3.296 \(\int (d+e x)^3 (3+2 x+5 x^2)^2 (2+x+3 x^2-5 x^3+4 x^4) \, dx\)

Optimal. Leaf size=391 \[ \frac{\left (2800 d^2+315 d e+111 e^2\right ) (d+e x)^{10}}{10 e^9}-\frac{\left (945 d^2 e+5600 d^3+666 d e^2+37 e^3\right ) (d+e x)^9}{9 e^9}+\frac{\left (1665 d^2 e^2+1575 d^3 e+7000 d^4+185 d e^3+148 e^4\right ) (d+e x)^8}{8 e^9}-\frac{\left (2220 d^3 e^2+370 d^2 e^3+1575 d^4 e+5600 d^5+592 d e^4-65 e^5\right ) (d+e x)^7}{7 e^9}+\frac{\left (1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4+945 d^5 e+2800 d^6-195 d e^5+107 e^6\right ) (d+e x)^6}{6 e^9}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (88 d^3 e^2-4 d^2 e^3+127 d^4 e+160 d^5+64 d e^4-11 e^5\right ) (d+e x)^5}{5 e^9}+\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) (d+e x)^4}{4 e^9}+\frac{25 (d+e x)^{12}}{3 e^9}-\frac{5 (160 d+9 e) (d+e x)^{11}}{11 e^9} \]

[Out]

((5*d^2 - 2*d*e + 3*e^2)^2*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*(d + e*x)^4)/(4*e^9) - ((5*d^2 - 2*d*
e + 3*e^2)*(160*d^5 + 127*d^4*e + 88*d^3*e^2 - 4*d^2*e^3 + 64*d*e^4 - 11*e^5)*(d + e*x)^5)/(5*e^9) + ((2800*d^
6 + 945*d^5*e + 1665*d^4*e^2 + 370*d^3*e^3 + 888*d^2*e^4 - 195*d*e^5 + 107*e^6)*(d + e*x)^6)/(6*e^9) - ((5600*
d^5 + 1575*d^4*e + 2220*d^3*e^2 + 370*d^2*e^3 + 592*d*e^4 - 65*e^5)*(d + e*x)^7)/(7*e^9) + ((7000*d^4 + 1575*d
^3*e + 1665*d^2*e^2 + 185*d*e^3 + 148*e^4)*(d + e*x)^8)/(8*e^9) - ((5600*d^3 + 945*d^2*e + 666*d*e^2 + 37*e^3)
*(d + e*x)^9)/(9*e^9) + ((2800*d^2 + 315*d*e + 111*e^2)*(d + e*x)^10)/(10*e^9) - (5*(160*d + 9*e)*(d + e*x)^11
)/(11*e^9) + (25*(d + e*x)^12)/(3*e^9)

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Rubi [A]  time = 0.385555, antiderivative size = 391, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {1628} \[ \frac{\left (2800 d^2+315 d e+111 e^2\right ) (d+e x)^{10}}{10 e^9}-\frac{\left (945 d^2 e+5600 d^3+666 d e^2+37 e^3\right ) (d+e x)^9}{9 e^9}+\frac{\left (1665 d^2 e^2+1575 d^3 e+7000 d^4+185 d e^3+148 e^4\right ) (d+e x)^8}{8 e^9}-\frac{\left (2220 d^3 e^2+370 d^2 e^3+1575 d^4 e+5600 d^5+592 d e^4-65 e^5\right ) (d+e x)^7}{7 e^9}+\frac{\left (1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4+945 d^5 e+2800 d^6-195 d e^5+107 e^6\right ) (d+e x)^6}{6 e^9}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (88 d^3 e^2-4 d^2 e^3+127 d^4 e+160 d^5+64 d e^4-11 e^5\right ) (d+e x)^5}{5 e^9}+\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) (d+e x)^4}{4 e^9}+\frac{25 (d+e x)^{12}}{3 e^9}-\frac{5 (160 d+9 e) (d+e x)^{11}}{11 e^9} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]

[Out]

