### 3.289 $$\int (d+e x)^3 (3+2 x+5 x^2) (2+x+3 x^2-5 x^3+4 x^4) \, dx$$

Optimal. Leaf size=258 $\frac{\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^8}{8 e^7}-\frac{2 \left (85 d^2 e+200 d^3+34 d e^2+2 e^3\right ) (d+e x)^7}{7 e^7}+\frac{\left (102 d^2 e^2+170 d^3 e+300 d^4+12 d e^3+21 e^4\right ) (d+e x)^6}{6 e^7}-\frac{\left (68 d^3 e^2+12 d^2 e^3+85 d^4 e+120 d^5+42 d e^4-7 e^5\right ) (d+e x)^5}{5 e^7}+\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) (d+e x)^4}{4 e^7}+\frac{2 (d+e x)^{10}}{e^7}-\frac{(120 d+17 e) (d+e x)^9}{9 e^7}$

[Out]

((5*d^2 - 2*d*e + 3*e^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^7) - ((120*d^5 + 85*d
^4*e + 68*d^3*e^2 + 12*d^2*e^3 + 42*d*e^4 - 7*e^5)*(d + e*x)^5)/(5*e^7) + ((300*d^4 + 170*d^3*e + 102*d^2*e^2
+ 12*d*e^3 + 21*e^4)*(d + e*x)^6)/(6*e^7) - (2*(200*d^3 + 85*d^2*e + 34*d*e^2 + 2*e^3)*(d + e*x)^7)/(7*e^7) +
((300*d^2 + 85*d*e + 17*e^2)*(d + e*x)^8)/(8*e^7) - ((120*d + 17*e)*(d + e*x)^9)/(9*e^7) + (2*(d + e*x)^10)/e^
7

________________________________________________________________________________________

Rubi [A]  time = 0.256623, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 36, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.028, Rules used = {1628} $\frac{\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^8}{8 e^7}-\frac{2 \left (85 d^2 e+200 d^3+34 d e^2+2 e^3\right ) (d+e x)^7}{7 e^7}+\frac{\left (102 d^2 e^2+170 d^3 e+300 d^4+12 d e^3+21 e^4\right ) (d+e x)^6}{6 e^7}-\frac{\left (68 d^3 e^2+12 d^2 e^3+85 d^4 e+120 d^5+42 d e^4-7 e^5\right ) (d+e x)^5}{5 e^7}+\frac{\left (5 d^2-2 d e+3 e^2\right ) \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) (d+e x)^4}{4 e^7}+\frac{2 (d+e x)^{10}}{e^7}-\frac{(120 d+17 e) (d+e x)^9}{9 e^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((5*d^2 - 2*d*e + 3*e^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^7) - ((120*d^5 + 85*d
^4*e + 68*d^3*e^2 + 12*d^2*e^3 + 42*d*e^4 - 7*e^5)*(d + e*x)^5)/(5*e^7) + ((300*d^4 + 170*d^3*e + 102*d^2*e^2
+ 12*d*e^3 + 21*e^4)*(d + e*x)^6)/(6*e^7) - (2*(200*d^3 + 85*d^2*e + 34*d*e^2 + 2*e^3)*(d + e*x)^7)/(7*e^7) +
((300*d^2 + 85*d*e + 17*e^2)*(d + e*x)^8)/(8*e^7) - ((120*d + 17*e)*(d + e*x)^9)/(9*e^7) + (2*(d + e*x)^10)/e^
7

