\(\int (d+e x)^6 (a^2+2 a b x+b^2 x^2)^3 \, dx\) [1483]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.24 (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} \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {e^5 (a+b x)^{12} (b d-a e)}{2 b^7}+\frac {15 e^4 (a+b x)^{11} (b d-a e)^2}{11 b^7}+\frac {2 e^3 (a+b x)^{10} (b d-a e)^3}{b^7}+\frac {5 e^2 (a+b x)^9 (b d-a e)^4}{3 b^7}+\frac {3 e (a+b x)^8 (b d-a e)^5}{4 b^7}+\frac {(a+b x)^7 (b d-a e)^6}{7 b^7}+\frac {e^6 (a+b x)^{13}}{13 b^7} \]

[In]

Int[(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(573\) vs. \(2(171)=342\).

Time = 0.05 (sec) , antiderivative size = 573, normalized size of antiderivative = 3.35 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^6 d^6 x+3 a^5 d^5 (b d+a e) x^2+a^4 d^4 \left (5 b^2 d^2+12 a b d e+5 a^2 e^2\right ) x^3+\frac {5}{2} a^3 d^3 \left (2 b^3 d^3+9 a b^2 d^2 e+9 a^2 b d e^2+2 a^3 e^3\right ) x^4+3 a^2 d^2 \left (b^4 d^4+8 a b^3 d^3 e+15 a^2 b^2 d^2 e^2+8 a^3 b d e^3+a^4 e^4\right ) x^5+a d \left (b^5 d^5+15 a b^4 d^4 e+50 a^2 b^3 d^3 e^2+50 a^3 b^2 d^2 e^3+15 a^4 b d e^4+a^5 e^5\right ) x^6+\frac {1}{7} \left (b^6 d^6+36 a b^5 d^5 e+225 a^2 b^4 d^4 e^2+400 a^3 b^3 d^3 e^3+225 a^4 b^2 d^2 e^4+36 a^5 b d e^5+a^6 e^6\right ) x^7+\frac {3}{4} b e \left (b^5 d^5+15 a b^4 d^4 e+50 a^2 b^3 d^3 e^2+50 a^3 b^2 d^2 e^3+15 a^4 b d e^4+a^5 e^5\right ) x^8+\frac {5}{3} b^2 e^2 \left (b^4 d^4+8 a b^3 d^3 e+15 a^2 b^2 d^2 e^2+8 a^3 b d e^3+a^4 e^4\right ) x^9+b^3 e^3 \left (2 b^3 d^3+9 a b^2 d^2 e+9 a^2 b d e^2+2 a^3 e^3\right ) x^{10}+\frac {3}{11} b^4 e^4 \left (5 b^2 d^2+12 a b d e+5 a^2 e^2\right ) x^{11}+\frac {1}{2} b^5 e^5 (b d+a e) x^{12}+\frac {1}{13} b^6 e^6 x^{13} \]

[In]

Integrate[(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

a^6*d^6*x + 3*a^5*d^5*(b*d + a*e)*x^2 + a^4*d^4*(5*b^2*d^2 + 12*a*b*d*e + 5*a^2*e^2)*x^3 + (5*a^3*d^3*(2*b^3*d
^3 + 9*a*b^2*d^2*e + 9*a^2*b*d*e^2 + 2*a^3*e^3)*x^4)/2 + 3*a^2*d^2*(b^4*d^4 + 8*a*b^3*d^3*e + 15*a^2*b^2*d^2*e
^2 + 8*a^3*b*d*e^3 + a^4*e^4)*x^5 + a*d*(b^5*d^5 + 15*a*b^4*d^4*e + 50*a^2*b^3*d^3*e^2 + 50*a^3*b^2*d^2*e^3 +
15*a^4*b*d*e^4 + a^5*e^5)*x^6 + ((b^6*d^6 + 36*a*b^5*d^5*e + 225*a^2*b^4*d^4*e^2 + 400*a^3*b^3*d^3*e^3 + 225*a
^4*b^2*d^2*e^4 + 36*a^5*b*d*e^5 + a^6*e^6)*x^7)/7 + (3*b*e*(b^5*d^5 + 15*a*b^4*d^4*e + 50*a^2*b^3*d^3*e^2 + 50
*a^3*b^2*d^2*e^3 + 15*a^4*b*d*e^4 + a^5*e^5)*x^8)/4 + (5*b^2*e^2*(b^4*d^4 + 8*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2
+ 8*a^3*b*d*e^3 + a^4*e^4)*x^9)/3 + b^3*e^3*(2*b^3*d^3 + 9*a*b^2*d^2*e + 9*a^2*b*d*e^2 + 2*a^3*e^3)*x^10 + (3
*b^4*e^4*(5*b^2*d^2 + 12*a*b*d*e + 5*a^2*e^2)*x^11)/11 + (b^5*e^5*(b*d + a*e)*x^12)/2 + (b^6*e^6*x^13)/13

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(603\) vs. \(2(159)=318\).

Time = 2.29 (sec) , antiderivative size = 604, normalized size of antiderivative = 3.53