((5*d^2 - 2*d*e + 3*e^2)^2*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*(d + e*x)^4)/(4*e^9) - ((5*d^2 - 2*d*
e + 3*e^2)*(160*d^5 + 127*d^4*e + 88*d^3*e^2 - 4*d^2*e^3 + 64*d*e^4 - 11*e^5)*(d + e*x)^5)/(5*e^9) + ((2800*d^
6 + 945*d^5*e + 1665*d^4*e^2 + 370*d^3*e^3 + 888*d^2*e^4 - 195*d*e^5 + 107*e^6)*(d + e*x)^6)/(6*e^9) - ((5600*
d^5 + 1575*d^4*e + 2220*d^3*e^2 + 370*d^2*e^3 + 592*d*e^4 - 65*e^5)*(d + e*x)^7)/(7*e^9) + ((7000*d^4 + 1575*d
^3*e + 1665*d^2*e^2 + 185*d*e^3 + 148*e^4)*(d + e*x)^8)/(8*e^9) - ((5600*d^3 + 945*d^2*e + 666*d*e^2 + 37*e^3)
*(d + e*x)^9)/(9*e^9) + ((2800*d^2 + 315*d*e + 111*e^2)*(d + e*x)^10)/(10*e^9) - (5*(160*d + 9*e)*(d + e*x)^11
)/(11*e^9) + (25*(d + e*x)^12)/(3*e^9)

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int (d+e x)^3 \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx &=\int \left (\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) (d+e x)^3}{e^8}+\frac{\left (-800 d^7-315 d^6 e-666 d^5 e^2-185 d^4 e^3-592 d^3 e^4+195 d^2 e^5-214 d e^6+33 e^7\right ) (d+e x)^4}{e^8}+\frac{\left (2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6\right ) (d+e x)^5}{e^8}+\frac{\left (-5600 d^5-1575 d^4 e-2220 d^3 e^2-370 d^2 e^3-592 d e^4+65 e^5\right ) (d+e x)^6}{e^8}+\frac{\left (7000 d^4+1575 d^3 e+1665 d^2 e^2+185 d e^3+148 e^4\right ) (d+e x)^7}{e^8}+\frac{\left (-5600 d^3-945 d^2 e-666 d e^2-37 e^3\right ) (d+e x)^8}{e^8}+\frac{\left (2800 d^2+315 d e+111 e^2\right ) (d+e x)^9}{e^8}-\frac{5 (160 d+9 e) (d+e x)^{10}}{e^8}+\frac{100 (d+e x)^{11}}{e^8}\right ) \, dx\\ &=\frac{\left (5 d^2-2 d e+3 e^2\right )^2 \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) (d+e x)^4}{4 e^9}-\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right ) (d+e x)^5}{5 e^9}+\frac{\left (2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6\right ) (d+e x)^6}{6 e^9}-\frac{\left (5600 d^5+1575 d^4 e+2220 d^3 e^2+370 d^2 e^3+592 d e^4-65 e^5\right ) (d+e x)^7}{7 e^9}+\frac{\left (7000 d^4+1575 d^3 e+1665 d^2 e^2+185 d e^3+148 e^4\right ) (d+e x)^8}{8 e^9}-\frac{\left (5600 d^3+945 d^2 e+666 d e^2+37 e^3\right ) (d+e x)^9}{9 e^9}+\frac{\left (2800 d^2+315 d e+111 e^2\right ) (d+e x)^{10}}{10 e^9}-\frac{5 (160 d+9 e) (d+e x)^{11}}{11 e^9}+\frac{25 (d+e x)^{12}}{3 e^9}\\ \end{align*}

