\(\int (d+e x)^3 (a^2+2 a b x+b^2 x^2)^3 \, dx\) [1486]

Optimal result

Integrand size = 26, antiderivative size = 92 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(b d-a e)^3 (a+b x)^7}{7 b^4}+\frac {3 e (b d-a e)^2 (a+b x)^8}{8 b^4}+\frac {e^2 (b d-a e) (a+b x)^9}{3 b^4}+\frac {e^3 (a+b x)^{10}}{10 b^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 92, 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} \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {e^2 (a+b x)^9 (b d-a e)}{3 b^4}+\frac {3 e (a+b x)^8 (b d-a e)^2}{8 b^4}+\frac {(a+b x)^7 (b d-a e)^3}{7 b^4}+\frac {e^3 (a+b x)^{10}}{10 b^4} \]

[In]

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

[Out]

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(276\) vs. \(2(92)=184\).

Time = 0.06 (sec) , antiderivative size = 276, normalized size of antiderivative = 3.00 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{840} x \left (210 a^6 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+252 a^5 b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+210 a^4 b^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+120 a^3 b^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+45 a^2 b^4 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+10 a b^5 x^5 \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )+b^6 x^6 \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )\right ) \]

[In]

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

[Out]

(x*(210*a^6*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 252*a^5*b*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*
e^3*x^3) + 210*a^4*b^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 120*a^3*b^3*x^3*(35*d^3 + 84*d^
2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 45*a^2*b^4*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + 10*a
*b^5*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + b^6*x^6*(120*d^3 + 315*d^2*e*x + 280*d*e^2*x^2
+ 84*e^3*x^3)))/840

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(324\) vs. \(2(84)=168\).

Time = 2.30 (sec) , antiderivative size = 325, normalized size of antiderivative = 3.53