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 ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx &=\int \left (\frac{\left (20 d^6+17 d^5 e+17 d^4 e^2+4 d^3 e^3+21 d^2 e^4-7 d e^5+6 e^6\right ) (d+e x)^3}{e^6}+\frac{\left (-120 d^5-85 d^4 e-68 d^3 e^2-12 d^2 e^3-42 d e^4+7 e^5\right ) (d+e x)^4}{e^6}+\frac{\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^5}{e^6}-\frac{2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^6}{e^6}+\frac{\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^7}{e^6}+\frac{(-120 d-17 e) (d+e x)^8}{e^6}+\frac{20 (d+e x)^9}{e^6}\right ) \, dx\\ &=\frac{\left (5 d^2-2 d e+3 e^2\right ) \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^7}-\frac{\left (120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right ) (d+e x)^5}{5 e^7}+\frac{\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^6}{6 e^7}-\frac{2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^7}{7 e^7}+\frac{\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^8}{8 e^7}-\frac{(120 d+17 e) (d+e x)^9}{9 e^7}+\frac{2 (d+e x)^{10}}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.0406716, size = 212, normalized size = 0.82 $\frac{1}{8} e x^8 \left (60 d^2-51 d e+17 e^2\right )+\frac{1}{7} x^7 \left (-51 d^2 e+20 d^3+51 d e^2-4 e^3\right )+\frac{1}{6} x^6 \left (51 d^2 e-17 d^3-12 d e^2+21 e^3\right )+\frac{1}{5} x^5 \left (-12 d^2 e+17 d^3+63 d e^2+7 e^3\right )+\frac{1}{4} x^4 \left (63 d^2 e-4 d^3+21 d e^2+6 e^3\right )+d x^3 \left (7 d^2+7 d e+6 e^2\right )+\frac{1}{2} d^2 x^2 (7 d+18 e)+6 d^3 x+\frac{1}{9} e^2 x^9 (60 d-17 e)+2 e^3 x^{10}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

6*d^3*x + (d^2*(7*d + 18*e)*x^2)/2 + d*(7*d^2 + 7*d*e + 6*e^2)*x^3 + ((-4*d^3 + 63*d^2*e + 21*d*e^2 + 6*e^3)*x
^4)/4 + ((17*d^3 - 12*d^2*e + 63*d*e^2 + 7*e^3)*x^5)/5 + ((-17*d^3 + 51*d^2*e - 12*d*e^2 + 21*e^3)*x^6)/6 + ((
20*d^3 - 51*d^2*e + 51*d*e^2 - 4*e^3)*x^7)/7 + (e*(60*d^2 - 51*d*e + 17*e^2)*x^8)/8 + ((60*d - 17*e)*e^2*x^9)/
9 + 2*e^3*x^10

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 208, normalized size = 0.8 \begin{align*} 2\,{e}^{3}{x}^{10}+{\frac{ \left ( 60\,d{e}^{2}-17\,{e}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 60\,{d}^{2}e-51\,d{e}^{2}+17\,{e}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 20\,{d}^{3}-51\,{d}^{2}e+51\,d{e}^{2}-4\,{e}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( -17\,{d}^{3}+51\,{d}^{2}e-12\,d{e}^{2}+21\,{e}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 17\,{d}^{3}-12\,{d}^{2}e+63\,d{e}^{2}+7\,{e}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( -4\,{d}^{3}+63\,{d}^{2}e+21\,d{e}^{2}+6\,{e}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,{d}^{3}+21\,{d}^{2}e+18\,d{e}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,{d}^{3}+18\,{d}^{2}e \right ){x}^{2}}{2}}+6\,{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)*(4*x^4-5*x^3+3*x^2+x+2),x)

[Out]

2*e^3*x^10+1/9*(60*d*e^2-17*e^3)*x^9+1/8*(60*d^2*e-51*d*e^2+17*e^3)*x^8+1/7*(20*d^3-51*d^2*e+51*d*e^2-4*e^3)*x
^7+1/6*(-17*d^3+51*d^2*e-12*d*e^2+21*e^3)*x^6+1/5*(17*d^3-12*d^2*e+63*d*e^2+7*e^3)*x^5+1/4*(-4*d^3+63*d^2*e+21
*d*e^2+6*e^3)*x^4+1/3*(21*d^3+21*d^2*e+18*d*e^2)*x^3+1/2*(7*d^3+18*d^2*e)*x^2+6*d^3*x

________________________________________________________________________________________

Maxima [A]  time = 0.984523, size = 278, normalized size = 1.08 \begin{align*} 2 \, e^{3} x^{10} + \frac{1}{9} \,{\left (60 \, d e^{2} - 17 \, e^{3}\right )} x^{9} + \frac{1}{8} \,{\left (60 \, d^{2} e - 51 \, d e^{2} + 17 \, e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (20 \, d^{3} - 51 \, d^{2} e + 51 \, d e^{2} - 4 \, e^{3}\right )} x^{7} - \frac{1}{6} \,{\left (17 \, d^{3} - 51 \, d^{2} e + 12 \, d e^{2} - 21 \, e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (17 \, d^{3} - 12 \, d^{2} e + 63 \, d e^{2} + 7 \, e^{3}\right )} x^{5} - \frac{1}{4} \,{\left (4 \, d^{3} - 63 \, d^{2} e - 21 \, d e^{2} - 6 \, e^{3}\right )} x^{4} + 6 \, d^{3} x +{\left (7 \, d^{3} + 7 \, d^{2} e + 6 \, d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (7 \, 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)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="maxima")