Mathematica [A]  time = 0.0430987, size = 277, normalized size = 0.71 \[ \frac{3}{10} e x^{10} \left (100 d^2-45 d e+37 e^2\right )+\frac{1}{9} x^9 \left (-135 d^2 e+100 d^3+333 d e^2-37 e^3\right )+\frac{1}{8} x^8 \left (333 d^2 e-45 d^3-111 d e^2+148 e^3\right )+\frac{1}{7} x^7 \left (-111 d^2 e+111 d^3+444 d e^2+65 e^3\right )+\frac{1}{6} x^6 \left (444 d^2 e-37 d^3+195 d e^2+107 e^3\right )+\frac{1}{5} x^5 \left (195 d^2 e+148 d^3+321 d e^2+33 e^3\right )+\frac{1}{4} x^4 \left (321 d^2 e+65 d^3+99 d e^2+18 e^3\right )+\frac{1}{3} d x^3 \left (107 d^2+99 d e+54 e^2\right )+\frac{3}{2} d^2 x^2 (11 d+18 e)+18 d^3 x+\frac{15}{11} e^2 x^{11} (20 d-3 e)+\frac{25 e^3 x^{12}}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]

[Out]

18*d^3*x + (3*d^2*(11*d + 18*e)*x^2)/2 + (d*(107*d^2 + 99*d*e + 54*e^2)*x^3)/3 + ((65*d^3 + 321*d^2*e + 99*d*e
^2 + 18*e^3)*x^4)/4 + ((148*d^3 + 195*d^2*e + 321*d*e^2 + 33*e^3)*x^5)/5 + ((-37*d^3 + 444*d^2*e + 195*d*e^2 +
107*e^3)*x^6)/6 + ((111*d^3 - 111*d^2*e + 444*d*e^2 + 65*e^3)*x^7)/7 + ((-45*d^3 + 333*d^2*e - 111*d*e^2 + 14
8*e^3)*x^8)/8 + ((100*d^3 - 135*d^2*e + 333*d*e^2 - 37*e^3)*x^9)/9 + (3*e*(100*d^2 - 45*d*e + 37*e^2)*x^10)/10
+ (15*(20*d - 3*e)*e^2*x^11)/11 + (25*e^3*x^12)/3

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Maple [A]  time = 0.058, size = 264, normalized size = 0.7 \begin{align*}{\frac{25\,{e}^{3}{x}^{12}}{3}}+{\frac{ \left ( 300\,d{e}^{2}-45\,{e}^{3} \right ){x}^{11}}{11}}+{\frac{ \left ( 300\,{d}^{2}e-135\,d{e}^{2}+111\,{e}^{3} \right ){x}^{10}}{10}}+{\frac{ \left ( 100\,{d}^{3}-135\,{d}^{2}e+333\,d{e}^{2}-37\,{e}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( -45\,{d}^{3}+333\,{d}^{2}e-111\,d{e}^{2}+148\,{e}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 111\,{d}^{3}-111\,{d}^{2}e+444\,d{e}^{2}+65\,{e}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( -37\,{d}^{3}+444\,{d}^{2}e+195\,d{e}^{2}+107\,{e}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 148\,{d}^{3}+195\,{d}^{2}e+321\,d{e}^{2}+33\,{e}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 65\,{d}^{3}+321\,{d}^{2}e+99\,d{e}^{2}+18\,{e}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 107\,{d}^{3}+99\,{d}^{2}e+54\,d{e}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 33\,{d}^{3}+54\,{d}^{2}e \right ){x}^{2}}{2}}+18\,{d}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(5*x^2+2*x+3)^2*(4*x^4-5*x^3+3*x^2+x+2),x)

[Out]

25/3*e^3*x^12+1/11*(300*d*e^2-45*e^3)*x^11+1/10*(300*d^2*e-135*d*e^2+111*e^3)*x^10+1/9*(100*d^3-135*d^2*e+333*
d*e^2-37*e^3)*x^9+1/8*(-45*d^3+333*d^2*e-111*d*e^2+148*e^3)*x^8+1/7*(111*d^3-111*d^2*e+444*d*e^2+65*e^3)*x^7+1
/6*(-37*d^3+444*d^2*e+195*d*e^2+107*e^3)*x^6+1/5*(148*d^3+195*d^2*e+321*d*e^2+33*e^3)*x^5+1/4*(65*d^3+321*d^2*
e+99*d*e^2+18*e^3)*x^4+1/3*(107*d^3+99*d^2*e+54*d*e^2)*x^3+1/2*(33*d^3+54*d^2*e)*x^2+18*d^3*x