method	result	size
norman	\(\frac {b^{6} e^{6} x^{13}}{13}+\left (\frac {1}{2} a \,b^{5} e^{6}+\frac {1}{2} b^{6} d \,e^{5}\right ) x^{12}+\left (\frac {15}{11} a^{2} b^{4} e^{6}+\frac {36}{11} a \,b^{5} d \,e^{5}+\frac {15}{11} b^{6} d^{2} e^{4}\right ) x^{11}+\left (2 a^{3} b^{3} e^{6}+9 a^{2} b^{4} d \,e^{5}+9 a \,b^{5} d^{2} e^{4}+2 b^{6} d^{3} e^{3}\right ) x^{10}+\left (\frac {5}{3} a^{4} b^{2} e^{6}+\frac {40}{3} a^{3} b^{3} d \,e^{5}+25 a^{2} b^{4} d^{2} e^{4}+\frac {40}{3} a \,b^{5} d^{3} e^{3}+\frac {5}{3} b^{6} d^{4} e^{2}\right ) x^{9}+\left (\frac {3}{4} a^{5} b \,e^{6}+\frac {45}{4} a^{4} b^{2} d \,e^{5}+\frac {75}{2} a^{3} b^{3} d^{2} e^{4}+\frac {75}{2} a^{2} b^{4} d^{3} e^{3}+\frac {45}{4} a \,b^{5} d^{4} e^{2}+\frac {3}{4} b^{6} d^{5} e \right ) x^{8}+\left (\frac {1}{7} a^{6} e^{6}+\frac {36}{7} a^{5} b d \,e^{5}+\frac {225}{7} a^{4} b^{2} d^{2} e^{4}+\frac {400}{7} a^{3} b^{3} d^{3} e^{3}+\frac {225}{7} a^{2} b^{4} d^{4} e^{2}+\frac {36}{7} a \,b^{5} d^{5} e +\frac {1}{7} b^{6} d^{6}\right ) x^{7}+\left (a^{6} d \,e^{5}+15 a^{5} b \,d^{2} e^{4}+50 a^{4} b^{2} d^{3} e^{3}+50 a^{3} b^{3} d^{4} e^{2}+15 a^{2} b^{4} d^{5} e +a \,b^{5} d^{6}\right ) x^{6}+\left (3 a^{6} d^{2} e^{4}+24 a^{5} b \,d^{3} e^{3}+45 a^{4} b^{2} d^{4} e^{2}+24 a^{3} b^{3} d^{5} e +3 a^{2} b^{4} d^{6}\right ) x^{5}+\left (5 a^{6} d^{3} e^{3}+\frac {45}{2} a^{5} b \,d^{4} e^{2}+\frac {45}{2} a^{4} b^{2} d^{5} e +5 a^{3} b^{3} d^{6}\right ) x^{4}+\left (5 a^{6} d^{4} e^{2}+12 a^{5} b \,d^{5} e +5 a^{4} b^{2} d^{6}\right ) x^{3}+\left (3 a^{6} d^{5} e +3 a^{5} b \,d^{6}\right ) x^{2}+a^{6} d^{6} x\)	\(604\)
default	\(\frac {b^{6} e^{6} x^{13}}{13}+\frac {\left (6 a \,b^{5} e^{6}+6 b^{6} d \,e^{5}\right ) x^{12}}{12}+\frac {\left (15 a^{2} b^{4} e^{6}+36 a \,b^{5} d \,e^{5}+15 b^{6} d^{2} e^{4}\right ) x^{11}}{11}+\frac {\left (20 a^{3} b^{3} e^{6}+90 a^{2} b^{4} d \,e^{5}+90 a \,b^{5} d^{2} e^{4}+20 b^{6} d^{3} e^{3}\right ) x^{10}}{10}+\frac {\left (15 a^{4} b^{2} e^{6}+120 a^{3} b^{3} d \,e^{5}+225 a^{2} b^{4} d^{2} e^{4}+120 a \,b^{5} d^{3} e^{3}+15 b^{6} d^{4} e^{2}\right ) x^{9}}{9}+\frac {\left (6 a^{5} b \,e^{6}+90 a^{4} b^{2} d \,e^{5}+300 a^{3} b^{3} d^{2} e^{4}+300 a^{2} b^{4} d^{3} e^{3}+90 a \,b^{5} d^{4} e^{2}+6 b^{6} d^{5} e \right ) x^{8}}{8}+\frac {\left (a^{6} e^{6}+36 a^{5} b d \,e^{5}+225 a^{4} b^{2} d^{2} e^{4}+400 a^{3} b^{3} d^{3} e^{3}+225 a^{2} b^{4} d^{4} e^{2}+36 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) x^{7}}{7}+\frac {\left (6 a^{6} d \,e^{5}+90 a^{5} b \,d^{2} e^{4}+300 a^{4} b^{2} d^{3} e^{3}+300 a^{3} b^{3} d^{4} e^{2}+90 a^{2} b^{4} d^{5} e +6 a \,b^{5} d^{6}\right ) x^{6}}{6}+\frac {\left (15 a^{6} d^{2} e^{4}+120 a^{5} b \,d^{3} e^{3}+225 a^{4} b^{2} d^{4} e^{2}+120 a^{3} b^{3} d^{5} e +15 a^{2} b^{4} d^{6}\right ) x^{5}}{5}+\frac {\left (20 a^{6} d^{3} e^{3}+90 a^{5} b \,d^{4} e^{2}+90 a^{4} b^{2} d^{5} e +20 a^{3} b^{3} d^{6}\right ) x^{4}}{4}+\frac {\left (15 a^{6} d^{4} e^{2}+36 a^{5} b \,d^{5} e +15 a^{4} b^{2} d^{6}\right ) x^{3}}{3}+\frac {\left (6 a^{6} d^{5} e +6 a^{5} b \,d^{6}\right ) x^{2}}{2}+a^{6} d^{6} x\)	\(615\)
risch	\(\frac {1}{2} x^{12} a \,b^{5} e^{6}+\frac {1}{13} b^{6} e^{6} x^{13}+24 a^{5} b \,d^{3} e^{3} x^{5}+45 a^{4} b^{2} d^{4} e^{2} x^{5}+24 a^{3} b^{3} d^{5} e \,x^{5}+12 a^{5} b \,d^{5} e \,x^{3}+a^{6} d^{6} x +\frac {1}{7} d^{6} x^{7} b^{6}+\frac {1}{2} x^{12} b^{6} d \,e^{5}+5 a^{4} b^{2} d^{6} x^{3}+3 a^{5} b \,d^{6} x^{2}+a \,b^{5} d^{6} x^{6}+3 a^{2} b^{4} d^{6} x^{5}+5 a^{3} b^{3} d^{6} x^{4}+9 a^{2} b^{4} d \,e^{5} x^{10}+\frac {36}{11} x^{11} a \,b^{5} d \,e^{5}+\frac {40}{3} x^{9} a^{3} b^{3} d \,e^{5}+25 x^{9} a^{2} b^{4} d^{2} e^{4}+\frac {40}{3} x^{9} a \,b^{5} d^{3} e^{3}+\frac {45}{4} x^{8} a^{4} b^{2} d \,e^{5}+\frac {75}{2} x^{8} a^{3} b^{3} d^{2} e^{4}+\frac {15}{11} x^{11} b^{6} d^{2} e^{4}+\frac {5}{3} x^{9} a^{4} b^{2} e^{6}+\frac {5}{3} x^{9} b^{6} d^{4} e^{2}+\frac {1}{7} x^{7} a^{6} e^{6}+\frac {3}{4} x^{8} a^{5} b \,e^{6}+\frac {3}{4} x^{8} b^{6} d^{5} e +\frac {15}{11} x^{11} a^{2} b^{4} e^{6}+\frac {75}{2} x^{8} a^{2} b^{4} d^{3} e^{3}+\frac {45}{4} x^{8} a \,b^{5} d^{4} e^{2}+\frac {36}{7} x^{7} a^{5} b d \,e^{5}+\frac {225}{7} x^{7} a^{4} b^{2} d^{2} e^{4}+\frac {400}{7} x^{7} a^{3} b^{3} d^{3} e^{3}+\frac {225}{7} x^{7} a^{2} b^{4} d^{4} e^{2}+\frac {36}{7} x^{7} a \,b^{5} d^{5} e +\frac {45}{2} x^{4} a^{5} b \,d^{4} e^{2}+\frac {45}{2} x^{4} a^{4} b^{2} d^{5} e +9 a \,b^{5} d^{2} e^{4} x^{10}+15 a^{5} b \,d^{2} e^{4} x^{6}+50 a^{4} b^{2} d^{3} e^{3} x^{6}+50 a^{3} b^{3} d^{4} e^{2} x^{6}+15 a^{2} b^{4} d^{5} e \,x^{6}+5 x^{4} a^{6} d^{3} e^{3}+2 a^{3} b^{3} e^{6} x^{10}+2 b^{6} d^{3} e^{3} x^{10}+a^{6} d \,e^{5} x^{6}+3 a^{6} d^{2} e^{4} x^{5}+5 a^{6} d^{4} e^{2} x^{3}+3 a^{6} d^{5} e \,x^{2}\)	\(690\)
parallelrisch	\(\frac {1}{2} x^{12} a \,b^{5} e^{6}+\frac {1}{13} b^{6} e^{6} x^{13}+24 a^{5} b \,d^{3} e^{3} x^{5}+45 a^{4} b^{2} d^{4} e^{2} x^{5}+24 a^{3} b^{3} d^{5} e \,x^{5}+12 a^{5} b \,d^{5} e \,x^{3}+a^{6} d^{6} x +\frac {1}{7} d^{6} x^{7} b^{6}+\frac {1}{2} x^{12} b^{6} d \,e^{5}+5 a^{4} b^{2} d^{6} x^{3}+3 a^{5} b \,d^{6} x^{2}+a \,b^{5} d^{6} x^{6}+3 a^{2} b^{4} d^{6} x^{5}+5 a^{3} b^{3} d^{6} x^{4}+9 a^{2} b^{4} d \,e^{5} x^{10}+\frac {36}{11} x^{11} a \,b^{5} d \,e^{5}+\frac {40}{3} x^{9} a^{3} b^{3} d \,e^{5}+25 x^{9} a^{2} b^{4} d^{2} e^{4}+\frac {40}{3} x^{9} a \,b^{5} d^{3} e^{3}+\frac {45}{4} x^{8} a^{4} b^{2} d \,e^{5}+\frac {75}{2} x^{8} a^{3} b^{3} d^{2} e^{4}+\frac {15}{11} x^{11} b^{6} d^{2} e^{4}+\frac {5}{3} x^{9} a^{4} b^{2} e^{6}+\frac {5}{3} x^{9} b^{6} d^{4} e^{2}+\frac {1}{7} x^{7} a^{6} e^{6}+\frac {3}{4} x^{8} a^{5} b \,e^{6}+\frac {3}{4} x^{8} b^{6} d^{5} e +\frac {15}{11} x^{11} a^{2} b^{4} e^{6}+\frac {75}{2} x^{8} a^{2} b^{4} d^{3} e^{3}+\frac {45}{4} x^{8} a \,b^{5} d^{4} e^{2}+\frac {36}{7} x^{7} a^{5} b d \,e^{5}+\frac {225}{7} x^{7} a^{4} b^{2} d^{2} e^{4}+\frac {400}{7} x^{7} a^{3} b^{3} d^{3} e^{3}+\frac {225}{7} x^{7} a^{2} b^{4} d^{4} e^{2}+\frac {36}{7} x^{7} a \,b^{5} d^{5} e +\frac {45}{2} x^{4} a^{5} b \,d^{4} e^{2}+\frac {45}{2} x^{4} a^{4} b^{2} d^{5} e +9 a \,b^{5} d^{2} e^{4} x^{10}+15 a^{5} b \,d^{2} e^{4} x^{6}+50 a^{4} b^{2} d^{3} e^{3} x^{6}+50 a^{3} b^{3} d^{4} e^{2} x^{6}+15 a^{2} b^{4} d^{5} e \,x^{6}+5 x^{4} a^{6} d^{3} e^{3}+2 a^{3} b^{3} e^{6} x^{10}+2 b^{6} d^{3} e^{3} x^{10}+a^{6} d \,e^{5} x^{6}+3 a^{6} d^{2} e^{4} x^{5}+5 a^{6} d^{4} e^{2} x^{3}+3 a^{6} d^{5} e \,x^{2}\)	\(690\)
gosper	\(\frac {x \left (924 b^{6} e^{6} x^{12}+6006 x^{11} a \,b^{5} e^{6}+6006 x^{11} b^{6} d \,e^{5}+16380 x^{10} a^{2} b^{4} e^{6}+39312 x^{10} a \,b^{5} d \,e^{5}+16380 x^{10} b^{6} d^{2} e^{4}+24024 a^{3} b^{3} e^{6} x^{9}+108108 a^{2} b^{4} d \,e^{5} x^{9}+108108 a \,b^{5} d^{2} e^{4} x^{9}+24024 b^{6} d^{3} e^{3} x^{9}+20020 x^{8} a^{4} b^{2} e^{6}+160160 x^{8} a^{3} b^{3} d \,e^{5}+300300 x^{8} a^{2} b^{4} d^{2} e^{4}+160160 x^{8} a \,b^{5} d^{3} e^{3}+20020 x^{8} b^{6} d^{4} e^{2}+9009 x^{7} a^{5} b \,e^{6}+135135 x^{7} a^{4} b^{2} d \,e^{5}+450450 x^{7} a^{3} b^{3} d^{2} e^{4}+450450 x^{7} a^{2} b^{4} d^{3} e^{3}+135135 x^{7} a \,b^{5} d^{4} e^{2}+9009 x^{7} b^{6} d^{5} e +1716 x^{6} a^{6} e^{6}+61776 x^{6} a^{5} b d \,e^{5}+386100 x^{6} a^{4} b^{2} d^{2} e^{4}+686400 x^{6} a^{3} b^{3} d^{3} e^{3}+386100 x^{6} a^{2} b^{4} d^{4} e^{2}+61776 x^{6} a \,b^{5} d^{5} e +1716 x^{6} b^{6} d^{6}+12012 a^{6} d \,e^{5} x^{5}+180180 a^{5} b \,d^{2} e^{4} x^{5}+600600 a^{4} b^{2} d^{3} e^{3} x^{5}+600600 a^{3} b^{3} d^{4} e^{2} x^{5}+180180 a^{2} b^{4} d^{5} e \,x^{5}+12012 a \,b^{5} d^{6} x^{5}+36036 a^{6} d^{2} e^{4} x^{4}+288288 a^{5} b \,d^{3} e^{3} x^{4}+540540 a^{4} b^{2} d^{4} e^{2} x^{4}+288288 a^{3} b^{3} d^{5} e \,x^{4}+36036 a^{2} b^{4} d^{6} x^{4}+60060 x^{3} a^{6} d^{3} e^{3}+270270 x^{3} a^{5} b \,d^{4} e^{2}+270270 x^{3} a^{4} b^{2} d^{5} e +60060 x^{3} a^{3} b^{3} d^{6}+60060 a^{6} d^{4} e^{2} x^{2}+144144 a^{5} b \,d^{5} e \,x^{2}+60060 a^{4} b^{2} d^{6} x^{2}+36036 a^{6} d^{5} e x +36036 a^{5} b \,d^{6} x +12012 a^{6} d^{6}\right )}{12012}\)	\(691\)

