{4}+1009008 a^{5} b \,d^{4} e^{3} x^{4}+1513512 a^{4} b^{2} d^{5} e^{2} x^{4}+672672 a^{3} b^{3} d^{6} e \,x^{4}+72072 a^{2} b^{4} d^{7} x^{4}+210210 x^{3} a^{6} d^{4} e^{3}+756756 x^{3} a^{5} b \,d^{5} e^{2}+630630 x^{3} a^{4} b^{2} d^{6} e +120120 x^{3} a^{3} b^{3} d^{7}+168168 a^{6} d^{5} e^{2} x^{2}+336336 a^{5} b \,d^{6} e \,x^{2}+120120 a^{4} b^{2} d^{7} x^{2}+84084 x \,a^{6} d^{6} e +72072 x \,a^{5} b \,d^{7}+24024 a^{6} d^{7}\right )}{24024}\)	\(800\)

[In]

int((e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

a^6*d^7*x+(7/2*a^6*d^6*e+3*a^5*b*d^7)*x^2+(7*a^6*d^5*e^2+14*a^5*b*d^6*e+5*a^4*b^2*d^7)*x^3+(35/4*a^6*d^4*e^3+6
3/2*a^5*b*d^5*e^2+105/4*a^4*b^2*d^6*e+5*a^3*b^3*d^7)*x^4+(7*a^6*d^3*e^4+42*a^5*b*d^4*e^3+63*a^4*b^2*d^5*e^2+28
*a^3*b^3*d^6*e+3*a^2*b^4*d^7)*x^5+(7/2*a^6*d^2*e^5+35*a^5*b*d^3*e^4+175/2*a^4*b^2*d^4*e^3+70*a^3*b^3*d^5*e^2+3
5/2*a^2*b^4*d^6*e+a*b^5*d^7)*x^6+(a^6*d*e^6+18*a^5*b*d^2*e^5+75*a^4*b^2*d^3*e^4+100*a^3*b^3*d^4*e^3+45*a^2*b^4
*d^5*e^2+6*a*b^5*d^6*e+1/7*b^6*d^7)*x^7+(1/8*a^6*e^7+21/4*a^5*b*d*e^6+315/8*a^4*b^2*d^2*e^5+175/2*a^3*b^3*d^3*
e^4+525/8*a^2*b^4*d^4*e^3+63/4*a*b^5*d^5*e^2+7/8*b^6*d^6*e)*x^8+(2/3*a^5*b*e^7+35/3*a^4*b^2*d*e^6+140/3*a^3*b^
3*d^2*e^5+175/3*a^2*b^4*d^3*e^4+70/3*a*b^5*d^4*e^3+7/3*b^6*d^5*e^2)*x^9+(3/2*a^4*b^2*e^7+14*a^3*b^3*d*e^6+63/2
*a^2*b^4*d^2*e^5+21*a*b^5*d^3*e^4+7/2*b^6*d^4*e^3)*x^10+(20/11*a^3*b^3*e^7+105/11*a^2*b^4*d*e^6+126/11*a*b^5*d
^2*e^5+35/11*b^6*d^3*e^4)*x^11+(5/4*a^2*b^4*e^7+7/2*a*b^5*d*e^6+7/4*b^6*d^2*e^5)*x^12+(6/13*a*b^5*e^7+7/13*b^6
*d*e^6)*x^13+1/14*b^6*e^7*x^14

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (159) = 318\).