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Maxima [A]  time = 0.962501, size = 355, normalized size = 0.91 \begin{align*} \frac{25}{3} \, e^{3} x^{12} + \frac{15}{11} \,{\left (20 \, d e^{2} - 3 \, e^{3}\right )} x^{11} + \frac{3}{10} \,{\left (100 \, d^{2} e - 45 \, d e^{2} + 37 \, e^{3}\right )} x^{10} + \frac{1}{9} \,{\left (100 \, d^{3} - 135 \, d^{2} e + 333 \, d e^{2} - 37 \, e^{3}\right )} x^{9} - \frac{1}{8} \,{\left (45 \, d^{3} - 333 \, d^{2} e + 111 \, d e^{2} - 148 \, e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (111 \, d^{3} - 111 \, d^{2} e + 444 \, d e^{2} + 65 \, e^{3}\right )} x^{7} - \frac{1}{6} \,{\left (37 \, d^{3} - 444 \, d^{2} e - 195 \, d e^{2} - 107 \, e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (148 \, d^{3} + 195 \, d^{2} e + 321 \, d e^{2} + 33 \, e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (65 \, d^{3} + 321 \, d^{2} e + 99 \, d e^{2} + 18 \, e^{3}\right )} x^{4} + 18 \, d^{3} x + \frac{1}{3} \,{\left (107 \, d^{3} + 99 \, d^{2} e + 54 \, d e^{2}\right )} x^{3} + \frac{3}{2} \,{\left (11 \, d^{3} + 18 \, d^{2} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(5*x^2+2*x+3)^2*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="maxima")

[Out]

25/3*e^3*x^12 + 15/11*(20*d*e^2 - 3*e^3)*x^11 + 3/10*(100*d^2*e - 45*d*e^2 + 37*e^3)*x^10 + 1/9*(100*d^3 - 135
*d^2*e + 333*d*e^2 - 37*e^3)*x^9 - 1/8*(45*d^3 - 333*d^2*e + 111*d*e^2 - 148*e^3)*x^8 + 1/7*(111*d^3 - 111*d^2
*e + 444*d*e^2 + 65*e^3)*x^7 - 1/6*(37*d^3 - 444*d^2*e - 195*d*e^2 - 107*e^3)*x^6 + 1/5*(148*d^3 + 195*d^2*e +
321*d*e^2 + 33*e^3)*x^5 + 1/4*(65*d^3 + 321*d^2*e + 99*d*e^2 + 18*e^3)*x^4 + 18*d^3*x + 1/3*(107*d^3 + 99*d^2
*e + 54*d*e^2)*x^3 + 3/2*(11*d^3 + 18*d^2*e)*x^2

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Fricas [A]  time = 0.92602, size = 778, normalized size = 1.99 \begin{align*} \frac{25}{3} x^{12} e^{3} - \frac{45}{11} x^{11} e^{3} + \frac{300}{11} x^{11} e^{2} d + \frac{111}{10} x^{10} e^{3} - \frac{27}{2} x^{10} e^{2} d + 30 x^{10} e d^{2} - \frac{37}{9} x^{9} e^{3} + 37 x^{9} e^{2} d - 15 x^{9} e d^{2} + \frac{100}{9} x^{9} d^{3} + \frac{37}{2} x^{8} e^{3} - \frac{111}{8} x^{8} e^{2} d + \frac{333}{8} x^{8} e d^{2} - \frac{45}{8} x^{8} d^{3} + \frac{65}{7} x^{7} e^{3} + \frac{444}{7} x^{7} e^{2} d - \frac{111}{7} x^{7} e d^{2} + \frac{111}{7} x^{7} d^{3} + \frac{107}{6} x^{6} e^{3} + \frac{65}{2} x^{6} e^{2} d + 74 x^{6} e d^{2} - \frac{37}{6} x^{6} d^{3} + \frac{33}{5} x^{5} e^{3} + \frac{321}{5} x^{5} e^{2} d + 39 x^{5} e d^{2} + \frac{148}{5} x^{5} d^{3} + \frac{9}{2} x^{4} e^{3} + \frac{99}{4} x^{4} e^{2} d + \frac{321}{4} x^{4} e d^{2} + \frac{65}{4} x^{4} d^{3} + 18 x^{3} e^{2} d + 33 x^{3} e d^{2} + \frac{107}{3} x^{3} d^{3} + 27 x^{2} e d^{2} + \frac{33}{2} x^{2} d^{3} + 18 x d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(5*x^2+2*x+3)^2*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="fricas")

