3.1907 \(\int (a+b x) (d+e x)^4 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=119 \[ \frac{4 e^3 (a+b x)^9 (b d-a e)}{9 b^5}+\frac{3 e^2 (a+b x)^8 (b d-a e)^2}{4 b^5}+\frac{4 e (a+b x)^7 (b d-a e)^3}{7 b^5}+\frac{(a+b x)^6 (b d-a e)^4}{6 b^5}+\frac{e^4 (a+b x)^{10}}{10 b^5} \]

[Out]

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

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Rubi [A]  time = 0.215815, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ \frac{4 e^3 (a+b x)^9 (b d-a e)}{9 b^5}+\frac{3 e^2 (a+b x)^8 (b d-a e)^2}{4 b^5}+\frac{4 e (a+b x)^7 (b d-a e)^3}{7 b^5}+\frac{(a+b x)^6 (b d-a e)^4}{6 b^5}+\frac{e^4 (a+b x)^{10}}{10 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - a*e)^4*(a + b*x)^6)/(6*b^5) + (4*e*(b*d - a*e)^3*(a + b*x)^7)/(7*b^5) + (3*e^2*(b*d - a*e)^2*(a + b*x)
^8)/(4*b^5) + (4*e^3*(b*d - a*e)*(a + b*x)^9)/(9*b^5) + (e^4*(a + b*x)^10)/(10*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 43

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

Mathematica [B]  time = 0.0892771, size = 301, normalized size = 2.53 \[ \frac{x \left (120 a^3 b^2 x^2 \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )+45 a^2 b^3 x^3 \left (280 d^2 e^2 x^2+224 d^3 e x+70 d^4+160 d e^3 x^3+35 e^4 x^4\right )+210 a^4 b x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+252 a^5 \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+10 a b^4 x^4 \left (540 d^2 e^2 x^2+420 d^3 e x+126 d^4+315 d e^3 x^3+70 e^4 x^4\right )+b^5 x^5 \left (945 d^2 e^2 x^2+720 d^3 e x+210 d^4+560 d e^3 x^3+126 e^4 x^4\right )\right )}{1260} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(252*a^5*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + 210*a^4*b*x*(15*d^4 + 40*d^3*e*x +
45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 120*a^3*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e
^3*x^3 + 15*e^4*x^4) + 45*a^2*b^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4) +
10*a*b^4*x^4*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4) + b^5*x^5*(210*d^4 + 720*d
^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4)))/1260