Time = 0.30 (sec) , antiderivative size = 706, normalized size of antiderivative = 4.08 \[ \int (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{14} \, b^{6} e^{7} x^{14} + a^{6} d^{7} x + \frac {1}{13} \, {\left (7 \, b^{6} d e^{6} + 6 \, a b^{5} e^{7}\right )} x^{13} + \frac {1}{4} \, {\left (7 \, b^{6} d^{2} e^{5} + 14 \, a b^{5} d e^{6} + 5 \, a^{2} b^{4} e^{7}\right )} x^{12} + \frac {1}{11} \, {\left (35 \, b^{6} d^{3} e^{4} + 126 \, a b^{5} d^{2} e^{5} + 105 \, a^{2} b^{4} d e^{6} + 20 \, a^{3} b^{3} e^{7}\right )} x^{11} + \frac {1}{2} \, {\left (7 \, b^{6} d^{4} e^{3} + 42 \, a b^{5} d^{3} e^{4} + 63 \, a^{2} b^{4} d^{2} e^{5} + 28 \, a^{3} b^{3} d e^{6} + 3 \, a^{4} b^{2} e^{7}\right )} x^{10} + \frac {1}{3} \, {\left (7 \, b^{6} d^{5} e^{2} + 70 \, a b^{5} d^{4} e^{3} + 175 \, a^{2} b^{4} d^{3} e^{4} + 140 \, a^{3} b^{3} d^{2} e^{5} + 35 \, a^{4} b^{2} d e^{6} + 2 \, a^{5} b e^{7}\right )} x^{9} + \frac {1}{8} \, {\left (7 \, b^{6} d^{6} e + 126 \, a b^{5} d^{5} e^{2} + 525 \, a^{2} b^{4} d^{4} e^{3} + 700 \, a^{3} b^{3} d^{3} e^{4} + 315 \, a^{4} b^{2} d^{2} e^{5} + 42 \, a^{5} b d e^{6} + a^{6} e^{7}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{7} + 42 \, a b^{5} d^{6} e + 315 \, a^{2} b^{4} d^{5} e^{2} + 700 \, a^{3} b^{3} d^{4} e^{3} + 525 \, a^{4} b^{2} d^{3} e^{4} + 126 \, a^{5} b d^{2} e^{5} + 7 \, a^{6} d e^{6}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d^{7} + 35 \, a^{2} b^{4} d^{6} e + 140 \, a^{3} b^{3} d^{5} e^{2} + 175 \, a^{4} b^{2} d^{4} e^{3} + 70 \, a^{5} b d^{3} e^{4} + 7 \, a^{6} d^{2} e^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{7} + 28 \, a^{3} b^{3} d^{6} e + 63 \, a^{4} b^{2} d^{5} e^{2} + 42 \, a^{5} b d^{4} e^{3} + 7 \, a^{6} d^{3} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} d^{7} + 105 \, a^{4} b^{2} d^{6} e + 126 \, a^{5} b d^{5} e^{2} + 35 \, a^{6} d^{4} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{7} + 14 \, a^{5} b d^{6} e + 7 \, a^{6} d^{5} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d^{7} + 7 \, a^{6} d^{6} e\right )} x^{2} \]

[In]

integrate((e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/14*b^6*e^7*x^14 + a^6*d^7*x + 1/13*(7*b^6*d*e^6 + 6*a*b^5*e^7)*x^13 + 1/4*(7*b^6*d^2*e^5 + 14*a*b^5*d*e^6 +
5*a^2*b^4*e^7)*x^12 + 1/11*(35*b^6*d^3*e^4 + 126*a*b^5*d^2*e^5 + 105*a^2*b^4*d*e^6 + 20*a^3*b^3*e^7)*x^11 + 1/
2*(7*b^6*d^4*e^3 + 42*a*b^5*d^3*e^4 + 63*a^2*b^4*d^2*e^5 + 28*a^3*b^3*d*e^6 + 3*a^4*b^2*e^7)*x^10 + 1/3*(7*b^6
*d^5*e^2 + 70*a*b^5*d^4*e^3 + 175*a^2*b^4*d^3*e^4 + 140*a^3*b^3*d^2*e^5 + 35*a^4*b^2*d*e^6 + 2*a^5*b*e^7)*x^9
+ 1/8*(7*b^6*d^6*e + 126*a*b^5*d^5*e^2 + 525*a^2*b^4*d^4*e^3 + 700*a^3*b^3*d^3*e^4 + 315*a^4*b^2*d^2*e^5 + 42*
a^5*b*d*e^6 + a^6*e^7)*x^8 + 1/7*(b^6*d^7 + 42*a*b^5*d^6*e + 315*a^2*b^4*d^5*e^2 + 700*a^3*b^3*d^4*e^3 + 525*a
^4*b^2*d^3*e^4 + 126*a^5*b*d^2*e^5 + 7*a^6*d*e^6)*x^7 + 1/2*(2*a*b^5*d^7 + 35*a^2*b^4*d^6*e + 140*a^3*b^3*d^5*
e^2 + 175*a^4*b^2*d^4*e^3 + 70*a^5*b*d^3*e^4 + 7*a^6*d^2*e^5)*x^6 + (3*a^2*b^4*d^7 + 28*a^3*b^3*d^6*e + 63*a^4
*b^2*d^5*e^2 + 42*a^5*b*d^4*e^3 + 7*a^6*d^3*e^4)*x^5 + 1/4*(20*a^3*b^3*d^7 + 105*a^4*b^2*d^6*e + 126*a^5*b*d^5
*e^2 + 35*a^6*d^4*e^3)*x^4 + (5*a^4*b^2*d^7 + 14*a^5*b*d^6*e + 7*a^6*d^5*e^2)*x^3 + 1/2*(6*a^5*b*d^7 + 7*a^6*d
^6*e)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 796 vs. \(2 (158) = 316\).

