\(\int (d+e x)^4 (a^2+2 a b x+b^2 x^2)^3 \, dx\) [1485]

Optimal result

Integrand size = 26, antiderivative size = 119 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(b d-a e)^4 (a+b x)^7}{7 b^5}+\frac {e (b d-a e)^3 (a+b x)^8}{2 b^5}+\frac {2 e^2 (b d-a e)^2 (a+b x)^9}{3 b^5}+\frac {2 e^3 (b d-a e) (a+b x)^{10}}{5 b^5}+\frac {e^4 (a+b x)^{11}}{11 b^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 119, 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} \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 e^3 (a+b x)^{10} (b d-a e)}{5 b^5}+\frac {2 e^2 (a+b x)^9 (b d-a e)^2}{3 b^5}+\frac {e (a+b x)^8 (b d-a e)^3}{2 b^5}+\frac {(a+b x)^7 (b d-a e)^4}{7 b^5}+\frac {e^4 (a+b x)^{11}}{11 b^5} \]

[In]

Int[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(398\) vs. \(2(119)=238\).

Time = 0.04 (sec) , antiderivative size = 398, normalized size of antiderivative = 3.34 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^6 d^4 x+a^5 d^3 (3 b d+2 a e) x^2+a^4 d^2 \left (5 b^2 d^2+8 a b d e+2 a^2 e^2\right ) x^3+a^3 d \left (5 b^3 d^3+15 a b^2 d^2 e+9 a^2 b d e^2+a^3 e^3\right ) x^4+\frac {1}{5} a^2 \left (15 b^4 d^4+80 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+24 a^3 b d e^3+a^4 e^4\right ) x^5+a b \left (b^4 d^4+10 a b^3 d^3 e+20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4\right ) x^6+\frac {1}{7} b^2 \left (b^4 d^4+24 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+80 a^3 b d e^3+15 a^4 e^4\right ) x^7+\frac {1}{2} b^3 e \left (b^3 d^3+9 a b^2 d^2 e+15 a^2 b d e^2+5 a^3 e^3\right ) x^8+\frac {1}{3} b^4 e^2 \left (2 b^2 d^2+8 a b d e+5 a^2 e^2\right ) x^9+\frac {1}{5} b^5 e^3 (2 b d+3 a e) x^{10}+\frac {1}{11} b^6 e^4 x^{11} \]

[In]

Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

a^6*d^4*x + a^5*d^3*(3*b*d + 2*a*e)*x^2 + a^4*d^2*(5*b^2*d^2 + 8*a*b*d*e + 2*a^2*e^2)*x^3 + a^3*d*(5*b^3*d^3 +
15*a*b^2*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3)*x^4 + (a^2*(15*b^4*d^4 + 80*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 24*a
^3*b*d*e^3 + a^4*e^4)*x^5)/5 + a*b*(b^4*d^4 + 10*a*b^3*d^3*e + 20*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4)*
x^6 + (b^2*(b^4*d^4 + 24*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 80*a^3*b*d*e^3 + 15*a^4*e^4)*x^7)/7 + (b^3*e*(b^3*
d^3 + 9*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 5*a^3*e^3)*x^8)/2 + (b^4*e^2*(2*b^2*d^2 + 8*a*b*d*e + 5*a^2*e^2)*x^9)/3
+ (b^5*e^3*(2*b*d + 3*a*e)*x^10)/5 + (b^6*e^4*x^11)/11

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(416\) vs. \(2(109)=218\).

Time = 2.26 (sec) , antiderivative size = 417, normalized size of antiderivative = 3.50