[Out]

25/3*x^12*e^3 - 45/11*x^11*e^3 + 300/11*x^11*e^2*d + 111/10*x^10*e^3 - 27/2*x^10*e^2*d + 30*x^10*e*d^2 - 37/9*
x^9*e^3 + 37*x^9*e^2*d - 15*x^9*e*d^2 + 100/9*x^9*d^3 + 37/2*x^8*e^3 - 111/8*x^8*e^2*d + 333/8*x^8*e*d^2 - 45/
8*x^8*d^3 + 65/7*x^7*e^3 + 444/7*x^7*e^2*d - 111/7*x^7*e*d^2 + 111/7*x^7*d^3 + 107/6*x^6*e^3 + 65/2*x^6*e^2*d
+ 74*x^6*e*d^2 - 37/6*x^6*d^3 + 33/5*x^5*e^3 + 321/5*x^5*e^2*d + 39*x^5*e*d^2 + 148/5*x^5*d^3 + 9/2*x^4*e^3 +
99/4*x^4*e^2*d + 321/4*x^4*e*d^2 + 65/4*x^4*d^3 + 18*x^3*e^2*d + 33*x^3*e*d^2 + 107/3*x^3*d^3 + 27*x^2*e*d^2 +
33/2*x^2*d^3 + 18*x*d^3

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Sympy [A]  time = 0.118886, size = 298, normalized size = 0.76 \begin{align*} 18 d^{3} x + \frac{25 e^{3} x^{12}}{3} + x^{11} \left (\frac{300 d e^{2}}{11} - \frac{45 e^{3}}{11}\right ) + x^{10} \left (30 d^{2} e - \frac{27 d e^{2}}{2} + \frac{111 e^{3}}{10}\right ) + x^{9} \left (\frac{100 d^{3}}{9} - 15 d^{2} e + 37 d e^{2} - \frac{37 e^{3}}{9}\right ) + x^{8} \left (- \frac{45 d^{3}}{8} + \frac{333 d^{2} e}{8} - \frac{111 d e^{2}}{8} + \frac{37 e^{3}}{2}\right ) + x^{7} \left (\frac{111 d^{3}}{7} - \frac{111 d^{2} e}{7} + \frac{444 d e^{2}}{7} + \frac{65 e^{3}}{7}\right ) + x^{6} \left (- \frac{37 d^{3}}{6} + 74 d^{2} e + \frac{65 d e^{2}}{2} + \frac{107 e^{3}}{6}\right ) + x^{5} \left (\frac{148 d^{3}}{5} + 39 d^{2} e + \frac{321 d e^{2}}{5} + \frac{33 e^{3}}{5}\right ) + x^{4} \left (\frac{65 d^{3}}{4} + \frac{321 d^{2} e}{4} + \frac{99 d e^{2}}{4} + \frac{9 e^{3}}{2}\right ) + x^{3} \left (\frac{107 d^{3}}{3} + 33 d^{2} e + 18 d e^{2}\right ) + x^{2} \left (\frac{33 d^{3}}{2} + 27 d^{2} e\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(5*x**2+2*x+3)**2*(4*x**4-5*x**3+3*x**2+x+2),x)

[Out]