Time = 0.07 (sec) , antiderivative size = 796, normalized size of antiderivative = 4.60 \[ \int (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{6} d^{7} x + \frac {b^{6} e^{7} x^{14}}{14} + x^{13} \cdot \left (\frac {6 a b^{5} e^{7}}{13} + \frac {7 b^{6} d e^{6}}{13}\right ) + x^{12} \cdot \left (\frac {5 a^{2} b^{4} e^{7}}{4} + \frac {7 a b^{5} d e^{6}}{2} + \frac {7 b^{6} d^{2} e^{5}}{4}\right ) + x^{11} \cdot \left (\frac {20 a^{3} b^{3} e^{7}}{11} + \frac {105 a^{2} b^{4} d e^{6}}{11} + \frac {126 a b^{5} d^{2} e^{5}}{11} + \frac {35 b^{6} d^{3} e^{4}}{11}\right ) + x^{10} \cdot \left (\frac {3 a^{4} b^{2} e^{7}}{2} + 14 a^{3} b^{3} d e^{6} + \frac {63 a^{2} b^{4} d^{2} e^{5}}{2} + 21 a b^{5} d^{3} e^{4} + \frac {7 b^{6} d^{4} e^{3}}{2}\right ) + x^{9} \cdot \left (\frac {2 a^{5} b e^{7}}{3} + \frac {35 a^{4} b^{2} d e^{6}}{3} + \frac {140 a^{3} b^{3} d^{2} e^{5}}{3} + \frac {175 a^{2} b^{4} d^{3} e^{4}}{3} + \frac {70 a b^{5} d^{4} e^{3}}{3} + \frac {7 b^{6} d^{5} e^{2}}{3}\right ) + x^{8} \left (\frac {a^{6} e^{7}}{8} + \frac {21 a^{5} b d e^{6}}{4} + \frac {315 a^{4} b^{2} d^{2} e^{5}}{8} + \frac {175 a^{3} b^{3} d^{3} e^{4}}{2} + \frac {525 a^{2} b^{4} d^{4} e^{3}}{8} + \frac {63 a b^{5} d^{5} e^{2}}{4} + \frac {7 b^{6} d^{6} e}{8}\right ) + x^{7} \left (a^{6} d e^{6} + 18 a^{5} b d^{2} e^{5} + 75 a^{4} b^{2} d^{3} e^{4} + 100 a^{3} b^{3} d^{4} e^{3} + 45 a^{2} b^{4} d^{5} e^{2} + 6 a b^{5} d^{6} e + \frac {b^{6} d^{7}}{7}\right ) + x^{6} \cdot \left (\frac {7 a^{6} d^{2} e^{5}}{2} + 35 a^{5} b d^{3} e^{4} + \frac {175 a^{4} b^{2} d^{4} e^{3}}{2} + 70 a^{3} b^{3} d^{5} e^{2} + \frac {35 a^{2} b^{4} d^{6} e}{2} + a b^{5} d^{7}\right ) + x^{5} \cdot \left (7 a^{6} d^{3} e^{4} + 42 a^{5} b d^{4} e^{3} + 63 a^{4} b^{2} d^{5} e^{2} + 28 a^{3} b^{3} d^{6} e + 3 a^{2} b^{4} d^{7}\right ) + x^{4} \cdot \left (\frac {35 a^{6} d^{4} e^{3}}{4} + \frac {63 a^{5} b d^{5} e^{2}}{2} + \frac {105 a^{4} b^{2} d^{6} e}{4} + 5 a^{3} b^{3} d^{7}\right ) + x^{3} \cdot \left (7 a^{6} d^{5} e^{2} + 14 a^{5} b d^{6} e + 5 a^{4} b^{2} d^{7}\right ) + x^{2} \cdot \left (\frac {7 a^{6} d^{6} e}{2} + 3 a^{5} b d^{7}\right ) \]