[In]

int((e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/13*b^6*e^6*x^13+(1/2*a*b^5*e^6+1/2*b^6*d*e^5)*x^12+(15/11*a^2*b^4*e^6+36/11*a*b^5*d*e^5+15/11*b^6*d^2*e^4)*x
^11+(2*a^3*b^3*e^6+9*a^2*b^4*d*e^5+9*a*b^5*d^2*e^4+2*b^6*d^3*e^3)*x^10+(5/3*a^4*b^2*e^6+40/3*a^3*b^3*d*e^5+25*
a^2*b^4*d^2*e^4+40/3*a*b^5*d^3*e^3+5/3*b^6*d^4*e^2)*x^9+(3/4*a^5*b*e^6+45/4*a^4*b^2*d*e^5+75/2*a^3*b^3*d^2*e^4
+75/2*a^2*b^4*d^3*e^3+45/4*a*b^5*d^4*e^2+3/4*b^6*d^5*e)*x^8+(1/7*a^6*e^6+36/7*a^5*b*d*e^5+225/7*a^4*b^2*d^2*e^
4+400/7*a^3*b^3*d^3*e^3+225/7*a^2*b^4*d^4*e^2+36/7*a*b^5*d^5*e+1/7*b^6*d^6)*x^7+(a^6*d*e^5+15*a^5*b*d^2*e^4+50
*a^4*b^2*d^3*e^3+50*a^3*b^3*d^4*e^2+15*a^2*b^4*d^5*e+a*b^5*d^6)*x^6+(3*a^6*d^2*e^4+24*a^5*b*d^3*e^3+45*a^4*b^2
*d^4*e^2+24*a^3*b^3*d^5*e+3*a^2*b^4*d^6)*x^5+(5*a^6*d^3*e^3+45/2*a^5*b*d^4*e^2+45/2*a^4*b^2*d^5*e+5*a^3*b^3*d^
6)*x^4+(5*a^6*d^4*e^2+12*a^5*b*d^5*e+5*a^4*b^2*d^6)*x^3+(3*a^6*d^5*e+3*a^5*b*d^6)*x^2+a^6*d^6*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (159) = 318\).