method	result	size
norman	\(\frac {e^{3} b^{6} x^{10}}{10}+\left (\frac {2}{3} e^{3} a \,b^{5}+\frac {1}{3} d \,e^{2} b^{6}\right ) x^{9}+\left (\frac {15}{8} e^{3} a^{2} b^{4}+\frac {9}{4} d \,e^{2} a \,b^{5}+\frac {3}{8} d^{2} e \,b^{6}\right ) x^{8}+\left (\frac {20}{7} e^{3} a^{3} b^{3}+\frac {45}{7} d \,e^{2} a^{2} b^{4}+\frac {18}{7} d^{2} e a \,b^{5}+\frac {1}{7} d^{3} b^{6}\right ) x^{7}+\left (\frac {5}{2} e^{3} a^{4} b^{2}+10 d \,e^{2} a^{3} b^{3}+\frac {15}{2} d^{2} e \,a^{2} b^{4}+d^{3} a \,b^{5}\right ) x^{6}+\left (\frac {6}{5} a^{5} b \,e^{3}+9 d \,e^{2} a^{4} b^{2}+12 d^{2} e \,a^{3} b^{3}+3 d^{3} a^{2} b^{4}\right ) x^{5}+\left (\frac {1}{4} a^{6} e^{3}+\frac {9}{2} d \,e^{2} a^{5} b +\frac {45}{4} d^{2} e \,a^{4} b^{2}+5 a^{3} b^{3} d^{3}\right ) x^{4}+\left (d \,e^{2} a^{6}+6 d^{2} e \,a^{5} b +5 d^{3} a^{4} b^{2}\right ) x^{3}+\left (\frac {3}{2} d^{2} e \,a^{6}+3 a^{5} b \,d^{3}\right ) x^{2}+d^{3} a^{6} x\)	\(325\)
default	\(\frac {e^{3} b^{6} x^{10}}{10}+\frac {\left (6 e^{3} a \,b^{5}+3 d \,e^{2} b^{6}\right ) x^{9}}{9}+\frac {\left (15 e^{3} a^{2} b^{4}+18 d \,e^{2} a \,b^{5}+3 d^{2} e \,b^{6}\right ) x^{8}}{8}+\frac {\left (20 e^{3} a^{3} b^{3}+45 d \,e^{2} a^{2} b^{4}+18 d^{2} e a \,b^{5}+d^{3} b^{6}\right ) x^{7}}{7}+\frac {\left (15 e^{3} a^{4} b^{2}+60 d \,e^{2} a^{3} b^{3}+45 d^{2} e \,a^{2} b^{4}+6 d^{3} a \,b^{5}\right ) x^{6}}{6}+\frac {\left (6 a^{5} b \,e^{3}+45 d \,e^{2} a^{4} b^{2}+60 d^{2} e \,a^{3} b^{3}+15 d^{3} a^{2} b^{4}\right ) x^{5}}{5}+\frac {\left (a^{6} e^{3}+18 d \,e^{2} a^{5} b +45 d^{2} e \,a^{4} b^{2}+20 a^{3} b^{3} d^{3}\right ) x^{4}}{4}+\frac {\left (3 d \,e^{2} a^{6}+18 d^{2} e \,a^{5} b +15 d^{3} a^{4} b^{2}\right ) x^{3}}{3}+\frac {\left (3 d^{2} e \,a^{6}+6 a^{5} b \,d^{3}\right ) x^{2}}{2}+d^{3} a^{6} x\)	\(333\)
risch	\(\frac {1}{3} x^{9} d \,e^{2} b^{6}+\frac {15}{8} x^{8} e^{3} a^{2} b^{4}+\frac {3}{8} x^{8} d^{2} e \,b^{6}+\frac {20}{7} x^{7} e^{3} a^{3} b^{3}+\frac {3}{2} x^{2} d^{2} e \,a^{6}+\frac {1}{7} x^{7} d^{3} b^{6}+x^{6} d^{3} a \,b^{5}+\frac {9}{4} x^{8} d \,e^{2} a \,b^{5}+\frac {45}{7} x^{7} d \,e^{2} a^{2} b^{4}+\frac {45}{4} x^{4} d^{2} e \,a^{4} b^{2}+\frac {18}{7} x^{7} d^{2} e a \,b^{5}+10 x^{6} d \,e^{2} a^{3} b^{3}+\frac {15}{2} x^{6} d^{2} e \,a^{2} b^{4}+9 x^{5} d \,e^{2} a^{4} b^{2}+12 x^{5} d^{2} e \,a^{3} b^{3}+\frac {9}{2} x^{4} d \,e^{2} a^{5} b +\frac {1}{10} e^{3} b^{6} x^{10}+d^{3} a^{6} x +\frac {2}{3} x^{9} e^{3} a \,b^{5}+\frac {1}{4} x^{4} a^{6} e^{3}+6 a^{5} b \,d^{2} e \,x^{3}+\frac {5}{2} x^{6} e^{3} a^{4} b^{2}+\frac {6}{5} x^{5} a^{5} b \,e^{3}+3 x^{5} d^{3} a^{2} b^{4}+5 x^{4} a^{3} b^{3} d^{3}+3 x^{2} a^{5} b \,d^{3}+a^{6} d \,e^{2} x^{3}+5 a^{4} b^{2} d^{3} x^{3}\)	\(363\)
parallelrisch	\(\frac {1}{3} x^{9} d \,e^{2} b^{6}+\frac {15}{8} x^{8} e^{3} a^{2} b^{4}+\frac {3}{8} x^{8} d^{2} e \,b^{6}+\frac {20}{7} x^{7} e^{3} a^{3} b^{3}+\frac {3}{2} x^{2} d^{2} e \,a^{6}+\frac {1}{7} x^{7} d^{3} b^{6}+x^{6} d^{3} a \,b^{5}+\frac {9}{4} x^{8} d \,e^{2} a \,b^{5}+\frac {45}{7} x^{7} d \,e^{2} a^{2} b^{4}+\frac {45}{4} x^{4} d^{2} e \,a^{4} b^{2}+\frac {18}{7} x^{7} d^{2} e a \,b^{5}+10 x^{6} d \,e^{2} a^{3} b^{3}+\frac {15}{2} x^{6} d^{2} e \,a^{2} b^{4}+9 x^{5} d \,e^{2} a^{4} b^{2}+12 x^{5} d^{2} e \,a^{3} b^{3}+\frac {9}{2} x^{4} d \,e^{2} a^{5} b +\frac {1}{10} e^{3} b^{6} x^{10}+d^{3} a^{6} x +\frac {2}{3} x^{9} e^{3} a \,b^{5}+\frac {1}{4} x^{4} a^{6} e^{3}+6 a^{5} b \,d^{2} e \,x^{3}+\frac {5}{2} x^{6} e^{3} a^{4} b^{2}+\frac {6}{5} x^{5} a^{5} b \,e^{3}+3 x^{5} d^{3} a^{2} b^{4}+5 x^{4} a^{3} b^{3} d^{3}+3 x^{2} a^{5} b \,d^{3}+a^{6} d \,e^{2} x^{3}+5 a^{4} b^{2} d^{3} x^{3}\)	\(363\)
gosper	\(\frac {x \left (84 e^{3} b^{6} x^{9}+560 x^{8} e^{3} a \,b^{5}+280 x^{8} d \,e^{2} b^{6}+1575 x^{7} e^{3} a^{2} b^{4}+1890 x^{7} d \,e^{2} a \,b^{5}+315 x^{7} d^{2} e \,b^{6}+2400 x^{6} e^{3} a^{3} b^{3}+5400 x^{6} d \,e^{2} a^{2} b^{4}+2160 x^{6} d^{2} e a \,b^{5}+120 x^{6} d^{3} b^{6}+2100 x^{5} e^{3} a^{4} b^{2}+8400 x^{5} d \,e^{2} a^{3} b^{3}+6300 x^{5} d^{2} e \,a^{2} b^{4}+840 x^{5} d^{3} a \,b^{5}+1008 x^{4} a^{5} b \,e^{3}+7560 x^{4} d \,e^{2} a^{4} b^{2}+10080 x^{4} d^{2} e \,a^{3} b^{3}+2520 x^{4} d^{3} a^{2} b^{4}+210 x^{3} a^{6} e^{3}+3780 x^{3} d \,e^{2} a^{5} b +9450 x^{3} d^{2} e \,a^{4} b^{2}+4200 x^{3} a^{3} b^{3} d^{3}+840 a^{6} d \,e^{2} x^{2}+5040 a^{5} b \,d^{2} e \,x^{2}+4200 a^{4} b^{2} d^{3} x^{2}+1260 x \,d^{2} e \,a^{6}+2520 x \,a^{5} b \,d^{3}+840 d^{3} a^{6}\right )}{840}\)	\(364\)

