### 3.298 $$\int (d+e x) (3+2 x+5 x^2)^2 (2+x+3 x^2-5 x^3+4 x^4) \, dx$$

Optimal. Leaf size=121 $\frac{5}{9} x^9 (20 d-9 e)-\frac{3}{8} x^8 (15 d-37 e)+\frac{37}{7} x^7 (3 d-e)-\frac{37}{6} x^6 (d-4 e)+\frac{1}{5} x^5 (148 d+65 e)+\frac{1}{4} x^4 (65 d+107 e)+\frac{1}{3} x^3 (107 d+33 e)+\frac{3}{2} x^2 (11 d+6 e)+18 d x+10 e x^{10}$

[Out]

18*d*x + (3*(11*d + 6*e)*x^2)/2 + ((107*d + 33*e)*x^3)/3 + ((65*d + 107*e)*x^4)/4 + ((148*d + 65*e)*x^5)/5 - (
37*(d - 4*e)*x^6)/6 + (37*(3*d - e)*x^7)/7 - (3*(15*d - 37*e)*x^8)/8 + (5*(20*d - 9*e)*x^9)/9 + 10*e*x^10

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Rubi [A]  time = 0.160346, antiderivative size = 121, 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} $\frac{5}{9} x^9 (20 d-9 e)-\frac{3}{8} x^8 (15 d-37 e)+\frac{37}{7} x^7 (3 d-e)-\frac{37}{6} x^6 (d-4 e)+\frac{1}{5} x^5 (148 d+65 e)+\frac{1}{4} x^4 (65 d+107 e)+\frac{1}{3} x^3 (107 d+33 e)+\frac{3}{2} x^2 (11 d+6 e)+18 d x+10 e x^{10}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

18*d*x + (3*(11*d + 6*e)*x^2)/2 + ((107*d + 33*e)*x^3)/3 + ((65*d + 107*e)*x^4)/4 + ((148*d + 65*e)*x^5)/5 - (
37*(d - 4*e)*x^6)/6 + (37*(3*d - e)*x^7)/7 - (3*(15*d - 37*e)*x^8)/8 + (5*(20*d - 9*e)*x^9)/9 + 10*e*x^10

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) \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 (18 d+3 (11 d+6 e) x+(107 d+33 e) x^2+(65 d+107 e) x^3+(148 d+65 e) x^4-37 (d-4 e) x^5+37 (3 d-e) x^6-3 (15 d-37 e) x^7+5 (20 d-9 e) x^8+100 e x^9\right ) \, dx\\ &=18 d x+\frac{3}{2} (11 d+6 e) x^2+\frac{1}{3} (107 d+33 e) x^3+\frac{1}{4} (65 d+107 e) x^4+\frac{1}{5} (148 d+65 e) x^5-\frac{37}{6} (d-4 e) x^6+\frac{37}{7} (3 d-e) x^7-\frac{3}{8} (15 d-37 e) x^8+\frac{5}{9} (20 d-9 e) x^9+10 e x^{10}\\ \end{align*}

Mathematica [A]  time = 0.0175363, size = 121, normalized size = 1. $\frac{5}{9} x^9 (20 d-9 e)-\frac{3}{8} x^8 (15 d-37 e)+\frac{37}{7} x^7 (3 d-e)-\frac{37}{6} x^6 (d-4 e)+\frac{1}{5} x^5 (148 d+65 e)+\frac{1}{4} x^4 (65 d+107 e)+\frac{1}{3} x^3 (107 d+33 e)+\frac{3}{2} x^2 (11 d+6 e)+18 d x+10 e x^{10}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

18*d*x + (3*(11*d + 6*e)*x^2)/2 + ((107*d + 33*e)*x^3)/3 + ((65*d + 107*e)*x^4)/4 + ((148*d + 65*e)*x^5)/5 - (
37*(d - 4*e)*x^6)/6 + (37*(3*d - e)*x^7)/7 - (3*(15*d - 37*e)*x^8)/8 + (5*(20*d - 9*e)*x^9)/9 + 10*e*x^10

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Maple [A]  time = 0.043, size = 108, normalized size = 0.9 \begin{align*} 10\,e{x}^{10}+{\frac{ \left ( 100\,d-45\,e \right ){x}^{9}}{9}}+{\frac{ \left ( -45\,d+111\,e \right ){x}^{8}}{8}}+{\frac{ \left ( 111\,d-37\,e \right ){x}^{7}}{7}}+{\frac{ \left ( -37\,d+148\,e \right ){x}^{6}}{6}}+{\frac{ \left ( 148\,d+65\,e \right ){x}^{5}}{5}}+{\frac{ \left ( 65\,d+107\,e \right ){x}^{4}}{4}}+{\frac{ \left ( 107\,d+33\,e \right ){x}^{3}}{3}}+{\frac{ \left ( 33\,d+18\,e \right ){x}^{2}}{2}}+18\,dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*e*x^10+1/9*(100*d-45*e)*x^9+1/8*(-45*d+111*e)*x^8+1/7*(111*d-37*e)*x^7+1/6*(-37*d+148*e)*x^6+1/5*(148*d+65*
e)*x^5+1/4*(65*d+107*e)*x^4+1/3*(107*d+33*e)*x^3+1/2*(33*d+18*e)*x^2+18*d*x