[In]

integrate((e*x+d)**7*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**6*d**7*x + b**6*e**7*x**14/14 + x**13*(6*a*b**5*e**7/13 + 7*b**6*d*e**6/13) + x**12*(5*a**2*b**4*e**7/4 + 7
*a*b**5*d*e**6/2 + 7*b**6*d**2*e**5/4) + x**11*(20*a**3*b**3*e**7/11 + 105*a**2*b**4*d*e**6/11 + 126*a*b**5*d*
*2*e**5/11 + 35*b**6*d**3*e**4/11) + x**10*(3*a**4*b**2*e**7/2 + 14*a**3*b**3*d*e**6 + 63*a**2*b**4*d**2*e**5/
2 + 21*a*b**5*d**3*e**4 + 7*b**6*d**4*e**3/2) + x**9*(2*a**5*b*e**7/3 + 35*a**4*b**2*d*e**6/3 + 140*a**3*b**3*
d**2*e**5/3 + 175*a**2*b**4*d**3*e**4/3 + 70*a*b**5*d**4*e**3/3 + 7*b**6*d**5*e**2/3) + x**8*(a**6*e**7/8 + 21
*a**5*b*d*e**6/4 + 315*a**4*b**2*d**2*e**5/8 + 175*a**3*b**3*d**3*e**4/2 + 525*a**2*b**4*d**4*e**3/8 + 63*a*b*
*5*d**5*e**2/4 + 7*b**6*d**6*e/8) + x**7*(a**6*d*e**6 + 18*a**5*b*d**2*e**5 + 75*a**4*b**2*d**3*e**4 + 100*a**
3*b**3*d**4*e**3 + 45*a**2*b**4*d**5*e**2 + 6*a*b**5*d**6*e + b**6*d**7/7) + x**6*(7*a**6*d**2*e**5/2 + 35*a**
5*b*d**3*e**4 + 175*a**4*b**2*d**4*e**3/2 + 70*a**3*b**3*d**5*e**2 + 35*a**2*b**4*d**6*e/2 + a*b**5*d**7) + x*
*5*(7*a**6*d**3*e**4 + 42*a**5*b*d**4*e**3 + 63*a**4*b**2*d**5*e**2 + 28*a**3*b**3*d**6*e + 3*a**2*b**4*d**7)
+ x**4*(35*a**6*d**4*e**3/4 + 63*a**5*b*d**5*e**2/2 + 105*a**4*b**2*d**6*e/4 + 5*a**3*b**3*d**7) + x**3*(7*a**
6*d**5*e**2 + 14*a**5*b*d**6*e + 5*a**4*b**2*d**7) + x**2*(7*a**6*d**6*e/2 + 3*a**5*b*d**7)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (159) = 318\).

Time = 0.22 (sec) , antiderivative size = 706, normalized size of antiderivative = 4.08 \[ \int (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{14} \, b^{6} e^{7} x^{14} + a^{6} d^{7} x + \frac {1}{13} \, {\left (7 \, b^{6} d e^{6} + 6 \, a b^{5} e^{7}\right )} x^{13} + \frac {1}{4} \, {\left (7 \, b^{6} d^{2} e^{5} + 14 \, a b^{5} d e^{6} + 5 \, a^{2} b^{4} e^{7}\right )} x^{12} + \frac {1}{11} \, {\left (35 \, b^{6} d^{3} e^{4} + 126 \, a b^{5} d^{2} e^{5} + 105 \, a^{2} b^{4} d e^{6} + 20 \, a^{3} b^{3} e^{7}\right )} x^{11} + \frac {1}{2} \, {\left (7 \, b^{6} d^{4} e^{3} + 42 \, a b^{5} d^{3} e^{4} + 63 \, a^{2} b^{4} d^{2} e^{5} + 28 \, a^{3} b^{3} d e^{6} + 3 \, a^{4} b^{2} e^{7}\right )} x^{10} + \frac {1}{3} \, {\left (7 \, b^{6} d^{5} e^{2} + 70 \, a b^{5} d^{4} e^{3} + 175 \, a^{2} b^{4} d^{3} e^{4} + 140 \, a^{3} b^{3} d^{2} e^{5} + 35 \, a^{4} b^{2} d e^{6} + 2 \, a^{5} b e^{7}\right )} x^{9} + \frac {1}{8} \, {\left (7 \, b^{6} d^{6} e + 126 \, a b^{5} d^{5} e^{2} + 525 \, a^{2} b^{4} d^{4} e^{3} + 700 \, a^{3} b^{3} d^{3} e^{4} + 315 \, a^{4} b^{2} d^{2} e^{5} + 42 \, a^{5} b d e^{6} + a^{6} e^{7}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{7} + 42 \, a b^{5} d^{6} e + 315 \, a^{2} b^{4} d^{5} e^{2} + 700 \, a^{3} b^{3} d^{4} e^{3} + 525 \, a^{4} b^{2} d^{3} e^{4} + 126 \, a^{5} b d^{2} e^{5} + 7 \, a^{6} d e^{6}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d^{7} + 35 \, a^{2} b^{4} d^{6} e + 140 \, a^{3} b^{3} d^{5} e^{2} + 175 \, a^{4} b^{2} d^{4} e^{3} + 70 \, a^{5} b d^{3} e^{4} + 7 \, a^{6} d^{2} e^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{7} + 28 \, a^{3} b^{3} d^{6} e + 63 \, a^{4} b^{2} d^{5} e^{2} + 42 \, a^{5} b d^{4} e^{3} + 7 \, a^{6} d^{3} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} d^{7} + 105 \, a^{4} b^{2} d^{6} e + 126 \, a^{5} b d^{5} e^{2} + 35 \, a^{6} d^{4} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{7} + 14 \, a^{5} b d^{6} e + 7 \, a^{6} d^{5} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d^{7} + 7 \, a^{6} d^{6} e\right )} x^{2} \]