18*d**3*x + 25*e**3*x**12/3 + x**11*(300*d*e**2/11 - 45*e**3/11) + x**10*(30*d**2*e - 27*d*e**2/2 + 111*e**3/1
0) + x**9*(100*d**3/9 - 15*d**2*e + 37*d*e**2 - 37*e**3/9) + x**8*(-45*d**3/8 + 333*d**2*e/8 - 111*d*e**2/8 +
37*e**3/2) + x**7*(111*d**3/7 - 111*d**2*e/7 + 444*d*e**2/7 + 65*e**3/7) + x**6*(-37*d**3/6 + 74*d**2*e + 65*d
*e**2/2 + 107*e**3/6) + x**5*(148*d**3/5 + 39*d**2*e + 321*d*e**2/5 + 33*e**3/5) + x**4*(65*d**3/4 + 321*d**2*
e/4 + 99*d*e**2/4 + 9*e**3/2) + x**3*(107*d**3/3 + 33*d**2*e + 18*d*e**2) + x**2*(33*d**3/2 + 27*d**2*e)

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Giac [A]  time = 1.14521, size = 400, normalized size = 1.02 \begin{align*} \frac{25}{3} \, x^{12} e^{3} + \frac{300}{11} \, d x^{11} e^{2} + 30 \, d^{2} x^{10} e + \frac{100}{9} \, d^{3} x^{9} - \frac{45}{11} \, x^{11} e^{3} - \frac{27}{2} \, d x^{10} e^{2} - 15 \, d^{2} x^{9} e - \frac{45}{8} \, d^{3} x^{8} + \frac{111}{10} \, x^{10} e^{3} + 37 \, d x^{9} e^{2} + \frac{333}{8} \, d^{2} x^{8} e + \frac{111}{7} \, d^{3} x^{7} - \frac{37}{9} \, x^{9} e^{3} - \frac{111}{8} \, d x^{8} e^{2} - \frac{111}{7} \, d^{2} x^{7} e - \frac{37}{6} \, d^{3} x^{6} + \frac{37}{2} \, x^{8} e^{3} + \frac{444}{7} \, d x^{7} e^{2} + 74 \, d^{2} x^{6} e + \frac{148}{5} \, d^{3} x^{5} + \frac{65}{7} \, x^{7} e^{3} + \frac{65}{2} \, d x^{6} e^{2} + 39 \, d^{2} x^{5} e + \frac{65}{4} \, d^{3} x^{4} + \frac{107}{6} \, x^{6} e^{3} + \frac{321}{5} \, d x^{5} e^{2} + \frac{321}{4} \, d^{2} x^{4} e + \frac{107}{3} \, d^{3} x^{3} + \frac{33}{5} \, x^{5} e^{3} + \frac{99}{4} \, d x^{4} e^{2} + 33 \, d^{2} x^{3} e + \frac{33}{2} \, d^{3} x^{2} + \frac{9}{2} \, x^{4} e^{3} + 18 \, d x^{3} e^{2} + 27 \, d^{2} x^{2} e + 18 \, d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(5*x^2+2*x+3)^2*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="giac")

[Out]

25/3*x^12*e^3 + 300/11*d*x^11*e^2 + 30*d^2*x^10*e + 100/9*d^3*x^9 - 45/11*x^11*e^3 - 27/2*d*x^10*e^2 - 15*d^2*
x^9*e - 45/8*d^3*x^8 + 111/10*x^10*e^3 + 37*d*x^9*e^2 + 333/8*d^2*x^8*e + 111/7*d^3*x^7 - 37/9*x^9*e^3 - 111/8
*d*x^8*e^2 - 111/7*d^2*x^7*e - 37/6*d^3*x^6 + 37/2*x^8*e^3 + 444/7*d*x^7*e^2 + 74*d^2*x^6*e + 148/5*d^3*x^5 +
65/7*x^7*e^3 + 65/2*d*x^6*e^2 + 39*d^2*x^5*e + 65/4*d^3*x^4 + 107/6*x^6*e^3 + 321/5*d*x^5*e^2 + 321/4*d^2*x^4*
e + 107/3*d^3*x^3 + 33/5*x^5*e^3 + 99/4*d*x^4*e^2 + 33*d^2*x^3*e + 33/2*d^3*x^2 + 9/2*x^4*e^3 + 18*d*x^3*e^2 +
27*d^2*x^2*e + 18*d^3*x