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Maxima [A]  time = 0.964713, size = 142, normalized size = 1.17 \begin{align*} 10 \, e x^{10} + \frac{5}{9} \,{\left (20 \, d - 9 \, e\right )} x^{9} - \frac{3}{8} \,{\left (15 \, d - 37 \, e\right )} x^{8} + \frac{37}{7} \,{\left (3 \, d - e\right )} x^{7} - \frac{37}{6} \,{\left (d - 4 \, e\right )} x^{6} + \frac{1}{5} \,{\left (148 \, d + 65 \, e\right )} x^{5} + \frac{1}{4} \,{\left (65 \, d + 107 \, e\right )} x^{4} + \frac{1}{3} \,{\left (107 \, d + 33 \, e\right )} x^{3} + \frac{3}{2} \,{\left (11 \, d + 6 \, e\right )} x^{2} + 18 \, d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*e*x^10 + 5/9*(20*d - 9*e)*x^9 - 3/8*(15*d - 37*e)*x^8 + 37/7*(3*d - e)*x^7 - 37/6*(d - 4*e)*x^6 + 1/5*(148*
d + 65*e)*x^5 + 1/4*(65*d + 107*e)*x^4 + 1/3*(107*d + 33*e)*x^3 + 3/2*(11*d + 6*e)*x^2 + 18*d*x

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Fricas [A]  time = 0.871077, size = 302, normalized size = 2.5 \begin{align*} 10 x^{10} e - 5 x^{9} e + \frac{100}{9} x^{9} d + \frac{111}{8} x^{8} e - \frac{45}{8} x^{8} d - \frac{37}{7} x^{7} e + \frac{111}{7} x^{7} d + \frac{74}{3} x^{6} e - \frac{37}{6} x^{6} d + 13 x^{5} e + \frac{148}{5} x^{5} d + \frac{107}{4} x^{4} e + \frac{65}{4} x^{4} d + 11 x^{3} e + \frac{107}{3} x^{3} d + 9 x^{2} e + \frac{33}{2} x^{2} d + 18 x d \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*x^10*e - 5*x^9*e + 100/9*x^9*d + 111/8*x^8*e - 45/8*x^8*d - 37/7*x^7*e + 111/7*x^7*d + 74/3*x^6*e - 37/6*x^
6*d + 13*x^5*e + 148/5*x^5*d + 107/4*x^4*e + 65/4*x^4*d + 11*x^3*e + 107/3*x^3*d + 9*x^2*e + 33/2*x^2*d + 18*x
*d

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Sympy [A]  time = 0.089445, size = 112, normalized size = 0.93 \begin{align*} 18 d x + 10 e x^{10} + x^{9} \left (\frac{100 d}{9} - 5 e\right ) + x^{8} \left (- \frac{45 d}{8} + \frac{111 e}{8}\right ) + x^{7} \left (\frac{111 d}{7} - \frac{37 e}{7}\right ) + x^{6} \left (- \frac{37 d}{6} + \frac{74 e}{3}\right ) + x^{5} \left (\frac{148 d}{5} + 13 e\right ) + x^{4} \left (\frac{65 d}{4} + \frac{107 e}{4}\right ) + x^{3} \left (\frac{107 d}{3} + 11 e\right ) + x^{2} \left (\frac{33 d}{2} + 9 e\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18*d*x + 10*e*x**10 + x**9*(100*d/9 - 5*e) + x**8*(-45*d/8 + 111*e/8) + x**7*(111*d/7 - 37*e/7) + x**6*(-37*d/
6 + 74*e/3) + x**5*(148*d/5 + 13*e) + x**4*(65*d/4 + 107*e/4) + x**3*(107*d/3 + 11*e) + x**2*(33*d/2 + 9*e)

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Giac [A]  time = 1.13902, size = 157, normalized size = 1.3 \begin{align*} 10 \, x^{10} e + \frac{100}{9} \, d x^{9} - 5 \, x^{9} e - \frac{45}{8} \, d x^{8} + \frac{111}{8} \, x^{8} e + \frac{111}{7} \, d x^{7} - \frac{37}{7} \, x^{7} e - \frac{37}{6} \, d x^{6} + \frac{74}{3} \, x^{6} e + \frac{148}{5} \, d x^{5} + 13 \, x^{5} e + \frac{65}{4} \, d x^{4} + \frac{107}{4} \, x^{4} e + \frac{107}{3} \, d x^{3} + 11 \, x^{3} e + \frac{33}{2} \, d x^{2} + 9 \, x^{2} e + 18 \, d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*x^10*e + 100/9*d*x^9 - 5*x^9*e - 45/8*d*x^8 + 111/8*x^8*e + 111/7*d*x^7 - 37/7*x^7*e - 37/6*d*x^6 + 74/3*x^
6*e + 148/5*d*x^5 + 13*x^5*e + 65/4*d*x^4 + 107/4*x^4*e + 107/3*d*x^3 + 11*x^3*e + 33/2*d*x^2 + 9*x^2*e + 18*d
*x