[Out]

2*e^3*x^10 + 1/9*(60*d*e^2 - 17*e^3)*x^9 + 1/8*(60*d^2*e - 51*d*e^2 + 17*e^3)*x^8 + 1/7*(20*d^3 - 51*d^2*e + 5
1*d*e^2 - 4*e^3)*x^7 - 1/6*(17*d^3 - 51*d^2*e + 12*d*e^2 - 21*e^3)*x^6 + 1/5*(17*d^3 - 12*d^2*e + 63*d*e^2 + 7
*e^3)*x^5 - 1/4*(4*d^3 - 63*d^2*e - 21*d*e^2 - 6*e^3)*x^4 + 6*d^3*x + (7*d^3 + 7*d^2*e + 6*d*e^2)*x^3 + 1/2*(7
*d^3 + 18*d^2*e)*x^2

________________________________________________________________________________________

Fricas [A]  time = 0.831358, size = 560, normalized size = 2.17 \begin{align*} 2 x^{10} e^{3} - \frac{17}{9} x^{9} e^{3} + \frac{20}{3} x^{9} e^{2} d + \frac{17}{8} x^{8} e^{3} - \frac{51}{8} x^{8} e^{2} d + \frac{15}{2} x^{8} e d^{2} - \frac{4}{7} x^{7} e^{3} + \frac{51}{7} x^{7} e^{2} d - \frac{51}{7} x^{7} e d^{2} + \frac{20}{7} x^{7} d^{3} + \frac{7}{2} x^{6} e^{3} - 2 x^{6} e^{2} d + \frac{17}{2} x^{6} e d^{2} - \frac{17}{6} x^{6} d^{3} + \frac{7}{5} x^{5} e^{3} + \frac{63}{5} x^{5} e^{2} d - \frac{12}{5} x^{5} e d^{2} + \frac{17}{5} x^{5} d^{3} + \frac{3}{2} x^{4} e^{3} + \frac{21}{4} x^{4} e^{2} d + \frac{63}{4} x^{4} e d^{2} - x^{4} d^{3} + 6 x^{3} e^{2} d + 7 x^{3} e d^{2} + 7 x^{3} d^{3} + 9 x^{2} e d^{2} + \frac{7}{2} x^{2} d^{3} + 6 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)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="fricas")

[Out]

2*x^10*e^3 - 17/9*x^9*e^3 + 20/3*x^9*e^2*d + 17/8*x^8*e^3 - 51/8*x^8*e^2*d + 15/2*x^8*e*d^2 - 4/7*x^7*e^3 + 51
/7*x^7*e^2*d - 51/7*x^7*e*d^2 + 20/7*x^7*d^3 + 7/2*x^6*e^3 - 2*x^6*e^2*d + 17/2*x^6*e*d^2 - 17/6*x^6*d^3 + 7/5
*x^5*e^3 + 63/5*x^5*e^2*d - 12/5*x^5*e*d^2 + 17/5*x^5*d^3 + 3/2*x^4*e^3 + 21/4*x^4*e^2*d + 63/4*x^4*e*d^2 - x^
4*d^3 + 6*x^3*e^2*d + 7*x^3*e*d^2 + 7*x^3*d^3 + 9*x^2*e*d^2 + 7/2*x^2*d^3 + 6*x*d^3