method	result	size
norman	\(\frac {e^{4} b^{6} x^{11}}{11}+\left (\frac {3}{5} e^{4} a \,b^{5}+\frac {2}{5} d \,e^{3} b^{6}\right ) x^{10}+\left (\frac {5}{3} e^{4} a^{2} b^{4}+\frac {8}{3} d \,e^{3} a \,b^{5}+\frac {2}{3} d^{2} e^{2} b^{6}\right ) x^{9}+\left (\frac {5}{2} e^{4} a^{3} b^{3}+\frac {15}{2} d \,e^{3} a^{2} b^{4}+\frac {9}{2} d^{2} e^{2} a \,b^{5}+\frac {1}{2} d^{3} e \,b^{6}\right ) x^{8}+\left (\frac {15}{7} e^{4} a^{4} b^{2}+\frac {80}{7} d \,e^{3} a^{3} b^{3}+\frac {90}{7} d^{2} e^{2} a^{2} b^{4}+\frac {24}{7} d^{3} e a \,b^{5}+\frac {1}{7} d^{4} b^{6}\right ) x^{7}+\left (e^{4} a^{5} b +10 d \,e^{3} a^{4} b^{2}+20 d^{2} e^{2} a^{3} b^{3}+10 d^{3} e \,a^{2} b^{4}+d^{4} a \,b^{5}\right ) x^{6}+\left (\frac {1}{5} e^{4} a^{6}+\frac {24}{5} d \,e^{3} a^{5} b +18 d^{2} e^{2} a^{4} b^{2}+16 d^{3} e \,a^{3} b^{3}+3 a^{2} b^{4} d^{4}\right ) x^{5}+\left (d \,e^{3} a^{6}+9 d^{2} e^{2} a^{5} b +15 d^{3} e \,a^{4} b^{2}+5 d^{4} a^{3} b^{3}\right ) x^{4}+\left (2 d^{2} e^{2} a^{6}+8 d^{3} e \,a^{5} b +5 d^{4} a^{4} b^{2}\right ) x^{3}+\left (2 d^{3} e \,a^{6}+3 a^{5} d^{4} b \right ) x^{2}+d^{4} a^{6} x\)	\(417\)
default	\(\frac {e^{4} b^{6} x^{11}}{11}+\frac {\left (6 e^{4} a \,b^{5}+4 d \,e^{3} b^{6}\right ) x^{10}}{10}+\frac {\left (15 e^{4} a^{2} b^{4}+24 d \,e^{3} a \,b^{5}+6 d^{2} e^{2} b^{6}\right ) x^{9}}{9}+\frac {\left (20 e^{4} a^{3} b^{3}+60 d \,e^{3} a^{2} b^{4}+36 d^{2} e^{2} a \,b^{5}+4 d^{3} e \,b^{6}\right ) x^{8}}{8}+\frac {\left (15 e^{4} a^{4} b^{2}+80 d \,e^{3} a^{3} b^{3}+90 d^{2} e^{2} a^{2} b^{4}+24 d^{3} e a \,b^{5}+d^{4} b^{6}\right ) x^{7}}{7}+\frac {\left (6 e^{4} a^{5} b +60 d \,e^{3} a^{4} b^{2}+120 d^{2} e^{2} a^{3} b^{3}+60 d^{3} e \,a^{2} b^{4}+6 d^{4} a \,b^{5}\right ) x^{6}}{6}+\frac {\left (e^{4} a^{6}+24 d \,e^{3} a^{5} b +90 d^{2} e^{2} a^{4} b^{2}+80 d^{3} e \,a^{3} b^{3}+15 a^{2} b^{4} d^{4}\right ) x^{5}}{5}+\frac {\left (4 d \,e^{3} a^{6}+36 d^{2} e^{2} a^{5} b +60 d^{3} e \,a^{4} b^{2}+20 d^{4} a^{3} b^{3}\right ) x^{4}}{4}+\frac {\left (6 d^{2} e^{2} a^{6}+24 d^{3} e \,a^{5} b +15 d^{4} a^{4} b^{2}\right ) x^{3}}{3}+\frac {\left (4 d^{3} e \,a^{6}+6 a^{5} d^{4} b \right ) x^{2}}{2}+d^{4} a^{6} x\)	\(427\)
risch	\(a^{6} d \,e^{3} x^{4}+5 a^{3} b^{3} d^{4} x^{4}+2 a^{6} d^{2} e^{2} x^{3}+5 a^{4} b^{2} d^{4} x^{3}+20 a^{3} b^{3} d^{2} e^{2} x^{6}+10 a^{2} b^{4} d^{3} e \,x^{6}+9 a^{5} b \,d^{2} e^{2} x^{4}+15 a^{4} b^{2} d^{3} e \,x^{4}+8 a^{5} b \,d^{3} e \,x^{3}+a^{5} b \,e^{4} x^{6}+a \,b^{5} d^{4} x^{6}+\frac {1}{7} x^{7} d^{4} b^{6}+d^{4} a^{6} x +\frac {1}{5} x^{5} e^{4} a^{6}+\frac {1}{11} e^{4} b^{6} x^{11}+\frac {5}{2} x^{8} e^{4} a^{3} b^{3}+\frac {1}{2} x^{8} d^{3} e \,b^{6}+\frac {15}{7} x^{7} e^{4} a^{4} b^{2}+3 x^{5} a^{2} b^{4} d^{4}+\frac {8}{3} x^{9} d \,e^{3} a \,b^{5}+\frac {15}{2} x^{8} d \,e^{3} a^{2} b^{4}+\frac {9}{2} x^{8} d^{2} e^{2} a \,b^{5}+\frac {80}{7} x^{7} d \,e^{3} a^{3} b^{3}+\frac {90}{7} x^{7} d^{2} e^{2} a^{2} b^{4}+\frac {24}{7} x^{7} d^{3} e a \,b^{5}+\frac {24}{5} x^{5} d \,e^{3} a^{5} b +18 x^{5} d^{2} e^{2} a^{4} b^{2}+16 x^{5} d^{3} e \,a^{3} b^{3}+10 a^{4} b^{2} d \,e^{3} x^{6}+\frac {3}{5} x^{10} e^{4} a \,b^{5}+\frac {2}{5} x^{10} d \,e^{3} b^{6}+\frac {5}{3} x^{9} e^{4} a^{2} b^{4}+\frac {2}{3} x^{9} d^{2} e^{2} b^{6}+2 a^{6} d^{3} e \,x^{2}+3 a^{5} b \,d^{4} x^{2}\)	\(471\)
parallelrisch	\(a^{6} d \,e^{3} x^{4}+5 a^{3} b^{3} d^{4} x^{4}+2 a^{6} d^{2} e^{2} x^{3}+5 a^{4} b^{2} d^{4} x^{3}+20 a^{3} b^{3} d^{2} e^{2} x^{6}+10 a^{2} b^{4} d^{3} e \,x^{6}+9 a^{5} b \,d^{2} e^{2} x^{4}+15 a^{4} b^{2} d^{3} e \,x^{4}+8 a^{5} b \,d^{3} e \,x^{3}+a^{5} b \,e^{4} x^{6}+a \,b^{5} d^{4} x^{6}+\frac {1}{7} x^{7} d^{4} b^{6}+d^{4} a^{6} x +\frac {1}{5} x^{5} e^{4} a^{6}+\frac {1}{11} e^{4} b^{6} x^{11}+\frac {5}{2} x^{8} e^{4} a^{3} b^{3}+\frac {1}{2} x^{8} d^{3} e \,b^{6}+\frac {15}{7} x^{7} e^{4} a^{4} b^{2}+3 x^{5} a^{2} b^{4} d^{4}+\frac {8}{3} x^{9} d \,e^{3} a \,b^{5}+\frac {15}{2} x^{8} d \,e^{3} a^{2} b^{4}+\frac {9}{2} x^{8} d^{2} e^{2} a \,b^{5}+\frac {80}{7} x^{7} d \,e^{3} a^{3} b^{3}+\frac {90}{7} x^{7} d^{2} e^{2} a^{2} b^{4}+\frac {24}{7} x^{7} d^{3} e a \,b^{5}+\frac {24}{5} x^{5} d \,e^{3} a^{5} b +18 x^{5} d^{2} e^{2} a^{4} b^{2}+16 x^{5} d^{3} e \,a^{3} b^{3}+10 a^{4} b^{2} d \,e^{3} x^{6}+\frac {3}{5} x^{10} e^{4} a \,b^{5}+\frac {2}{5} x^{10} d \,e^{3} b^{6}+\frac {5}{3} x^{9} e^{4} a^{2} b^{4}+\frac {2}{3} x^{9} d^{2} e^{2} b^{6}+2 a^{6} d^{3} e \,x^{2}+3 a^{5} b \,d^{4} x^{2}\)	\(471\)
gosper	\(\frac {x \left (210 e^{4} b^{6} x^{10}+1386 x^{9} e^{4} a \,b^{5}+924 x^{9} d \,e^{3} b^{6}+3850 x^{8} e^{4} a^{2} b^{4}+6160 x^{8} d \,e^{3} a \,b^{5}+1540 x^{8} d^{2} e^{2} b^{6}+5775 x^{7} e^{4} a^{3} b^{3}+17325 x^{7} d \,e^{3} a^{2} b^{4}+10395 x^{7} d^{2} e^{2} a \,b^{5}+1155 x^{7} d^{3} e \,b^{6}+4950 x^{6} e^{4} a^{4} b^{2}+26400 x^{6} d \,e^{3} a^{3} b^{3}+29700 x^{6} d^{2} e^{2} a^{2} b^{4}+7920 x^{6} d^{3} e a \,b^{5}+330 x^{6} d^{4} b^{6}+2310 a^{5} b \,e^{4} x^{5}+23100 a^{4} b^{2} d \,e^{3} x^{5}+46200 a^{3} b^{3} d^{2} e^{2} x^{5}+23100 a^{2} b^{4} d^{3} e \,x^{5}+2310 a \,b^{5} d^{4} x^{5}+462 x^{4} e^{4} a^{6}+11088 x^{4} d \,e^{3} a^{5} b +41580 x^{4} d^{2} e^{2} a^{4} b^{2}+36960 x^{4} d^{3} e \,a^{3} b^{3}+6930 x^{4} a^{2} b^{4} d^{4}+2310 a^{6} d \,e^{3} x^{3}+20790 a^{5} b \,d^{2} e^{2} x^{3}+34650 a^{4} b^{2} d^{3} e \,x^{3}+11550 a^{3} b^{3} d^{4} x^{3}+4620 a^{6} d^{2} e^{2} x^{2}+18480 a^{5} b \,d^{3} e \,x^{2}+11550 a^{4} b^{2} d^{4} x^{2}+4620 a^{6} d^{3} e x +6930 a^{5} b \,d^{4} x +2310 d^{4} a^{6}\right )}{2310}\)	\(473\)