Time = 0.30 (sec) , antiderivative size = 599, normalized size of antiderivative = 3.50 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{13} \, b^{6} e^{6} x^{13} + a^{6} d^{6} x + \frac {1}{2} \, {\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{12} + \frac {3}{11} \, {\left (5 \, b^{6} d^{2} e^{4} + 12 \, a b^{5} d e^{5} + 5 \, a^{2} b^{4} e^{6}\right )} x^{11} + {\left (2 \, b^{6} d^{3} e^{3} + 9 \, a b^{5} d^{2} e^{4} + 9 \, a^{2} b^{4} d e^{5} + 2 \, a^{3} b^{3} e^{6}\right )} x^{10} + \frac {5}{3} \, {\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 8 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{9} + \frac {3}{4} \, {\left (b^{6} d^{5} e + 15 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{6} + 36 \, a b^{5} d^{5} e + 225 \, a^{2} b^{4} d^{4} e^{2} + 400 \, a^{3} b^{3} d^{3} e^{3} + 225 \, a^{4} b^{2} d^{2} e^{4} + 36 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} x^{7} + {\left (a b^{5} d^{6} + 15 \, a^{2} b^{4} d^{5} e + 50 \, a^{3} b^{3} d^{4} e^{2} + 50 \, a^{4} b^{2} d^{3} e^{3} + 15 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x^{6} + 3 \, {\left (a^{2} b^{4} d^{6} + 8 \, a^{3} b^{3} d^{5} e + 15 \, a^{4} b^{2} d^{4} e^{2} + 8 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (2 \, a^{3} b^{3} d^{6} + 9 \, a^{4} b^{2} d^{5} e + 9 \, a^{5} b d^{4} e^{2} + 2 \, a^{6} d^{3} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{6} + 12 \, a^{5} b d^{5} e + 5 \, a^{6} d^{4} e^{2}\right )} x^{3} + 3 \, {\left (a^{5} b d^{6} + a^{6} d^{5} e\right )} x^{2} \]

[In]

integrate((e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/13*b^6*e^6*x^13 + a^6*d^6*x + 1/2*(b^6*d*e^5 + a*b^5*e^6)*x^12 + 3/11*(5*b^6*d^2*e^4 + 12*a*b^5*d*e^5 + 5*a^
2*b^4*e^6)*x^11 + (2*b^6*d^3*e^3 + 9*a*b^5*d^2*e^4 + 9*a^2*b^4*d*e^5 + 2*a^3*b^3*e^6)*x^10 + 5/3*(b^6*d^4*e^2
+ 8*a*b^5*d^3*e^3 + 15*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^9 + 3/4*(b^6*d^5*e + 15*a*b^5*d^4*e^
2 + 50*a^2*b^4*d^3*e^3 + 50*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 + a^5*b*e^6)*x^8 + 1/7*(b^6*d^6 + 36*a*b^5*d^5*
e + 225*a^2*b^4*d^4*e^2 + 400*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2*e^4 + 36*a^5*b*d*e^5 + a^6*e^6)*x^7 + (a*b^5*d
^6 + 15*a^2*b^4*d^5*e + 50*a^3*b^3*d^4*e^2 + 50*a^4*b^2*d^3*e^3 + 15*a^5*b*d^2*e^4 + a^6*d*e^5)*x^6 + 3*(a^2*b
^4*d^6 + 8*a^3*b^3*d^5*e + 15*a^4*b^2*d^4*e^2 + 8*a^5*b*d^3*e^3 + a^6*d^2*e^4)*x^5 + 5/2*(2*a^3*b^3*d^6 + 9*a^
4*b^2*d^5*e + 9*a^5*b*d^4*e^2 + 2*a^6*d^3*e^3)*x^4 + (5*a^4*b^2*d^6 + 12*a^5*b*d^5*e + 5*a^6*d^4*e^2)*x^3 + 3*
(a^5*b*d^6 + a^6*d^5*e)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 677 vs. \(2 (155) = 310\).

Time = 0.06 (sec) , antiderivative size = 677, normalized size of antiderivative = 3.96 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{6} d^{6} x + \frac {b^{6} e^{6} x^{13}}{13} + x^{12} \left (\frac {a b^{5} e^{6}}{2} + \frac {b^{6} d e^{5}}{2}\right ) + x^{11} \cdot \left (\frac {15 a^{2} b^{4} e^{6}}{11} + \frac {36 a b^{5} d e^{5}}{11} + \frac {15 b^{6} d^{2} e^{4}}{11}\right ) + x^{10} \cdot \left (2 a^{3} b^{3} e^{6} + 9 a^{2} b^{4} d e^{5} + 9 a b^{5} d^{2} e^{4} + 2 b^{6} d^{3} e^{3}\right ) + x^{9} \cdot \left (\frac {5 a^{4} b^{2} e^{6}}{3} + \frac {40 a^{3} b^{3} d e^{5}}{3} + 25 a^{2} b^{4} d^{2} e^{4} + \frac {40 a b^{5} d^{3} e^{3}}{3} + \frac {5 b^{6} d^{4} e^{2}}{3}\right ) + x^{8} \cdot \left (\frac {3 a^{5} b e^{6}}{4} + \frac {45 a^{4} b^{2} d e^{5}}{4} + \frac {75 a^{3} b^{3} d^{2} e^{4}}{2} + \frac {75 a^{2} b^{4} d^{3} e^{3}}{2} + \frac {45 a b^{5} d^{4} e^{2}}{4} + \frac {3 b^{6} d^{5} e}{4}\right ) + x^{7} \left (\frac {a^{6} e^{6}}{7} + \frac {36 a^{5} b d e^{5}}{7} + \frac {225 a^{4} b^{2} d^{2} e^{4}}{7} + \frac {400 a^{3} b^{3} d^{3} e^{3}}{7} + \frac {225 a^{2} b^{4} d^{4} e^{2}}{7} + \frac {36 a b^{5} d^{5} e}{7} + \frac {b^{6} d^{6}}{7}\right ) + x^{6} \left (a^{6} d e^{5} + 15 a^{5} b d^{2} e^{4} + 50 a^{4} b^{2} d^{3} e^{3} + 50 a^{3} b^{3} d^{4} e^{2} + 15 a^{2} b^{4} d^{5} e + a b^{5} d^{6}\right ) + x^{5} \cdot \left (3 a^{6} d^{2} e^{4} + 24 a^{5} b d^{3} e^{3} + 45 a^{4} b^{2} d^{4} e^{2} + 24 a^{3} b^{3} d^{5} e + 3 a^{2} b^{4} d^{6}\right ) + x^{4} \cdot \left (5 a^{6} d^{3} e^{3} + \frac {45 a^{5} b d^{4} e^{2}}{2} + \frac {45 a^{4} b^{2} d^{5} e}{2} + 5 a^{3} b^{3} d^{6}\right ) + x^{3} \cdot \left (5 a^{6} d^{4} e^{2} + 12 a^{5} b d^{5} e + 5 a^{4} b^{2} d^{6}\right ) + x^{2} \cdot \left (3 a^{6} d^{5} e + 3 a^{5} b d^{6}\right ) \]