[In]

integrate((e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/14*b^6*e^7*x^14 + a^6*d^7*x + 1/13*(7*b^6*d*e^6 + 6*a*b^5*e^7)*x^13 + 1/4*(7*b^6*d^2*e^5 + 14*a*b^5*d*e^6 +
5*a^2*b^4*e^7)*x^12 + 1/11*(35*b^6*d^3*e^4 + 126*a*b^5*d^2*e^5 + 105*a^2*b^4*d*e^6 + 20*a^3*b^3*e^7)*x^11 + 1/
2*(7*b^6*d^4*e^3 + 42*a*b^5*d^3*e^4 + 63*a^2*b^4*d^2*e^5 + 28*a^3*b^3*d*e^6 + 3*a^4*b^2*e^7)*x^10 + 1/3*(7*b^6
*d^5*e^2 + 70*a*b^5*d^4*e^3 + 175*a^2*b^4*d^3*e^4 + 140*a^3*b^3*d^2*e^5 + 35*a^4*b^2*d*e^6 + 2*a^5*b*e^7)*x^9
+ 1/8*(7*b^6*d^6*e + 126*a*b^5*d^5*e^2 + 525*a^2*b^4*d^4*e^3 + 700*a^3*b^3*d^3*e^4 + 315*a^4*b^2*d^2*e^5 + 42*
a^5*b*d*e^6 + a^6*e^7)*x^8 + 1/7*(b^6*d^7 + 42*a*b^5*d^6*e + 315*a^2*b^4*d^5*e^2 + 700*a^3*b^3*d^4*e^3 + 525*a
^4*b^2*d^3*e^4 + 126*a^5*b*d^2*e^5 + 7*a^6*d*e^6)*x^7 + 1/2*(2*a*b^5*d^7 + 35*a^2*b^4*d^6*e + 140*a^3*b^3*d^5*
e^2 + 175*a^4*b^2*d^4*e^3 + 70*a^5*b*d^3*e^4 + 7*a^6*d^2*e^5)*x^6 + (3*a^2*b^4*d^7 + 28*a^3*b^3*d^6*e + 63*a^4
*b^2*d^5*e^2 + 42*a^5*b*d^4*e^3 + 7*a^6*d^3*e^4)*x^5 + 1/4*(20*a^3*b^3*d^7 + 105*a^4*b^2*d^6*e + 126*a^5*b*d^5
*e^2 + 35*a^6*d^4*e^3)*x^4 + (5*a^4*b^2*d^7 + 14*a^5*b*d^6*e + 7*a^6*d^5*e^2)*x^3 + 1/2*(6*a^5*b*d^7 + 7*a^6*d
^6*e)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 798 vs. \(2 (159) = 318\).