[In]

int((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/11*e^4*b^6*x^11+(3/5*e^4*a*b^5+2/5*d*e^3*b^6)*x^10+(5/3*e^4*a^2*b^4+8/3*d*e^3*a*b^5+2/3*d^2*e^2*b^6)*x^9+(5/
2*e^4*a^3*b^3+15/2*d*e^3*a^2*b^4+9/2*d^2*e^2*a*b^5+1/2*d^3*e*b^6)*x^8+(15/7*e^4*a^4*b^2+80/7*d*e^3*a^3*b^3+90/
7*d^2*e^2*a^2*b^4+24/7*d^3*e*a*b^5+1/7*d^4*b^6)*x^7+(a^5*b*e^4+10*a^4*b^2*d*e^3+20*a^3*b^3*d^2*e^2+10*a^2*b^4*
d^3*e+a*b^5*d^4)*x^6+(1/5*e^4*a^6+24/5*d*e^3*a^5*b+18*d^2*e^2*a^4*b^2+16*d^3*e*a^3*b^3+3*a^2*b^4*d^4)*x^5+(a^6
*d*e^3+9*a^5*b*d^2*e^2+15*a^4*b^2*d^3*e+5*a^3*b^3*d^4)*x^4+(2*a^6*d^2*e^2+8*a^5*b*d^3*e+5*a^4*b^2*d^4)*x^3+(2*
a^6*d^3*e+3*a^5*b*d^4)*x^2+d^4*a^6*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (109) = 218\).

Time = 0.30 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.51 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{11} \, b^{6} e^{4} x^{11} + a^{6} d^{4} x + \frac {1}{5} \, {\left (2 \, b^{6} d e^{3} + 3 \, a b^{5} e^{4}\right )} x^{10} + \frac {1}{3} \, {\left (2 \, b^{6} d^{2} e^{2} + 8 \, a b^{5} d e^{3} + 5 \, a^{2} b^{4} e^{4}\right )} x^{9} + \frac {1}{2} \, {\left (b^{6} d^{3} e + 9 \, a b^{5} d^{2} e^{2} + 15 \, a^{2} b^{4} d e^{3} + 5 \, a^{3} b^{3} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{4} + 24 \, a b^{5} d^{3} e + 90 \, a^{2} b^{4} d^{2} e^{2} + 80 \, a^{3} b^{3} d e^{3} + 15 \, a^{4} b^{2} e^{4}\right )} x^{7} + {\left (a b^{5} d^{4} + 10 \, a^{2} b^{4} d^{3} e + 20 \, a^{3} b^{3} d^{2} e^{2} + 10 \, a^{4} b^{2} d e^{3} + a^{5} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (15 \, a^{2} b^{4} d^{4} + 80 \, a^{3} b^{3} d^{3} e + 90 \, a^{4} b^{2} d^{2} e^{2} + 24 \, a^{5} b d e^{3} + a^{6} e^{4}\right )} x^{5} + {\left (5 \, a^{3} b^{3} d^{4} + 15 \, a^{4} b^{2} d^{3} e + 9 \, a^{5} b d^{2} e^{2} + a^{6} d e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{4} + 8 \, a^{5} b d^{3} e + 2 \, a^{6} d^{2} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{4} + 2 \, a^{6} d^{3} e\right )} x^{2} \]