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Maple [B]  time = 0.001, size = 559, normalized size = 4.7 \begin{align*}{\frac{{b}^{5}{e}^{4}{x}^{10}}{10}}+{\frac{ \left ( \left ( a{e}^{4}+4\,bd{e}^{3} \right ){b}^{4}+4\,{b}^{4}{e}^{4}a \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 4\,ad{e}^{3}+6\,b{d}^{2}{e}^{2} \right ){b}^{4}+4\, \left ( a{e}^{4}+4\,bd{e}^{3} \right ) a{b}^{3}+6\,{b}^{3}{e}^{4}{a}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 6\,a{d}^{2}{e}^{2}+4\,b{d}^{3}e \right ){b}^{4}+4\, \left ( 4\,ad{e}^{3}+6\,b{d}^{2}{e}^{2} \right ) a{b}^{3}+6\, \left ( a{e}^{4}+4\,bd{e}^{3} \right ){a}^{2}{b}^{2}+4\,{b}^{2}{e}^{4}{a}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,a{d}^{3}e+b{d}^{4} \right ){b}^{4}+4\, \left ( 6\,a{d}^{2}{e}^{2}+4\,b{d}^{3}e \right ) a{b}^{3}+6\, \left ( 4\,ad{e}^{3}+6\,b{d}^{2}{e}^{2} \right ){a}^{2}{b}^{2}+4\, \left ( a{e}^{4}+4\,bd{e}^{3} \right ){a}^{3}b+b{e}^{4}{a}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( a{d}^{4}{b}^{4}+4\, \left ( 4\,a{d}^{3}e+b{d}^{4} \right ) a{b}^{3}+6\, \left ( 6\,a{d}^{2}{e}^{2}+4\,b{d}^{3}e \right ){a}^{2}{b}^{2}+4\, \left ( 4\,ad{e}^{3}+6\,b{d}^{2}{e}^{2} \right ){a}^{3}b+ \left ( a{e}^{4}+4\,bd{e}^{3} \right ){a}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{a}^{2}{d}^{4}{b}^{3}+6\, \left ( 4\,a{d}^{3}e+b{d}^{4} \right ){a}^{2}{b}^{2}+4\, \left ( 6\,a{d}^{2}{e}^{2}+4\,b{d}^{3}e \right ){a}^{3}b+ \left ( 4\,ad{e}^{3}+6\,b{d}^{2}{e}^{2} \right ){a}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{a}^{3}{d}^{4}{b}^{2}+4\, \left ( 4\,a{d}^{3}e+b{d}^{4} \right ){a}^{3}b+ \left ( 6\,a{d}^{2}{e}^{2}+4\,b{d}^{3}e \right ){a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{a}^{4}{d}^{4}b+ \left ( 4\,a{d}^{3}e+b{d}^{4} \right ){a}^{4} \right ){x}^{2}}{2}}+{a}^{5}{d}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.02433, size = 486, normalized size = 4.08 \begin{align*} \frac{1}{10} \, b^{5} e^{4} x^{10} + a^{5} d^{4} x + \frac{1}{9} \,{\left (4 \, b^{5} d e^{3} + 5 \, a b^{4} e^{4}\right )} x^{9} + \frac{1}{4} \,{\left (3 \, b^{5} d^{2} e^{2} + 10 \, a b^{4} d e^{3} + 5 \, a^{2} b^{3} e^{4}\right )} x^{8} + \frac{2}{7} \,{\left (2 \, b^{5} d^{3} e + 15 \, a b^{4} d^{2} e^{2} + 20 \, a^{2} b^{3} d e^{3} + 5 \, a^{3} b^{2} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} d^{4} + 20 \, a b^{4} d^{3} e + 60 \, a^{2} b^{3} d^{2} e^{2} + 40 \, a^{3} b^{2} d e^{3} + 5 \, a^{4} b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (5 \, a b^{4} d^{4} + 40 \, a^{2} b^{3} d^{3} e + 60 \, a^{3} b^{2} d^{2} e^{2} + 20 \, a^{4} b d e^{3} + a^{5} e^{4}\right )} x^{5} + \frac{1}{2} \,{\left (5 \, a^{2} b^{3} d^{4} + 20 \, a^{3} b^{2} d^{3} e + 15 \, a^{4} b d^{2} e^{2} + 2 \, a^{5} d e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (5 \, a^{3} b^{2} d^{4} + 10 \, a^{4} b d^{3} e + 3 \, a^{5} d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b d^{4} + 4 \, a^{5} d^{3} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^5*e^4*x^10 + a^5*d^4*x + 1/9*(4*b^5*d*e^3 + 5*a*b^4*e^4)*x^9 + 1/4*(3*b^5*d^2*e^2 + 10*a*b^4*d*e^3 + 5*
a^2*b^3*e^4)*x^8 + 2/7*(2*b^5*d^3*e + 15*a*b^4*d^2*e^2 + 20*a^2*b^3*d*e^3 + 5*a^3*b^2*e^4)*x^7 + 1/6*(b^5*d^4
+ 20*a*b^4*d^3*e + 60*a^2*b^3*d^2*e^2 + 40*a^3*b^2*d*e^3 + 5*a^4*b*e^4)*x^6 + 1/5*(5*a*b^4*d^4 + 40*a^2*b^3*d^
3*e + 60*a^3*b^2*d^2*e^2 + 20*a^4*b*d*e^3 + a^5*e^4)*x^5 + 1/2*(5*a^2*b^3*d^4 + 20*a^3*b^2*d^3*e + 15*a^4*b*d^
2*e^2 + 2*a^5*d*e^3)*x^4 + 2/3*(5*a^3*b^2*d^4 + 10*a^4*b*d^3*e + 3*a^5*d^2*e^2)*x^3 + 1/2*(5*a^4*b*d^4 + 4*a^5
*d^3*e)*x^2