Time = 0.26 (sec) , antiderivative size = 798, normalized size of antiderivative = 4.61 \[ \int (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{14} \, b^{6} e^{7} x^{14} + \frac {7}{13} \, b^{6} d e^{6} x^{13} + \frac {6}{13} \, a b^{5} e^{7} x^{13} + \frac {7}{4} \, b^{6} d^{2} e^{5} x^{12} + \frac {7}{2} \, a b^{5} d e^{6} x^{12} + \frac {5}{4} \, a^{2} b^{4} e^{7} x^{12} + \frac {35}{11} \, b^{6} d^{3} e^{4} x^{11} + \frac {126}{11} \, a b^{5} d^{2} e^{5} x^{11} + \frac {105}{11} \, a^{2} b^{4} d e^{6} x^{11} + \frac {20}{11} \, a^{3} b^{3} e^{7} x^{11} + \frac {7}{2} \, b^{6} d^{4} e^{3} x^{10} + 21 \, a b^{5} d^{3} e^{4} x^{10} + \frac {63}{2} \, a^{2} b^{4} d^{2} e^{5} x^{10} + 14 \, a^{3} b^{3} d e^{6} x^{10} + \frac {3}{2} \, a^{4} b^{2} e^{7} x^{10} + \frac {7}{3} \, b^{6} d^{5} e^{2} x^{9} + \frac {70}{3} \, a b^{5} d^{4} e^{3} x^{9} + \frac {175}{3} \, a^{2} b^{4} d^{3} e^{4} x^{9} + \frac {140}{3} \, a^{3} b^{3} d^{2} e^{5} x^{9} + \frac {35}{3} \, a^{4} b^{2} d e^{6} x^{9} + \frac {2}{3} \, a^{5} b e^{7} x^{9} + \frac {7}{8} \, b^{6} d^{6} e x^{8} + \frac {63}{4} \, a b^{5} d^{5} e^{2} x^{8} + \frac {525}{8} \, a^{2} b^{4} d^{4} e^{3} x^{8} + \frac {175}{2} \, a^{3} b^{3} d^{3} e^{4} x^{8} + \frac {315}{8} \, a^{4} b^{2} d^{2} e^{5} x^{8} + \frac {21}{4} \, a^{5} b d e^{6} x^{8} + \frac {1}{8} \, a^{6} e^{7} x^{8} + \frac {1}{7} \, b^{6} d^{7} x^{7} + 6 \, a b^{5} d^{6} e x^{7} + 45 \, a^{2} b^{4} d^{5} e^{2} x^{7} + 100 \, a^{3} b^{3} d^{4} e^{3} x^{7} + 75 \, a^{4} b^{2} d^{3} e^{4} x^{7} + 18 \, a^{5} b d^{2} e^{5} x^{7} + a^{6} d e^{6} x^{7} + a b^{5} d^{7} x^{6} + \frac {35}{2} \, a^{2} b^{4} d^{6} e x^{6} + 70 \, a^{3} b^{3} d^{5} e^{2} x^{6} + \frac {175}{2} \, a^{4} b^{2} d^{4} e^{3} x^{6} + 35 \, a^{5} b d^{3} e^{4} x^{6} + \frac {7}{2} \, a^{6} d^{2} e^{5} x^{6} + 3 \, a^{2} b^{4} d^{7} x^{5} + 28 \, a^{3} b^{3} d^{6} e x^{5} + 63 \, a^{4} b^{2} d^{5} e^{2} x^{5} + 42 \, a^{5} b d^{4} e^{3} x^{5} + 7 \, a^{6} d^{3} e^{4} x^{5} + 5 \, a^{3} b^{3} d^{7} x^{4} + \frac {105}{4} \, a^{4} b^{2} d^{6} e x^{4} + \frac {63}{2} \, a^{5} b d^{5} e^{2} x^{4} + \frac {35}{4} \, a^{6} d^{4} e^{3} x^{4} + 5 \, a^{4} b^{2} d^{7} x^{3} + 14 \, a^{5} b d^{6} e x^{3} + 7 \, a^{6} d^{5} e^{2} x^{3} + 3 \, a^{5} b d^{7} x^{2} + \frac {7}{2} \, a^{6} d^{6} e x^{2} + a^{6} d^{7} x \]