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/11*b^6*e^4*x^11 + a^6*d^4*x + 1/5*(2*b^6*d*e^3 + 3*a*b^5*e^4)*x^10 + 1/3*(2*b^6*d^2*e^2 + 8*a*b^5*d*e^3 + 5*
a^2*b^4*e^4)*x^9 + 1/2*(b^6*d^3*e + 9*a*b^5*d^2*e^2 + 15*a^2*b^4*d*e^3 + 5*a^3*b^3*e^4)*x^8 + 1/7*(b^6*d^4 + 2
4*a*b^5*d^3*e + 90*a^2*b^4*d^2*e^2 + 80*a^3*b^3*d*e^3 + 15*a^4*b^2*e^4)*x^7 + (a*b^5*d^4 + 10*a^2*b^4*d^3*e +
20*a^3*b^3*d^2*e^2 + 10*a^4*b^2*d*e^3 + a^5*b*e^4)*x^6 + 1/5*(15*a^2*b^4*d^4 + 80*a^3*b^3*d^3*e + 90*a^4*b^2*d
^2*e^2 + 24*a^5*b*d*e^3 + a^6*e^4)*x^5 + (5*a^3*b^3*d^4 + 15*a^4*b^2*d^3*e + 9*a^5*b*d^2*e^2 + a^6*d*e^3)*x^4
+ (5*a^4*b^2*d^4 + 8*a^5*b*d^3*e + 2*a^6*d^2*e^2)*x^3 + (3*a^5*b*d^4 + 2*a^6*d^3*e)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (105) = 210\).

Time = 0.05 (sec) , antiderivative size = 462, normalized size of antiderivative = 3.88 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{6} d^{4} x + \frac {b^{6} e^{4} x^{11}}{11} + x^{10} \cdot \left (\frac {3 a b^{5} e^{4}}{5} + \frac {2 b^{6} d e^{3}}{5}\right ) + x^{9} \cdot \left (\frac {5 a^{2} b^{4} e^{4}}{3} + \frac {8 a b^{5} d e^{3}}{3} + \frac {2 b^{6} d^{2} e^{2}}{3}\right ) + x^{8} \cdot \left (\frac {5 a^{3} b^{3} e^{4}}{2} + \frac {15 a^{2} b^{4} d e^{3}}{2} + \frac {9 a b^{5} d^{2} e^{2}}{2} + \frac {b^{6} d^{3} e}{2}\right ) + x^{7} \cdot \left (\frac {15 a^{4} b^{2} e^{4}}{7} + \frac {80 a^{3} b^{3} d e^{3}}{7} + \frac {90 a^{2} b^{4} d^{2} e^{2}}{7} + \frac {24 a b^{5} d^{3} e}{7} + \frac {b^{6} d^{4}}{7}\right ) + x^{6} \left (a^{5} b e^{4} + 10 a^{4} b^{2} d e^{3} + 20 a^{3} b^{3} d^{2} e^{2} + 10 a^{2} b^{4} d^{3} e + a b^{5} d^{4}\right ) + x^{5} \left (\frac {a^{6} e^{4}}{5} + \frac {24 a^{5} b d e^{3}}{5} + 18 a^{4} b^{2} d^{2} e^{2} + 16 a^{3} b^{3} d^{3} e + 3 a^{2} b^{4} d^{4}\right ) + x^{4} \left (a^{6} d e^{3} + 9 a^{5} b d^{2} e^{2} + 15 a^{4} b^{2} d^{3} e + 5 a^{3} b^{3} d^{4}\right ) + x^{3} \cdot \left (2 a^{6} d^{2} e^{2} + 8 a^{5} b d^{3} e + 5 a^{4} b^{2} d^{4}\right ) + x^{2} \cdot \left (2 a^{6} d^{3} e + 3 a^{5} b d^{4}\right ) \]