[In]

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

[Out]

1/10*e^3*b^6*x^10+(2/3*e^3*a*b^5+1/3*d*e^2*b^6)*x^9+(15/8*e^3*a^2*b^4+9/4*d*e^2*a*b^5+3/8*d^2*e*b^6)*x^8+(20/7
*e^3*a^3*b^3+45/7*d*e^2*a^2*b^4+18/7*d^2*e*a*b^5+1/7*d^3*b^6)*x^7+(5/2*e^3*a^4*b^2+10*d*e^2*a^3*b^3+15/2*d^2*e
*a^2*b^4+d^3*a*b^5)*x^6+(6/5*a^5*b*e^3+9*d*e^2*a^4*b^2+12*d^2*e*a^3*b^3+3*d^3*a^2*b^4)*x^5+(1/4*a^6*e^3+9/2*d*
e^2*a^5*b+45/4*d^2*e*a^4*b^2+5*a^3*b^3*d^3)*x^4+(a^6*d*e^2+6*a^5*b*d^2*e+5*a^4*b^2*d^3)*x^3+(3/2*d^2*e*a^6+3*a
^5*b*d^3)*x^2+d^3*a^6*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (84) = 168\).

Time = 0.28 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.55 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{10} \, b^{6} e^{3} x^{10} + a^{6} d^{3} x + \frac {1}{3} \, {\left (b^{6} d e^{2} + 2 \, a b^{5} e^{3}\right )} x^{9} + \frac {3}{8} \, {\left (b^{6} d^{2} e + 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{3} + 18 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} + 20 \, a^{3} b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d^{3} + 15 \, a^{2} b^{4} d^{2} e + 20 \, a^{3} b^{3} d e^{2} + 5 \, a^{4} b^{2} e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (5 \, a^{2} b^{4} d^{3} + 20 \, a^{3} b^{3} d^{2} e + 15 \, a^{4} b^{2} d e^{2} + 2 \, a^{5} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} d^{3} + 45 \, a^{4} b^{2} d^{2} e + 18 \, a^{5} b d e^{2} + a^{6} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{3} + 6 \, a^{5} b d^{2} e + a^{6} d e^{2}\right )} x^{3} + \frac {3}{2} \, {\left (2 \, a^{5} b d^{3} + a^{6} d^{2} e\right )} x^{2} \]

[In]