[In]

integrate((e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/14*b^6*e^7*x^14 + 7/13*b^6*d*e^6*x^13 + 6/13*a*b^5*e^7*x^13 + 7/4*b^6*d^2*e^5*x^12 + 7/2*a*b^5*d*e^6*x^12 +
5/4*a^2*b^4*e^7*x^12 + 35/11*b^6*d^3*e^4*x^11 + 126/11*a*b^5*d^2*e^5*x^11 + 105/11*a^2*b^4*d*e^6*x^11 + 20/11*
a^3*b^3*e^7*x^11 + 7/2*b^6*d^4*e^3*x^10 + 21*a*b^5*d^3*e^4*x^10 + 63/2*a^2*b^4*d^2*e^5*x^10 + 14*a^3*b^3*d*e^6
*x^10 + 3/2*a^4*b^2*e^7*x^10 + 7/3*b^6*d^5*e^2*x^9 + 70/3*a*b^5*d^4*e^3*x^9 + 175/3*a^2*b^4*d^3*e^4*x^9 + 140/
3*a^3*b^3*d^2*e^5*x^9 + 35/3*a^4*b^2*d*e^6*x^9 + 2/3*a^5*b*e^7*x^9 + 7/8*b^6*d^6*e*x^8 + 63/4*a*b^5*d^5*e^2*x^
8 + 525/8*a^2*b^4*d^4*e^3*x^8 + 175/2*a^3*b^3*d^3*e^4*x^8 + 315/8*a^4*b^2*d^2*e^5*x^8 + 21/4*a^5*b*d*e^6*x^8 +
1/8*a^6*e^7*x^8 + 1/7*b^6*d^7*x^7 + 6*a*b^5*d^6*e*x^7 + 45*a^2*b^4*d^5*e^2*x^7 + 100*a^3*b^3*d^4*e^3*x^7 + 75
*a^4*b^2*d^3*e^4*x^7 + 18*a^5*b*d^2*e^5*x^7 + a^6*d*e^6*x^7 + a*b^5*d^7*x^6 + 35/2*a^2*b^4*d^6*e*x^6 + 70*a^3*
b^3*d^5*e^2*x^6 + 175/2*a^4*b^2*d^4*e^3*x^6 + 35*a^5*b*d^3*e^4*x^6 + 7/2*a^6*d^2*e^5*x^6 + 3*a^2*b^4*d^7*x^5 +
28*a^3*b^3*d^6*e*x^5 + 63*a^4*b^2*d^5*e^2*x^5 + 42*a^5*b*d^4*e^3*x^5 + 7*a^6*d^3*e^4*x^5 + 5*a^3*b^3*d^7*x^4
+ 105/4*a^4*b^2*d^6*e*x^4 + 63/2*a^5*b*d^5*e^2*x^4 + 35/4*a^6*d^4*e^3*x^4 + 5*a^4*b^2*d^7*x^3 + 14*a^5*b*d^6*e
*x^3 + 7*a^6*d^5*e^2*x^3 + 3*a^5*b*d^7*x^2 + 7/2*a^6*d^6*e*x^2 + a^6*d^7*x

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 683, normalized size of antiderivative = 3.95 \[ \int (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^5\,\left (7\,a^6\,d^3\,e^4+42\,a^5\,b\,d^4\,e^3+63\,a^4\,b^2\,d^5\,e^2+28\,a^3\,b^3\,d^6\,e+3\,a^2\,b^4\,d^7\right )+x^{10}\,\left (\frac {3\,a^4\,b^2\,e^7}{2}+14\,a^3\,b^3\,d\,e^6+\frac {63\,a^2\,b^4\,d^2\,e^5}{2}+21\,a\,b^5\,d^3\,e^4+\frac {7\,b^6\,d^4\,e^3}{2}\right )+x^6\,\left (\frac {7\,a^6\,d^2\,e^5}{2}+35\,a^5\,b\,d^3\,e^4+\frac {175\,a^4\,b^2\,d^4\,e^3}{2}+70\,a^3\,b^3\,d^5\,e^2+\frac {35\,a^2\,b^4\,d^6\,e}{2}+a\,b^5\,d^7\right )+x^9\,\left (\frac {2\,a^5\,b\,e^7}{3}+\frac {35\,a^4\,b^2\,d\,e^6}{3}+\frac {140\,a^3\,b^3\,d^2\,e^5}{3}+\frac {175\,a^2\,b^4\,d^3\,e^4}{3}+\frac {70\,a\,b^5\,d^4\,e^3}{3}+\frac {7\,b^6\,d^5\,e^2}{3}\right )+x^7\,\left (a^6\,d\,e^6+18\,a^5\,b\,d^2\,e^5+75\,a^4\,b^2\,d^3\,e^4+100\,a^3\,b^3\,d^4\,e^3+45\,a^2\,b^4\,d^5\,e^2+6\,a\,b^5\,d^6\,e+\frac {b^6\,d^7}{7}\right )+x^8\,\left (\frac {a^6\,e^7}{8}+\frac {21\,a^5\,b\,d\,e^6}{4}+\frac {315\,a^4\,b^2\,d^2\,e^5}{8}+\frac {175\,a^3\,b^3\,d^3\,e^4}{2}+\frac {525\,a^2\,b^4\,d^4\,e^3}{8}+\frac {63\,a\,b^5\,d^5\,e^2}{4}+\frac {7\,b^6\,d^6\,e}{8}\right )+x^4\,\left (\frac {35\,a^6\,d^4\,e^3}{4}+\frac {63\,a^5\,b\,d^5\,e^2}{2}+\frac {105\,a^4\,b^2\,d^6\,e}{4}+5\,a^3\,b^3\,d^7\right )+x^{11}\,\left (\frac {20\,a^3\,b^3\,e^7}{11}+\frac {105\,a^2\,b^4\,d\,e^6}{11}+\frac {126\,a\,b^5\,d^2\,e^5}{11}+\frac {35\,b^6\,d^3\,e^4}{11}\right )+a^6\,d^7\,x+\frac {b^6\,e^7\,x^{14}}{14}+\frac {a^5\,d^6\,x^2\,\left (7\,a\,e+6\,b\,d\right )}{2}+\frac {b^5\,e^6\,x^{13}\,\left (6\,a\,e+7\,b\,d\right )}{13}+a^4\,d^5\,x^3\,\left (7\,a^2\,e^2+14\,a\,b\,d\,e+5\,b^2\,d^2\right )+\frac {b^4\,e^5\,x^{12}\,\left (5\,a^2\,e^2+14\,a\,b\,d\,e+7\,b^2\,d^2\right )}{4} \]