[In]

integrate((e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**6*d**4*x + b**6*e**4*x**11/11 + x**10*(3*a*b**5*e**4/5 + 2*b**6*d*e**3/5) + x**9*(5*a**2*b**4*e**4/3 + 8*a*
b**5*d*e**3/3 + 2*b**6*d**2*e**2/3) + x**8*(5*a**3*b**3*e**4/2 + 15*a**2*b**4*d*e**3/2 + 9*a*b**5*d**2*e**2/2
+ b**6*d**3*e/2) + x**7*(15*a**4*b**2*e**4/7 + 80*a**3*b**3*d*e**3/7 + 90*a**2*b**4*d**2*e**2/7 + 24*a*b**5*d*
*3*e/7 + b**6*d**4/7) + x**6*(a**5*b*e**4 + 10*a**4*b**2*d*e**3 + 20*a**3*b**3*d**2*e**2 + 10*a**2*b**4*d**3*e
+ a*b**5*d**4) + x**5*(a**6*e**4/5 + 24*a**5*b*d*e**3/5 + 18*a**4*b**2*d**2*e**2 + 16*a**3*b**3*d**3*e + 3*a*
*2*b**4*d**4) + x**4*(a**6*d*e**3 + 9*a**5*b*d**2*e**2 + 15*a**4*b**2*d**3*e + 5*a**3*b**3*d**4) + x**3*(2*a**
6*d**2*e**2 + 8*a**5*b*d**3*e + 5*a**4*b**2*d**4) + x**2*(2*a**6*d**3*e + 3*a**5*b*d**4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (109) = 218\).

Time = 0.19 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.51 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{11} \, b^{6} e^{4} x^{11} + a^{6} d^{4} x + \frac {1}{5} \, {\left (2 \, b^{6} d e^{3} + 3 \, a b^{5} e^{4}\right )} x^{10} + \frac {1}{3} \, {\left (2 \, b^{6} d^{2} e^{2} + 8 \, a b^{5} d e^{3} + 5 \, a^{2} b^{4} e^{4}\right )} x^{9} + \frac {1}{2} \, {\left (b^{6} d^{3} e + 9 \, a b^{5} d^{2} e^{2} + 15 \, a^{2} b^{4} d e^{3} + 5 \, a^{3} b^{3} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{4} + 24 \, a b^{5} d^{3} e + 90 \, a^{2} b^{4} d^{2} e^{2} + 80 \, a^{3} b^{3} d e^{3} + 15 \, a^{4} b^{2} e^{4}\right )} x^{7} + {\left (a b^{5} d^{4} + 10 \, a^{2} b^{4} d^{3} e + 20 \, a^{3} b^{3} d^{2} e^{2} + 10 \, a^{4} b^{2} d e^{3} + a^{5} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (15 \, a^{2} b^{4} d^{4} + 80 \, a^{3} b^{3} d^{3} e + 90 \, a^{4} b^{2} d^{2} e^{2} + 24 \, a^{5} b d e^{3} + a^{6} e^{4}\right )} x^{5} + {\left (5 \, a^{3} b^{3} d^{4} + 15 \, a^{4} b^{2} d^{3} e + 9 \, a^{5} b d^{2} e^{2} + a^{6} d e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{4} + 8 \, a^{5} b d^{3} e + 2 \, a^{6} d^{2} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{4} + 2 \, a^{6} d^{3} e\right )} x^{2} \]

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/11*b^6*e^4*x^11 + a^6*d^4*x + 1/5*(2*b^6*d*e^3 + 3*a*b^5*e^4)*x^10 + 1/3*(2*b^6*d^2*e^2 + 8*a*b^5*d*e^3 + 5*
a^2*b^4*e^4)*x^9 + 1/2*(b^6*d^3*e + 9*a*b^5*d^2*e^2 + 15*a^2*b^4*d*e^3 + 5*a^3*b^3*e^4)*x^8 + 1/7*(b^6*d^4 + 2
4*a*b^5*d^3*e + 90*a^2*b^4*d^2*e^2 + 80*a^3*b^3*d*e^3 + 15*a^4*b^2*e^4)*x^7 + (a*b^5*d^4 + 10*a^2*b^4*d^3*e +
20*a^3*b^3*d^2*e^2 + 10*a^4*b^2*d*e^3 + a^5*b*e^4)*x^6 + 1/5*(15*a^2*b^4*d^4 + 80*a^3*b^3*d^3*e + 90*a^4*b^2*d
^2*e^2 + 24*a^5*b*d*e^3 + a^6*e^4)*x^5 + (5*a^3*b^3*d^4 + 15*a^4*b^2*d^3*e + 9*a^5*b*d^2*e^2 + a^6*d*e^3)*x^4
+ (5*a^4*b^2*d^4 + 8*a^5*b*d^3*e + 2*a^6*d^2*e^2)*x^3 + (3*a^5*b*d^4 + 2*a^6*d^3*e)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (109) = 218\).