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

[Out]

1/10*b^6*e^3*x^10 + a^6*d^3*x + 1/3*(b^6*d*e^2 + 2*a*b^5*e^3)*x^9 + 3/8*(b^6*d^2*e + 6*a*b^5*d*e^2 + 5*a^2*b^4
*e^3)*x^8 + 1/7*(b^6*d^3 + 18*a*b^5*d^2*e + 45*a^2*b^4*d*e^2 + 20*a^3*b^3*e^3)*x^7 + 1/2*(2*a*b^5*d^3 + 15*a^2
*b^4*d^2*e + 20*a^3*b^3*d*e^2 + 5*a^4*b^2*e^3)*x^6 + 3/5*(5*a^2*b^4*d^3 + 20*a^3*b^3*d^2*e + 15*a^4*b^2*d*e^2
+ 2*a^5*b*e^3)*x^5 + 1/4*(20*a^3*b^3*d^3 + 45*a^4*b^2*d^2*e + 18*a^5*b*d*e^2 + a^6*e^3)*x^4 + (5*a^4*b^2*d^3 +
6*a^5*b*d^2*e + a^6*d*e^2)*x^3 + 3/2*(2*a^5*b*d^3 + a^6*d^2*e)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (80) = 160\).

Time = 0.04 (sec) , antiderivative size = 364, normalized size of antiderivative = 3.96 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{6} d^{3} x + \frac {b^{6} e^{3} x^{10}}{10} + x^{9} \cdot \left (\frac {2 a b^{5} e^{3}}{3} + \frac {b^{6} d e^{2}}{3}\right ) + x^{8} \cdot \left (\frac {15 a^{2} b^{4} e^{3}}{8} + \frac {9 a b^{5} d e^{2}}{4} + \frac {3 b^{6} d^{2} e}{8}\right ) + x^{7} \cdot \left (\frac {20 a^{3} b^{3} e^{3}}{7} + \frac {45 a^{2} b^{4} d e^{2}}{7} + \frac {18 a b^{5} d^{2} e}{7} + \frac {b^{6} d^{3}}{7}\right ) + x^{6} \cdot \left (\frac {5 a^{4} b^{2} e^{3}}{2} + 10 a^{3} b^{3} d e^{2} + \frac {15 a^{2} b^{4} d^{2} e}{2} + a b^{5} d^{3}\right ) + x^{5} \cdot \left (\frac {6 a^{5} b e^{3}}{5} + 9 a^{4} b^{2} d e^{2} + 12 a^{3} b^{3} d^{2} e + 3 a^{2} b^{4} d^{3}\right ) + x^{4} \left (\frac {a^{6} e^{3}}{4} + \frac {9 a^{5} b d e^{2}}{2} + \frac {45 a^{4} b^{2} d^{2} e}{4} + 5 a^{3} b^{3} d^{3}\right ) + x^{3} \left (a^{6} d e^{2} + 6 a^{5} b d^{2} e + 5 a^{4} b^{2} d^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{6} d^{2} e}{2} + 3 a^{5} b d^{3}\right ) \]

[In]

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

[Out]

a**6*d**3*x + b**6*e**3*x**10/10 + x**9*(2*a*b**5*e**3/3 + b**6*d*e**2/3) + x**8*(15*a**2*b**4*e**3/8 + 9*a*b*
*5*d*e**2/4 + 3*b**6*d**2*e/8) + x**7*(20*a**3*b**3*e**3/7 + 45*a**2*b**4*d*e**2/7 + 18*a*b**5*d**2*e/7 + b**6
*d**3/7) + x**6*(5*a**4*b**2*e**3/2 + 10*a**3*b**3*d*e**2 + 15*a**2*b**4*d**2*e/2 + a*b**5*d**3) + x**5*(6*a**
5*b*e**3/5 + 9*a**4*b**2*d*e**2 + 12*a**3*b**3*d**2*e + 3*a**2*b**4*d**3) + x**4*(a**6*e**3/4 + 9*a**5*b*d*e**
2/2 + 45*a**4*b**2*d**2*e/4 + 5*a**3*b**3*d**3) + x**3*(a**6*d*e**2 + 6*a**5*b*d**2*e + 5*a**4*b**2*d**3) + x*
*2*(3*a**6*d**2*e/2 + 3*a**5*b*d**3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (84) = 168\).