[In]

integrate((e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**6*d**6*x + b**6*e**6*x**13/13 + x**12*(a*b**5*e**6/2 + b**6*d*e**5/2) + x**11*(15*a**2*b**4*e**6/11 + 36*a*
b**5*d*e**5/11 + 15*b**6*d**2*e**4/11) + x**10*(2*a**3*b**3*e**6 + 9*a**2*b**4*d*e**5 + 9*a*b**5*d**2*e**4 + 2
*b**6*d**3*e**3) + x**9*(5*a**4*b**2*e**6/3 + 40*a**3*b**3*d*e**5/3 + 25*a**2*b**4*d**2*e**4 + 40*a*b**5*d**3*
e**3/3 + 5*b**6*d**4*e**2/3) + x**8*(3*a**5*b*e**6/4 + 45*a**4*b**2*d*e**5/4 + 75*a**3*b**3*d**2*e**4/2 + 75*a
**2*b**4*d**3*e**3/2 + 45*a*b**5*d**4*e**2/4 + 3*b**6*d**5*e/4) + x**7*(a**6*e**6/7 + 36*a**5*b*d*e**5/7 + 225
*a**4*b**2*d**2*e**4/7 + 400*a**3*b**3*d**3*e**3/7 + 225*a**2*b**4*d**4*e**2/7 + 36*a*b**5*d**5*e/7 + b**6*d**
6/7) + x**6*(a**6*d*e**5 + 15*a**5*b*d**2*e**4 + 50*a**4*b**2*d**3*e**3 + 50*a**3*b**3*d**4*e**2 + 15*a**2*b**
4*d**5*e + a*b**5*d**6) + x**5*(3*a**6*d**2*e**4 + 24*a**5*b*d**3*e**3 + 45*a**4*b**2*d**4*e**2 + 24*a**3*b**3
*d**5*e + 3*a**2*b**4*d**6) + x**4*(5*a**6*d**3*e**3 + 45*a**5*b*d**4*e**2/2 + 45*a**4*b**2*d**5*e/2 + 5*a**3*
b**3*d**6) + x**3*(5*a**6*d**4*e**2 + 12*a**5*b*d**5*e + 5*a**4*b**2*d**6) + x**2*(3*a**6*d**5*e + 3*a**5*b*d*
*6)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (159) = 318\).

Time = 0.20 (sec) , antiderivative size = 599, normalized size of antiderivative = 3.50 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{13} \, b^{6} e^{6} x^{13} + a^{6} d^{6} x + \frac {1}{2} \, {\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{12} + \frac {3}{11} \, {\left (5 \, b^{6} d^{2} e^{4} + 12 \, a b^{5} d e^{5} + 5 \, a^{2} b^{4} e^{6}\right )} x^{11} + {\left (2 \, b^{6} d^{3} e^{3} + 9 \, a b^{5} d^{2} e^{4} + 9 \, a^{2} b^{4} d e^{5} + 2 \, a^{3} b^{3} e^{6}\right )} x^{10} + \frac {5}{3} \, {\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 8 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{9} + \frac {3}{4} \, {\left (b^{6} d^{5} e + 15 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{6} + 36 \, a b^{5} d^{5} e + 225 \, a^{2} b^{4} d^{4} e^{2} + 400 \, a^{3} b^{3} d^{3} e^{3} + 225 \, a^{4} b^{2} d^{2} e^{4} + 36 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} x^{7} + {\left (a b^{5} d^{6} + 15 \, a^{2} b^{4} d^{5} e + 50 \, a^{3} b^{3} d^{4} e^{2} + 50 \, a^{4} b^{2} d^{3} e^{3} + 15 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x^{6} + 3 \, {\left (a^{2} b^{4} d^{6} + 8 \, a^{3} b^{3} d^{5} e + 15 \, a^{4} b^{2} d^{4} e^{2} + 8 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (2 \, a^{3} b^{3} d^{6} + 9 \, a^{4} b^{2} d^{5} e + 9 \, a^{5} b d^{4} e^{2} + 2 \, a^{6} d^{3} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{6} + 12 \, a^{5} b d^{5} e + 5 \, a^{6} d^{4} e^{2}\right )} x^{3} + 3 \, {\left (a^{5} b d^{6} + a^{6} d^{5} e\right )} x^{2} \]

[In]

integrate((e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/13*b^6*e^6*x^13 + a^6*d^6*x + 1/2*(b^6*d*e^5 + a*b^5*e^6)*x^12 + 3/11*(5*b^6*d^2*e^4 + 12*a*b^5*d*e^5 + 5*a^
2*b^4*e^6)*x^11 + (2*b^6*d^3*e^3 + 9*a*b^5*d^2*e^4 + 9*a^2*b^4*d*e^5 + 2*a^3*b^3*e^6)*x^10 + 5/3*(b^6*d^4*e^2
+ 8*a*b^5*d^3*e^3 + 15*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^9 + 3/4*(b^6*d^5*e + 15*a*b^5*d^4*e^
2 + 50*a^2*b^4*d^3*e^3 + 50*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 + a^5*b*e^6)*x^8 + 1/7*(b^6*d^6 + 36*a*b^5*d^5*
e + 225*a^2*b^4*d^4*e^2 + 400*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2*e^4 + 36*a^5*b*d*e^5 + a^6*e^6)*x^7 + (a*b^5*d
^6 + 15*a^2*b^4*d^5*e + 50*a^3*b^3*d^4*e^2 + 50*a^4*b^2*d^3*e^3 + 15*a^5*b*d^2*e^4 + a^6*d*e^5)*x^6 + 3*(a^2*b
^4*d^6 + 8*a^3*b^3*d^5*e + 15*a^4*b^2*d^4*e^2 + 8*a^5*b*d^3*e^3 + a^6*d^2*e^4)*x^5 + 5/2*(2*a^3*b^3*d^6 + 9*a^
4*b^2*d^5*e + 9*a^5*b*d^4*e^2 + 2*a^6*d^3*e^3)*x^4 + (5*a^4*b^2*d^6 + 12*a^5*b*d^5*e + 5*a^6*d^4*e^2)*x^3 + 3*
(a^5*b*d^6 + a^6*d^5*e)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 689 vs. \(2 (159) = 318\).