Time = 0.25 (sec) , antiderivative size = 470, normalized size of antiderivative = 3.95 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{11} \, b^{6} e^{4} x^{11} + \frac {2}{5} \, b^{6} d e^{3} x^{10} + \frac {3}{5} \, a b^{5} e^{4} x^{10} + \frac {2}{3} \, b^{6} d^{2} e^{2} x^{9} + \frac {8}{3} \, a b^{5} d e^{3} x^{9} + \frac {5}{3} \, a^{2} b^{4} e^{4} x^{9} + \frac {1}{2} \, b^{6} d^{3} e x^{8} + \frac {9}{2} \, a b^{5} d^{2} e^{2} x^{8} + \frac {15}{2} \, a^{2} b^{4} d e^{3} x^{8} + \frac {5}{2} \, a^{3} b^{3} e^{4} x^{8} + \frac {1}{7} \, b^{6} d^{4} x^{7} + \frac {24}{7} \, a b^{5} d^{3} e x^{7} + \frac {90}{7} \, a^{2} b^{4} d^{2} e^{2} x^{7} + \frac {80}{7} \, a^{3} b^{3} d e^{3} x^{7} + \frac {15}{7} \, a^{4} b^{2} e^{4} x^{7} + a b^{5} d^{4} x^{6} + 10 \, a^{2} b^{4} d^{3} e x^{6} + 20 \, a^{3} b^{3} d^{2} e^{2} x^{6} + 10 \, a^{4} b^{2} d e^{3} x^{6} + a^{5} b e^{4} x^{6} + 3 \, a^{2} b^{4} d^{4} x^{5} + 16 \, a^{3} b^{3} d^{3} e x^{5} + 18 \, a^{4} b^{2} d^{2} e^{2} x^{5} + \frac {24}{5} \, a^{5} b d e^{3} x^{5} + \frac {1}{5} \, a^{6} e^{4} x^{5} + 5 \, a^{3} b^{3} d^{4} x^{4} + 15 \, a^{4} b^{2} d^{3} e x^{4} + 9 \, a^{5} b d^{2} e^{2} x^{4} + a^{6} d e^{3} x^{4} + 5 \, a^{4} b^{2} d^{4} x^{3} + 8 \, a^{5} b d^{3} e x^{3} + 2 \, a^{6} d^{2} e^{2} x^{3} + 3 \, a^{5} b d^{4} x^{2} + 2 \, a^{6} d^{3} e x^{2} + a^{6} d^{4} x \]

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/11*b^6*e^4*x^11 + 2/5*b^6*d*e^3*x^10 + 3/5*a*b^5*e^4*x^10 + 2/3*b^6*d^2*e^2*x^9 + 8/3*a*b^5*d*e^3*x^9 + 5/3*
a^2*b^4*e^4*x^9 + 1/2*b^6*d^3*e*x^8 + 9/2*a*b^5*d^2*e^2*x^8 + 15/2*a^2*b^4*d*e^3*x^8 + 5/2*a^3*b^3*e^4*x^8 + 1
/7*b^6*d^4*x^7 + 24/7*a*b^5*d^3*e*x^7 + 90/7*a^2*b^4*d^2*e^2*x^7 + 80/7*a^3*b^3*d*e^3*x^7 + 15/7*a^4*b^2*e^4*x
^7 + a*b^5*d^4*x^6 + 10*a^2*b^4*d^3*e*x^6 + 20*a^3*b^3*d^2*e^2*x^6 + 10*a^4*b^2*d*e^3*x^6 + a^5*b*e^4*x^6 + 3*
a^2*b^4*d^4*x^5 + 16*a^3*b^3*d^3*e*x^5 + 18*a^4*b^2*d^2*e^2*x^5 + 24/5*a^5*b*d*e^3*x^5 + 1/5*a^6*e^4*x^5 + 5*a
^3*b^3*d^4*x^4 + 15*a^4*b^2*d^3*e*x^4 + 9*a^5*b*d^2*e^2*x^4 + a^6*d*e^3*x^4 + 5*a^4*b^2*d^4*x^3 + 8*a^5*b*d^3*
e*x^3 + 2*a^6*d^2*e^2*x^3 + 3*a^5*b*d^4*x^2 + 2*a^6*d^3*e*x^2 + a^6*d^4*x