Time = 0.19 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.55 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{10} \, b^{6} e^{3} x^{10} + a^{6} d^{3} x + \frac {1}{3} \, {\left (b^{6} d e^{2} + 2 \, a b^{5} e^{3}\right )} x^{9} + \frac {3}{8} \, {\left (b^{6} d^{2} e + 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{3} + 18 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} + 20 \, a^{3} b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d^{3} + 15 \, a^{2} b^{4} d^{2} e + 20 \, a^{3} b^{3} d e^{2} + 5 \, a^{4} b^{2} e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (5 \, a^{2} b^{4} d^{3} + 20 \, a^{3} b^{3} d^{2} e + 15 \, a^{4} b^{2} d e^{2} + 2 \, a^{5} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} d^{3} + 45 \, a^{4} b^{2} d^{2} e + 18 \, a^{5} b d e^{2} + a^{6} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{3} + 6 \, a^{5} b d^{2} e + a^{6} d e^{2}\right )} x^{3} + \frac {3}{2} \, {\left (2 \, a^{5} b d^{3} + a^{6} d^{2} e\right )} x^{2} \]

[In]

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

[Out]

1/10*b^6*e^3*x^10 + a^6*d^3*x + 1/3*(b^6*d*e^2 + 2*a*b^5*e^3)*x^9 + 3/8*(b^6*d^2*e + 6*a*b^5*d*e^2 + 5*a^2*b^4
*e^3)*x^8 + 1/7*(b^6*d^3 + 18*a*b^5*d^2*e + 45*a^2*b^4*d*e^2 + 20*a^3*b^3*e^3)*x^7 + 1/2*(2*a*b^5*d^3 + 15*a^2
*b^4*d^2*e + 20*a^3*b^3*d*e^2 + 5*a^4*b^2*e^3)*x^6 + 3/5*(5*a^2*b^4*d^3 + 20*a^3*b^3*d^2*e + 15*a^4*b^2*d*e^2
+ 2*a^5*b*e^3)*x^5 + 1/4*(20*a^3*b^3*d^3 + 45*a^4*b^2*d^2*e + 18*a^5*b*d*e^2 + a^6*e^3)*x^4 + (5*a^4*b^2*d^3 +
6*a^5*b*d^2*e + a^6*d*e^2)*x^3 + 3/2*(2*a^5*b*d^3 + a^6*d^2*e)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (84) = 168\).

Time = 0.26 (sec) , antiderivative size = 362, normalized size of antiderivative = 3.93 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{10} \, b^{6} e^{3} x^{10} + \frac {1}{3} \, b^{6} d e^{2} x^{9} + \frac {2}{3} \, a b^{5} e^{3} x^{9} + \frac {3}{8} \, b^{6} d^{2} e x^{8} + \frac {9}{4} \, a b^{5} d e^{2} x^{8} + \frac {15}{8} \, a^{2} b^{4} e^{3} x^{8} + \frac {1}{7} \, b^{6} d^{3} x^{7} + \frac {18}{7} \, a b^{5} d^{2} e x^{7} + \frac {45}{7} \, a^{2} b^{4} d e^{2} x^{7} + \frac {20}{7} \, a^{3} b^{3} e^{3} x^{7} + a b^{5} d^{3} x^{6} + \frac {15}{2} \, a^{2} b^{4} d^{2} e x^{6} + 10 \, a^{3} b^{3} d e^{2} x^{6} + \frac {5}{2} \, a^{4} b^{2} e^{3} x^{6} + 3 \, a^{2} b^{4} d^{3} x^{5} + 12 \, a^{3} b^{3} d^{2} e x^{5} + 9 \, a^{4} b^{2} d e^{2} x^{5} + \frac {6}{5} \, a^{5} b e^{3} x^{5} + 5 \, a^{3} b^{3} d^{3} x^{4} + \frac {45}{4} \, a^{4} b^{2} d^{2} e x^{4} + \frac {9}{2} \, a^{5} b d e^{2} x^{4} + \frac {1}{4} \, a^{6} e^{3} x^{4} + 5 \, a^{4} b^{2} d^{3} x^{3} + 6 \, a^{5} b d^{2} e x^{3} + a^{6} d e^{2} x^{3} + 3 \, a^{5} b d^{3} x^{2} + \frac {3}{2} \, a^{6} d^{2} e x^{2} + a^{6} d^{3} x \]

[In]