[In]

int((d + e*x)^7*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

x^5*(3*a^2*b^4*d^7 + 7*a^6*d^3*e^4 + 28*a^3*b^3*d^6*e + 42*a^5*b*d^4*e^3 + 63*a^4*b^2*d^5*e^2) + x^10*((3*a^4*
b^2*e^7)/2 + (7*b^6*d^4*e^3)/2 + 21*a*b^5*d^3*e^4 + 14*a^3*b^3*d*e^6 + (63*a^2*b^4*d^2*e^5)/2) + x^6*(a*b^5*d^
7 + (7*a^6*d^2*e^5)/2 + (35*a^2*b^4*d^6*e)/2 + 35*a^5*b*d^3*e^4 + 70*a^3*b^3*d^5*e^2 + (175*a^4*b^2*d^4*e^3)/2
) + x^9*((2*a^5*b*e^7)/3 + (7*b^6*d^5*e^2)/3 + (70*a*b^5*d^4*e^3)/3 + (35*a^4*b^2*d*e^6)/3 + (175*a^2*b^4*d^3*
e^4)/3 + (140*a^3*b^3*d^2*e^5)/3) + x^7*((b^6*d^7)/7 + a^6*d*e^6 + 18*a^5*b*d^2*e^5 + 45*a^2*b^4*d^5*e^2 + 100
*a^3*b^3*d^4*e^3 + 75*a^4*b^2*d^3*e^4 + 6*a*b^5*d^6*e) + x^8*((a^6*e^7)/8 + (7*b^6*d^6*e)/8 + (63*a*b^5*d^5*e^
2)/4 + (525*a^2*b^4*d^4*e^3)/8 + (175*a^3*b^3*d^3*e^4)/2 + (315*a^4*b^2*d^2*e^5)/8 + (21*a^5*b*d*e^6)/4) + x^4
*(5*a^3*b^3*d^7 + (35*a^6*d^4*e^3)/4 + (105*a^4*b^2*d^6*e)/4 + (63*a^5*b*d^5*e^2)/2) + x^11*((20*a^3*b^3*e^7)/
11 + (35*b^6*d^3*e^4)/11 + (126*a*b^5*d^2*e^5)/11 + (105*a^2*b^4*d*e^6)/11) + a^6*d^7*x + (b^6*e^7*x^14)/14 +
(a^5*d^6*x^2*(7*a*e + 6*b*d))/2 + (b^5*e^6*x^13*(6*a*e + 7*b*d))/13 + a^4*d^5*x^3*(7*a^2*e^2 + 5*b^2*d^2 + 14*
a*b*d*e) + (b^4*e^5*x^12*(5*a^2*e^2 + 7*b^2*d^2 + 14*a*b*d*e))/4