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Fricas [B]  time = 1.32268, size = 853, normalized size = 7.17 \begin{align*} \frac{1}{10} x^{10} e^{4} b^{5} + \frac{4}{9} x^{9} e^{3} d b^{5} + \frac{5}{9} x^{9} e^{4} b^{4} a + \frac{3}{4} x^{8} e^{2} d^{2} b^{5} + \frac{5}{2} x^{8} e^{3} d b^{4} a + \frac{5}{4} x^{8} e^{4} b^{3} a^{2} + \frac{4}{7} x^{7} e d^{3} b^{5} + \frac{30}{7} x^{7} e^{2} d^{2} b^{4} a + \frac{40}{7} x^{7} e^{3} d b^{3} a^{2} + \frac{10}{7} x^{7} e^{4} b^{2} a^{3} + \frac{1}{6} x^{6} d^{4} b^{5} + \frac{10}{3} x^{6} e d^{3} b^{4} a + 10 x^{6} e^{2} d^{2} b^{3} a^{2} + \frac{20}{3} x^{6} e^{3} d b^{2} a^{3} + \frac{5}{6} x^{6} e^{4} b a^{4} + x^{5} d^{4} b^{4} a + 8 x^{5} e d^{3} b^{3} a^{2} + 12 x^{5} e^{2} d^{2} b^{2} a^{3} + 4 x^{5} e^{3} d b a^{4} + \frac{1}{5} x^{5} e^{4} a^{5} + \frac{5}{2} x^{4} d^{4} b^{3} a^{2} + 10 x^{4} e d^{3} b^{2} a^{3} + \frac{15}{2} x^{4} e^{2} d^{2} b a^{4} + x^{4} e^{3} d a^{5} + \frac{10}{3} x^{3} d^{4} b^{2} a^{3} + \frac{20}{3} x^{3} e d^{3} b a^{4} + 2 x^{3} e^{2} d^{2} a^{5} + \frac{5}{2} x^{2} d^{4} b a^{4} + 2 x^{2} e d^{3} a^{5} + x d^{4} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*e^4*b^5 + 4/9*x^9*e^3*d*b^5 + 5/9*x^9*e^4*b^4*a + 3/4*x^8*e^2*d^2*b^5 + 5/2*x^8*e^3*d*b^4*a + 5/4*x^
8*e^4*b^3*a^2 + 4/7*x^7*e*d^3*b^5 + 30/7*x^7*e^2*d^2*b^4*a + 40/7*x^7*e^3*d*b^3*a^2 + 10/7*x^7*e^4*b^2*a^3 + 1
/6*x^6*d^4*b^5 + 10/3*x^6*e*d^3*b^4*a + 10*x^6*e^2*d^2*b^3*a^2 + 20/3*x^6*e^3*d*b^2*a^3 + 5/6*x^6*e^4*b*a^4 +
x^5*d^4*b^4*a + 8*x^5*e*d^3*b^3*a^2 + 12*x^5*e^2*d^2*b^2*a^3 + 4*x^5*e^3*d*b*a^4 + 1/5*x^5*e^4*a^5 + 5/2*x^4*d
^4*b^3*a^2 + 10*x^4*e*d^3*b^2*a^3 + 15/2*x^4*e^2*d^2*b*a^4 + x^4*e^3*d*a^5 + 10/3*x^3*d^4*b^2*a^3 + 20/3*x^3*e
*d^3*b*a^4 + 2*x^3*e^2*d^2*a^5 + 5/2*x^2*d^4*b*a^4 + 2*x^2*e*d^3*a^5 + x*d^4*a^5

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Sympy [B]  time = 0.117115, size = 401, normalized size = 3.37 \begin{align*} a^{5} d^{4} x + \frac{b^{5} e^{4} x^{10}}{10} + x^{9} \left (\frac{5 a b^{4} e^{4}}{9} + \frac{4 b^{5} d e^{3}}{9}\right ) + x^{8} \left (\frac{5 a^{2} b^{3} e^{4}}{4} + \frac{5 a b^{4} d e^{3}}{2} + \frac{3 b^{5} d^{2} e^{2}}{4}\right ) + x^{7} \left (\frac{10 a^{3} b^{2} e^{4}}{7} + \frac{40 a^{2} b^{3} d e^{3}}{7} + \frac{30 a b^{4} d^{2} e^{2}}{7} + \frac{4 b^{5} d^{3} e}{7}\right ) + x^{6} \left (\frac{5 a^{4} b e^{4}}{6} + \frac{20 a^{3} b^{2} d e^{3}}{3} + 10 a^{2} b^{3} d^{2} e^{2} + \frac{10 a b^{4} d^{3} e}{3} + \frac{b^{5} d^{4}}{6}\right ) + x^{5} \left (\frac{a^{5} e^{4}}{5} + 4 a^{4} b d e^{3} + 12 a^{3} b^{2} d^{2} e^{2} + 8 a^{2} b^{3} d^{3} e + a b^{4} d^{4}\right ) + x^{4} \left (a^{5} d e^{3} + \frac{15 a^{4} b d^{2} e^{2}}{2} + 10 a^{3} b^{2} d^{3} e + \frac{5 a^{2} b^{3} d^{4}}{2}\right ) + x^{3} \left (2 a^{5} d^{2} e^{2} + \frac{20 a^{4} b d^{3} e}{3} + \frac{10 a^{3} b^{2} d^{4}}{3}\right ) + x^{2} \left (2 a^{5} d^{3} e + \frac{5 a^{4} b d^{4}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*d**4*x + b**5*e**4*x**10/10 + x**9*(5*a*b**4*e**4/9 + 4*b**5*d*e**3/9) + x**8*(5*a**2*b**3*e**4/4 + 5*a*b
**4*d*e**3/2 + 3*b**5*d**2*e**2/4) + x**7*(10*a**3*b**2*e**4/7 + 40*a**2*b**3*d*e**3/7 + 30*a*b**4*d**2*e**2/7
+ 4*b**5*d**3*e/7) + x**6*(5*a**4*b*e**4/6 + 20*a**3*b**2*d*e**3/3 + 10*a**2*b**3*d**2*e**2 + 10*a*b**4*d**3*
e/3 + b**5*d**4/6) + x**5*(a**5*e**4/5 + 4*a**4*b*d*e**3 + 12*a**3*b**2*d**2*e**2 + 8*a**2*b**3*d**3*e + a*b**
4*d**4) + x**4*(a**5*d*e**3 + 15*a**4*b*d**2*e**2/2 + 10*a**3*b**2*d**3*e + 5*a**2*b**3*d**4/2) + x**3*(2*a**5
*d**2*e**2 + 20*a**4*b*d**3*e/3 + 10*a**3*b**2*d**4/3) + x**2*(2*a**5*d**3*e + 5*a**4*b*d**4/2)