Time = 0.27 (sec) , antiderivative size = 689, normalized size of antiderivative = 4.03 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{13} \, b^{6} e^{6} x^{13} + \frac {1}{2} \, b^{6} d e^{5} x^{12} + \frac {1}{2} \, a b^{5} e^{6} x^{12} + \frac {15}{11} \, b^{6} d^{2} e^{4} x^{11} + \frac {36}{11} \, a b^{5} d e^{5} x^{11} + \frac {15}{11} \, a^{2} b^{4} e^{6} x^{11} + 2 \, b^{6} d^{3} e^{3} x^{10} + 9 \, a b^{5} d^{2} e^{4} x^{10} + 9 \, a^{2} b^{4} d e^{5} x^{10} + 2 \, a^{3} b^{3} e^{6} x^{10} + \frac {5}{3} \, b^{6} d^{4} e^{2} x^{9} + \frac {40}{3} \, a b^{5} d^{3} e^{3} x^{9} + 25 \, a^{2} b^{4} d^{2} e^{4} x^{9} + \frac {40}{3} \, a^{3} b^{3} d e^{5} x^{9} + \frac {5}{3} \, a^{4} b^{2} e^{6} x^{9} + \frac {3}{4} \, b^{6} d^{5} e x^{8} + \frac {45}{4} \, a b^{5} d^{4} e^{2} x^{8} + \frac {75}{2} \, a^{2} b^{4} d^{3} e^{3} x^{8} + \frac {75}{2} \, a^{3} b^{3} d^{2} e^{4} x^{8} + \frac {45}{4} \, a^{4} b^{2} d e^{5} x^{8} + \frac {3}{4} \, a^{5} b e^{6} x^{8} + \frac {1}{7} \, b^{6} d^{6} x^{7} + \frac {36}{7} \, a b^{5} d^{5} e x^{7} + \frac {225}{7} \, a^{2} b^{4} d^{4} e^{2} x^{7} + \frac {400}{7} \, a^{3} b^{3} d^{3} e^{3} x^{7} + \frac {225}{7} \, a^{4} b^{2} d^{2} e^{4} x^{7} + \frac {36}{7} \, a^{5} b d e^{5} x^{7} + \frac {1}{7} \, a^{6} e^{6} x^{7} + a b^{5} d^{6} x^{6} + 15 \, a^{2} b^{4} d^{5} e x^{6} + 50 \, a^{3} b^{3} d^{4} e^{2} x^{6} + 50 \, a^{4} b^{2} d^{3} e^{3} x^{6} + 15 \, a^{5} b d^{2} e^{4} x^{6} + a^{6} d e^{5} x^{6} + 3 \, a^{2} b^{4} d^{6} x^{5} + 24 \, a^{3} b^{3} d^{5} e x^{5} + 45 \, a^{4} b^{2} d^{4} e^{2} x^{5} + 24 \, a^{5} b d^{3} e^{3} x^{5} + 3 \, a^{6} d^{2} e^{4} x^{5} + 5 \, a^{3} b^{3} d^{6} x^{4} + \frac {45}{2} \, a^{4} b^{2} d^{5} e x^{4} + \frac {45}{2} \, a^{5} b d^{4} e^{2} x^{4} + 5 \, a^{6} d^{3} e^{3} x^{4} + 5 \, a^{4} b^{2} d^{6} x^{3} + 12 \, a^{5} b d^{5} e x^{3} + 5 \, a^{6} d^{4} e^{2} x^{3} + 3 \, a^{5} b d^{6} x^{2} + 3 \, a^{6} d^{5} e x^{2} + a^{6} d^{6} x \]

[In]

integrate((e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/13*b^6*e^6*x^13 + 1/2*b^6*d*e^5*x^12 + 1/2*a*b^5*e^6*x^12 + 15/11*b^6*d^2*e^4*x^11 + 36/11*a*b^5*d*e^5*x^11
+ 15/11*a^2*b^4*e^6*x^11 + 2*b^6*d^3*e^3*x^10 + 9*a*b^5*d^2*e^4*x^10 + 9*a^2*b^4*d*e^5*x^10 + 2*a^3*b^3*e^6*x^
10 + 5/3*b^6*d^4*e^2*x^9 + 40/3*a*b^5*d^3*e^3*x^9 + 25*a^2*b^4*d^2*e^4*x^9 + 40/3*a^3*b^3*d*e^5*x^9 + 5/3*a^4*
b^2*e^6*x^9 + 3/4*b^6*d^5*e*x^8 + 45/4*a*b^5*d^4*e^2*x^8 + 75/2*a^2*b^4*d^3*e^3*x^8 + 75/2*a^3*b^3*d^2*e^4*x^8
+ 45/4*a^4*b^2*d*e^5*x^8 + 3/4*a^5*b*e^6*x^8 + 1/7*b^6*d^6*x^7 + 36/7*a*b^5*d^5*e*x^7 + 225/7*a^2*b^4*d^4*e^2
*x^7 + 400/7*a^3*b^3*d^3*e^3*x^7 + 225/7*a^4*b^2*d^2*e^4*x^7 + 36/7*a^5*b*d*e^5*x^7 + 1/7*a^6*e^6*x^7 + a*b^5*
d^6*x^6 + 15*a^2*b^4*d^5*e*x^6 + 50*a^3*b^3*d^4*e^2*x^6 + 50*a^4*b^2*d^3*e^3*x^6 + 15*a^5*b*d^2*e^4*x^6 + a^6*
d*e^5*x^6 + 3*a^2*b^4*d^6*x^5 + 24*a^3*b^3*d^5*e*x^5 + 45*a^4*b^2*d^4*e^2*x^5 + 24*a^5*b*d^3*e^3*x^5 + 3*a^6*d
^2*e^4*x^5 + 5*a^3*b^3*d^6*x^4 + 45/2*a^4*b^2*d^5*e*x^4 + 45/2*a^5*b*d^4*e^2*x^4 + 5*a^6*d^3*e^3*x^4 + 5*a^4*b
^2*d^6*x^3 + 12*a^5*b*d^5*e*x^3 + 5*a^6*d^4*e^2*x^3 + 3*a^5*b*d^6*x^2 + 3*a^6*d^5*e*x^2 + a^6*d^6*x