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

[Out]

1/10*b^6*e^3*x^10 + 1/3*b^6*d*e^2*x^9 + 2/3*a*b^5*e^3*x^9 + 3/8*b^6*d^2*e*x^8 + 9/4*a*b^5*d*e^2*x^8 + 15/8*a^2
*b^4*e^3*x^8 + 1/7*b^6*d^3*x^7 + 18/7*a*b^5*d^2*e*x^7 + 45/7*a^2*b^4*d*e^2*x^7 + 20/7*a^3*b^3*e^3*x^7 + a*b^5*
d^3*x^6 + 15/2*a^2*b^4*d^2*e*x^6 + 10*a^3*b^3*d*e^2*x^6 + 5/2*a^4*b^2*e^3*x^6 + 3*a^2*b^4*d^3*x^5 + 12*a^3*b^3
*d^2*e*x^5 + 9*a^4*b^2*d*e^2*x^5 + 6/5*a^5*b*e^3*x^5 + 5*a^3*b^3*d^3*x^4 + 45/4*a^4*b^2*d^2*e*x^4 + 9/2*a^5*b*
d*e^2*x^4 + 1/4*a^6*e^3*x^4 + 5*a^4*b^2*d^3*x^3 + 6*a^5*b*d^2*e*x^3 + a^6*d*e^2*x^3 + 3*a^5*b*d^3*x^2 + 3/2*a^
6*d^2*e*x^2 + a^6*d^3*x

Mupad [B] (verification not implemented)

Time = 10.16 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.35 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^5\,\left (\frac {6\,a^5\,b\,e^3}{5}+9\,a^4\,b^2\,d\,e^2+12\,a^3\,b^3\,d^2\,e+3\,a^2\,b^4\,d^3\right )+x^6\,\left (\frac {5\,a^4\,b^2\,e^3}{2}+10\,a^3\,b^3\,d\,e^2+\frac {15\,a^2\,b^4\,d^2\,e}{2}+a\,b^5\,d^3\right )+x^4\,\left (\frac {a^6\,e^3}{4}+\frac {9\,a^5\,b\,d\,e^2}{2}+\frac {45\,a^4\,b^2\,d^2\,e}{4}+5\,a^3\,b^3\,d^3\right )+x^7\,\left (\frac {20\,a^3\,b^3\,e^3}{7}+\frac {45\,a^2\,b^4\,d\,e^2}{7}+\frac {18\,a\,b^5\,d^2\,e}{7}+\frac {b^6\,d^3}{7}\right )+a^6\,d^3\,x+\frac {b^6\,e^3\,x^{10}}{10}+\frac {3\,a^5\,d^2\,x^2\,\left (a\,e+2\,b\,d\right )}{2}+\frac {b^5\,e^2\,x^9\,\left (2\,a\,e+b\,d\right )}{3}+a^4\,d\,x^3\,\left (a^2\,e^2+6\,a\,b\,d\,e+5\,b^2\,d^2\right )+\frac {3\,b^4\,e\,x^8\,\left (5\,a^2\,e^2+6\,a\,b\,d\,e+b^2\,d^2\right )}{8} \]

[In]

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

[Out]

x^5*((6*a^5*b*e^3)/5 + 3*a^2*b^4*d^3 + 12*a^3*b^3*d^2*e + 9*a^4*b^2*d*e^2) + x^6*(a*b^5*d^3 + (5*a^4*b^2*e^3)/
2 + (15*a^2*b^4*d^2*e)/2 + 10*a^3*b^3*d*e^2) + x^4*((a^6*e^3)/4 + 5*a^3*b^3*d^3 + (45*a^4*b^2*d^2*e)/4 + (9*a^
5*b*d*e^2)/2) + x^7*((b^6*d^3)/7 + (20*a^3*b^3*e^3)/7 + (45*a^2*b^4*d*e^2)/7 + (18*a*b^5*d^2*e)/7) + a^6*d^3*x
+ (b^6*e^3*x^10)/10 + (3*a^5*d^2*x^2*(a*e + 2*b*d))/2 + (b^5*e^2*x^9*(2*a*e + b*d))/3 + a^4*d*x^3*(a^2*e^2 +
5*b^2*d^2 + 6*a*b*d*e) + (3*b^4*e*x^8*(5*a^2*e^2 + b^2*d^2 + 6*a*b*d*e))/8