________________________________________________________________________________________

Sympy [A]  time = 0.104844, size = 230, normalized size = 0.89 \begin{align*} 6 d^{3} x + 2 e^{3} x^{10} + x^{9} \left (\frac{20 d e^{2}}{3} - \frac{17 e^{3}}{9}\right ) + x^{8} \left (\frac{15 d^{2} e}{2} - \frac{51 d e^{2}}{8} + \frac{17 e^{3}}{8}\right ) + x^{7} \left (\frac{20 d^{3}}{7} - \frac{51 d^{2} e}{7} + \frac{51 d e^{2}}{7} - \frac{4 e^{3}}{7}\right ) + x^{6} \left (- \frac{17 d^{3}}{6} + \frac{17 d^{2} e}{2} - 2 d e^{2} + \frac{7 e^{3}}{2}\right ) + x^{5} \left (\frac{17 d^{3}}{5} - \frac{12 d^{2} e}{5} + \frac{63 d e^{2}}{5} + \frac{7 e^{3}}{5}\right ) + x^{4} \left (- d^{3} + \frac{63 d^{2} e}{4} + \frac{21 d e^{2}}{4} + \frac{3 e^{3}}{2}\right ) + x^{3} \left (7 d^{3} + 7 d^{2} e + 6 d e^{2}\right ) + x^{2} \left (\frac{7 d^{3}}{2} + 9 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)*(4*x**4-5*x**3+3*x**2+x+2),x)

[Out]

6*d**3*x + 2*e**3*x**10 + x**9*(20*d*e**2/3 - 17*e**3/9) + x**8*(15*d**2*e/2 - 51*d*e**2/8 + 17*e**3/8) + x**7
*(20*d**3/7 - 51*d**2*e/7 + 51*d*e**2/7 - 4*e**3/7) + x**6*(-17*d**3/6 + 17*d**2*e/2 - 2*d*e**2 + 7*e**3/2) +
x**5*(17*d**3/5 - 12*d**2*e/5 + 63*d*e**2/5 + 7*e**3/5) + x**4*(-d**3 + 63*d**2*e/4 + 21*d*e**2/4 + 3*e**3/2)
+ x**3*(7*d**3 + 7*d**2*e + 6*d*e**2) + x**2*(7*d**3/2 + 9*d**2*e)

________________________________________________________________________________________

Giac [A]  time = 1.12525, size = 311, normalized size = 1.21 \begin{align*} 2 \, x^{10} e^{3} + \frac{20}{3} \, d x^{9} e^{2} + \frac{15}{2} \, d^{2} x^{8} e + \frac{20}{7} \, d^{3} x^{7} - \frac{17}{9} \, x^{9} e^{3} - \frac{51}{8} \, d x^{8} e^{2} - \frac{51}{7} \, d^{2} x^{7} e - \frac{17}{6} \, d^{3} x^{6} + \frac{17}{8} \, x^{8} e^{3} + \frac{51}{7} \, d x^{7} e^{2} + \frac{17}{2} \, d^{2} x^{6} e + \frac{17}{5} \, d^{3} x^{5} - \frac{4}{7} \, x^{7} e^{3} - 2 \, d x^{6} e^{2} - \frac{12}{5} \, d^{2} x^{5} e - d^{3} x^{4} + \frac{7}{2} \, x^{6} e^{3} + \frac{63}{5} \, d x^{5} e^{2} + \frac{63}{4} \, d^{2} x^{4} e + 7 \, d^{3} x^{3} + \frac{7}{5} \, x^{5} e^{3} + \frac{21}{4} \, d x^{4} e^{2} + 7 \, d^{2} x^{3} e + \frac{7}{2} \, d^{3} x^{2} + \frac{3}{2} \, x^{4} e^{3} + 6 \, d x^{3} e^{2} + 9 \, d^{2} x^{2} e + 6 \, 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)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="giac")

[Out]

2*x^10*e^3 + 20/3*d*x^9*e^2 + 15/2*d^2*x^8*e + 20/7*d^3*x^7 - 17/9*x^9*e^3 - 51/8*d*x^8*e^2 - 51/7*d^2*x^7*e -
17/6*d^3*x^6 + 17/8*x^8*e^3 + 51/7*d*x^7*e^2 + 17/2*d^2*x^6*e + 17/5*d^3*x^5 - 4/7*x^7*e^3 - 2*d*x^6*e^2 - 12
/5*d^2*x^5*e - d^3*x^4 + 7/2*x^6*e^3 + 63/5*d*x^5*e^2 + 63/4*d^2*x^4*e + 7*d^3*x^3 + 7/5*x^5*e^3 + 21/4*d*x^4*
e^2 + 7*d^2*x^3*e + 7/2*d^3*x^2 + 3/2*x^4*e^3 + 6*d*x^3*e^2 + 9*d^2*x^2*e + 6*d^3*x