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Giac [B]  time = 1.19306, size = 518, normalized size = 4.35 \begin{align*} \frac{1}{10} \, b^{5} x^{10} e^{4} + \frac{4}{9} \, b^{5} d x^{9} e^{3} + \frac{3}{4} \, b^{5} d^{2} x^{8} e^{2} + \frac{4}{7} \, b^{5} d^{3} x^{7} e + \frac{1}{6} \, b^{5} d^{4} x^{6} + \frac{5}{9} \, a b^{4} x^{9} e^{4} + \frac{5}{2} \, a b^{4} d x^{8} e^{3} + \frac{30}{7} \, a b^{4} d^{2} x^{7} e^{2} + \frac{10}{3} \, a b^{4} d^{3} x^{6} e + a b^{4} d^{4} x^{5} + \frac{5}{4} \, a^{2} b^{3} x^{8} e^{4} + \frac{40}{7} \, a^{2} b^{3} d x^{7} e^{3} + 10 \, a^{2} b^{3} d^{2} x^{6} e^{2} + 8 \, a^{2} b^{3} d^{3} x^{5} e + \frac{5}{2} \, a^{2} b^{3} d^{4} x^{4} + \frac{10}{7} \, a^{3} b^{2} x^{7} e^{4} + \frac{20}{3} \, a^{3} b^{2} d x^{6} e^{3} + 12 \, a^{3} b^{2} d^{2} x^{5} e^{2} + 10 \, a^{3} b^{2} d^{3} x^{4} e + \frac{10}{3} \, a^{3} b^{2} d^{4} x^{3} + \frac{5}{6} \, a^{4} b x^{6} e^{4} + 4 \, a^{4} b d x^{5} e^{3} + \frac{15}{2} \, a^{4} b d^{2} x^{4} e^{2} + \frac{20}{3} \, a^{4} b d^{3} x^{3} e + \frac{5}{2} \, a^{4} b d^{4} x^{2} + \frac{1}{5} \, a^{5} x^{5} e^{4} + a^{5} d x^{4} e^{3} + 2 \, a^{5} d^{2} x^{3} e^{2} + 2 \, a^{5} d^{3} x^{2} e + a^{5} d^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^5*x^10*e^4 + 4/9*b^5*d*x^9*e^3 + 3/4*b^5*d^2*x^8*e^2 + 4/7*b^5*d^3*x^7*e + 1/6*b^5*d^4*x^6 + 5/9*a*b^4*
x^9*e^4 + 5/2*a*b^4*d*x^8*e^3 + 30/7*a*b^4*d^2*x^7*e^2 + 10/3*a*b^4*d^3*x^6*e + a*b^4*d^4*x^5 + 5/4*a^2*b^3*x^
8*e^4 + 40/7*a^2*b^3*d*x^7*e^3 + 10*a^2*b^3*d^2*x^6*e^2 + 8*a^2*b^3*d^3*x^5*e + 5/2*a^2*b^3*d^4*x^4 + 10/7*a^3
*b^2*x^7*e^4 + 20/3*a^3*b^2*d*x^6*e^3 + 12*a^3*b^2*d^2*x^5*e^2 + 10*a^3*b^2*d^3*x^4*e + 10/3*a^3*b^2*d^4*x^3 +
5/6*a^4*b*x^6*e^4 + 4*a^4*b*d*x^5*e^3 + 15/2*a^4*b*d^2*x^4*e^2 + 20/3*a^4*b*d^3*x^3*e + 5/2*a^4*b*d^4*x^2 + 1
/5*a^5*x^5*e^4 + a^5*d*x^4*e^3 + 2*a^5*d^2*x^3*e^2 + 2*a^5*d^3*x^2*e + a^5*d^4*x