Mupad [B] (verification not implemented)

Time = 10.16 (sec) , antiderivative size = 402, normalized size of antiderivative = 3.38 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^5\,\left (\frac {a^6\,e^4}{5}+\frac {24\,a^5\,b\,d\,e^3}{5}+18\,a^4\,b^2\,d^2\,e^2+16\,a^3\,b^3\,d^3\,e+3\,a^2\,b^4\,d^4\right )+x^7\,\left (\frac {15\,a^4\,b^2\,e^4}{7}+\frac {80\,a^3\,b^3\,d\,e^3}{7}+\frac {90\,a^2\,b^4\,d^2\,e^2}{7}+\frac {24\,a\,b^5\,d^3\,e}{7}+\frac {b^6\,d^4}{7}\right )+x^4\,\left (a^6\,d\,e^3+9\,a^5\,b\,d^2\,e^2+15\,a^4\,b^2\,d^3\,e+5\,a^3\,b^3\,d^4\right )+x^8\,\left (\frac {5\,a^3\,b^3\,e^4}{2}+\frac {15\,a^2\,b^4\,d\,e^3}{2}+\frac {9\,a\,b^5\,d^2\,e^2}{2}+\frac {b^6\,d^3\,e}{2}\right )+x^6\,\left (a^5\,b\,e^4+10\,a^4\,b^2\,d\,e^3+20\,a^3\,b^3\,d^2\,e^2+10\,a^2\,b^4\,d^3\,e+a\,b^5\,d^4\right )+a^6\,d^4\,x+\frac {b^6\,e^4\,x^{11}}{11}+a^5\,d^3\,x^2\,\left (2\,a\,e+3\,b\,d\right )+\frac {b^5\,e^3\,x^{10}\,\left (3\,a\,e+2\,b\,d\right )}{5}+a^4\,d^2\,x^3\,\left (2\,a^2\,e^2+8\,a\,b\,d\,e+5\,b^2\,d^2\right )+\frac {b^4\,e^2\,x^9\,\left (5\,a^2\,e^2+8\,a\,b\,d\,e+2\,b^2\,d^2\right )}{3} \]

[In]

int((d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

x^5*((a^6*e^4)/5 + 3*a^2*b^4*d^4 + 16*a^3*b^3*d^3*e + 18*a^4*b^2*d^2*e^2 + (24*a^5*b*d*e^3)/5) + x^7*((b^6*d^4
)/7 + (15*a^4*b^2*e^4)/7 + (80*a^3*b^3*d*e^3)/7 + (90*a^2*b^4*d^2*e^2)/7 + (24*a*b^5*d^3*e)/7) + x^4*(a^6*d*e^
3 + 5*a^3*b^3*d^4 + 15*a^4*b^2*d^3*e + 9*a^5*b*d^2*e^2) + x^8*((b^6*d^3*e)/2 + (5*a^3*b^3*e^4)/2 + (9*a*b^5*d^
2*e^2)/2 + (15*a^2*b^4*d*e^3)/2) + x^6*(a*b^5*d^4 + a^5*b*e^4 + 10*a^2*b^4*d^3*e + 10*a^4*b^2*d*e^3 + 20*a^3*b
^3*d^2*e^2) + a^6*d^4*x + (b^6*e^4*x^11)/11 + a^5*d^3*x^2*(2*a*e + 3*b*d) + (b^5*e^3*x^10*(3*a*e + 2*b*d))/5 +
a^4*d^2*x^3*(2*a^2*e^2 + 5*b^2*d^2 + 8*a*b*d*e) + (b^4*e^2*x^9*(5*a^2*e^2 + 2*b^2*d^2 + 8*a*b*d*e))/3