Mupad [B] (verification not implemented)

Time = 10.29 (sec) , antiderivative size = 579, normalized size of antiderivative = 3.39 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^6\,\left (a^6\,d\,e^5+15\,a^5\,b\,d^2\,e^4+50\,a^4\,b^2\,d^3\,e^3+50\,a^3\,b^3\,d^4\,e^2+15\,a^2\,b^4\,d^5\,e+a\,b^5\,d^6\right )+x^8\,\left (\frac {3\,a^5\,b\,e^6}{4}+\frac {45\,a^4\,b^2\,d\,e^5}{4}+\frac {75\,a^3\,b^3\,d^2\,e^4}{2}+\frac {75\,a^2\,b^4\,d^3\,e^3}{2}+\frac {45\,a\,b^5\,d^4\,e^2}{4}+\frac {3\,b^6\,d^5\,e}{4}\right )+x^5\,\left (3\,a^6\,d^2\,e^4+24\,a^5\,b\,d^3\,e^3+45\,a^4\,b^2\,d^4\,e^2+24\,a^3\,b^3\,d^5\,e+3\,a^2\,b^4\,d^6\right )+x^9\,\left (\frac {5\,a^4\,b^2\,e^6}{3}+\frac {40\,a^3\,b^3\,d\,e^5}{3}+25\,a^2\,b^4\,d^2\,e^4+\frac {40\,a\,b^5\,d^3\,e^3}{3}+\frac {5\,b^6\,d^4\,e^2}{3}\right )+x^7\,\left (\frac {a^6\,e^6}{7}+\frac {36\,a^5\,b\,d\,e^5}{7}+\frac {225\,a^4\,b^2\,d^2\,e^4}{7}+\frac {400\,a^3\,b^3\,d^3\,e^3}{7}+\frac {225\,a^2\,b^4\,d^4\,e^2}{7}+\frac {36\,a\,b^5\,d^5\,e}{7}+\frac {b^6\,d^6}{7}\right )+a^6\,d^6\,x+\frac {b^6\,e^6\,x^{13}}{13}+\frac {5\,a^3\,d^3\,x^4\,\left (2\,a^3\,e^3+9\,a^2\,b\,d\,e^2+9\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )}{2}+b^3\,e^3\,x^{10}\,\left (2\,a^3\,e^3+9\,a^2\,b\,d\,e^2+9\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )+3\,a^5\,d^5\,x^2\,\left (a\,e+b\,d\right )+\frac {b^5\,e^5\,x^{12}\,\left (a\,e+b\,d\right )}{2}+a^4\,d^4\,x^3\,\left (5\,a^2\,e^2+12\,a\,b\,d\,e+5\,b^2\,d^2\right )+\frac {3\,b^4\,e^4\,x^{11}\,\left (5\,a^2\,e^2+12\,a\,b\,d\,e+5\,b^2\,d^2\right )}{11} \]

[In]

int((d + e*x)^6*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

x^6*(a*b^5*d^6 + a^6*d*e^5 + 15*a^2*b^4*d^5*e + 15*a^5*b*d^2*e^4 + 50*a^3*b^3*d^4*e^2 + 50*a^4*b^2*d^3*e^3) +
x^8*((3*a^5*b*e^6)/4 + (3*b^6*d^5*e)/4 + (45*a*b^5*d^4*e^2)/4 + (45*a^4*b^2*d*e^5)/4 + (75*a^2*b^4*d^3*e^3)/2
+ (75*a^3*b^3*d^2*e^4)/2) + x^5*(3*a^2*b^4*d^6 + 3*a^6*d^2*e^4 + 24*a^3*b^3*d^5*e + 24*a^5*b*d^3*e^3 + 45*a^4*
b^2*d^4*e^2) + x^9*((5*a^4*b^2*e^6)/3 + (5*b^6*d^4*e^2)/3 + (40*a*b^5*d^3*e^3)/3 + (40*a^3*b^3*d*e^5)/3 + 25*a
^2*b^4*d^2*e^4) + x^7*((a^6*e^6)/7 + (b^6*d^6)/7 + (225*a^2*b^4*d^4*e^2)/7 + (400*a^3*b^3*d^3*e^3)/7 + (225*a^
4*b^2*d^2*e^4)/7 + (36*a*b^5*d^5*e)/7 + (36*a^5*b*d*e^5)/7) + a^6*d^6*x + (b^6*e^6*x^13)/13 + (5*a^3*d^3*x^4*(
2*a^3*e^3 + 2*b^3*d^3 + 9*a*b^2*d^2*e + 9*a^2*b*d*e^2))/2 + b^3*e^3*x^10*(2*a^3*e^3 + 2*b^3*d^3 + 9*a*b^2*d^2*
e + 9*a^2*b*d*e^2) + 3*a^5*d^5*x^2*(a*e + b*d) + (b^5*e^5*x^12*(a*e + b*d))/2 + a^4*d^4*x^3*(5*a^2*e^2 + 5*b^2
*d^2 + 12*a*b*d*e) + (3*b^4*e^4*x^11*(5*a^2*e^2 + 5*b^2*d^2 + 12*a*b*d*e))/11