### 3.291 $$\int (d+e x) (3+2 x+5 x^2) (2+x+3 x^2-5 x^3+4 x^4) \, dx$$

Optimal. Leaf size=93 $\frac{1}{7} x^7 (20 d-17 e)-\frac{17}{6} x^6 (d-e)+\frac{1}{5} x^5 (17 d-4 e)-\frac{1}{4} x^4 (4 d-21 e)+\frac{7}{3} x^3 (3 d+e)+\frac{1}{2} x^2 (7 d+6 e)+6 d x+\frac{5 e x^8}{2}$

[Out]

6*d*x + ((7*d + 6*e)*x^2)/2 + (7*(3*d + e)*x^3)/3 - ((4*d - 21*e)*x^4)/4 + ((17*d - 4*e)*x^5)/5 - (17*(d - e)*
x^6)/6 + ((20*d - 17*e)*x^7)/7 + (5*e*x^8)/2

________________________________________________________________________________________

Rubi [A]  time = 0.107599, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 34, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.029, Rules used = {1628} $\frac{1}{7} x^7 (20 d-17 e)-\frac{17}{6} x^6 (d-e)+\frac{1}{5} x^5 (17 d-4 e)-\frac{1}{4} x^4 (4 d-21 e)+\frac{7}{3} x^3 (3 d+e)+\frac{1}{2} x^2 (7 d+6 e)+6 d x+\frac{5 e x^8}{2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

6*d*x + ((7*d + 6*e)*x^2)/2 + (7*(3*d + e)*x^3)/3 - ((4*d - 21*e)*x^4)/4 + ((17*d - 4*e)*x^5)/5 - (17*(d - e)*
x^6)/6 + ((20*d - 17*e)*x^7)/7 + (5*e*x^8)/2

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 ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx &=\int \left (6 d+(7 d+6 e) x+7 (3 d+e) x^2-(4 d-21 e) x^3+(17 d-4 e) x^4-17 (d-e) x^5+(20 d-17 e) x^6+20 e x^7\right ) \, dx\\ &=6 d x+\frac{1}{2} (7 d+6 e) x^2+\frac{7}{3} (3 d+e) x^3-\frac{1}{4} (4 d-21 e) x^4+\frac{1}{5} (17 d-4 e) x^5-\frac{17}{6} (d-e) x^6+\frac{1}{7} (20 d-17 e) x^7+\frac{5 e x^8}{2}\\ \end{align*}

Mathematica [A]  time = 0.0144416, size = 93, normalized size = 1. $\frac{1}{7} x^7 (20 d-17 e)-\frac{17}{6} x^6 (d-e)+\frac{1}{5} x^5 (17 d-4 e)+\frac{1}{4} x^4 (21 e-4 d)+\frac{7}{3} x^3 (3 d+e)+\frac{1}{2} x^2 (7 d+6 e)+6 d x+\frac{5 e x^8}{2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

6*d*x + ((7*d + 6*e)*x^2)/2 + (7*(3*d + e)*x^3)/3 + ((-4*d + 21*e)*x^4)/4 + ((17*d - 4*e)*x^5)/5 - (17*(d - e)
*x^6)/6 + ((20*d - 17*e)*x^7)/7 + (5*e*x^8)/2

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 84, normalized size = 0.9 \begin{align*}{\frac{5\,e{x}^{8}}{2}}+{\frac{ \left ( 20\,d-17\,e \right ){x}^{7}}{7}}+{\frac{ \left ( -17\,d+17\,e \right ){x}^{6}}{6}}+{\frac{ \left ( 17\,d-4\,e \right ){x}^{5}}{5}}+{\frac{ \left ( -4\,d+21\,e \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,d+7\,e \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,d+6\,e \right ){x}^{2}}{2}}+6\,dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/2*e*x^8+1/7*(20*d-17*e)*x^7+1/6*(-17*d+17*e)*x^6+1/5*(17*d-4*e)*x^5+1/4*(-4*d+21*e)*x^4+1/3*(21*d+7*e)*x^3+1
/2*(7*d+6*e)*x^2+6*d*x

________________________________________________________________________________________

Maxima [A]  time = 0.991202, size = 107, normalized size = 1.15 \begin{align*} \frac{5}{2} \, e x^{8} + \frac{1}{7} \,{\left (20 \, d - 17 \, e\right )} x^{7} - \frac{17}{6} \,{\left (d - e\right )} x^{6} + \frac{1}{5} \,{\left (17 \, d - 4 \, e\right )} x^{5} - \frac{1}{4} \,{\left (4 \, d - 21 \, e\right )} x^{4} + \frac{7}{3} \,{\left (3 \, d + e\right )} x^{3} + \frac{1}{2} \,{\left (7 \, d + 6 \, e\right )} x^{2} + 6 \, 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)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="maxima")

[Out]

5/2*e*x^8 + 1/7*(20*d - 17*e)*x^7 - 17/6*(d - e)*x^6 + 1/5*(17*d - 4*e)*x^5 - 1/4*(4*d - 21*e)*x^4 + 7/3*(3*d
+ e)*x^3 + 1/2*(7*d + 6*e)*x^2 + 6*d*x

________________________________________________________________________________________

Fricas [A]  time = 0.839921, size = 217, normalized size = 2.33 \begin{align*} \frac{5}{2} x^{8} e - \frac{17}{7} x^{7} e + \frac{20}{7} x^{7} d + \frac{17}{6} x^{6} e - \frac{17}{6} x^{6} d - \frac{4}{5} x^{5} e + \frac{17}{5} x^{5} d + \frac{21}{4} x^{4} e - x^{4} d + \frac{7}{3} x^{3} e + 7 x^{3} d + 3 x^{2} e + \frac{7}{2} x^{2} d + 6 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)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="fricas")

[Out]

5/2*x^8*e - 17/7*x^7*e + 20/7*x^7*d + 17/6*x^6*e - 17/6*x^6*d - 4/5*x^5*e + 17/5*x^5*d + 21/4*x^4*e - x^4*d +
7/3*x^3*e + 7*x^3*d + 3*x^2*e + 7/2*x^2*d + 6*x*d

________________________________________________________________________________________

Sympy [A]  time = 0.083282, size = 87, normalized size = 0.94 \begin{align*} 6 d x + \frac{5 e x^{8}}{2} + x^{7} \left (\frac{20 d}{7} - \frac{17 e}{7}\right ) + x^{6} \left (- \frac{17 d}{6} + \frac{17 e}{6}\right ) + x^{5} \left (\frac{17 d}{5} - \frac{4 e}{5}\right ) + x^{4} \left (- d + \frac{21 e}{4}\right ) + x^{3} \left (7 d + \frac{7 e}{3}\right ) + x^{2} \left (\frac{7 d}{2} + 3 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)*(4*x**4-5*x**3+3*x**2+x+2),x)

[Out]

6*d*x + 5*e*x**8/2 + x**7*(20*d/7 - 17*e/7) + x**6*(-17*d/6 + 17*e/6) + x**5*(17*d/5 - 4*e/5) + x**4*(-d + 21*
e/4) + x**3*(7*d + 7*e/3) + x**2*(7*d/2 + 3*e)

________________________________________________________________________________________

Giac [A]  time = 1.14264, size = 122, normalized size = 1.31 \begin{align*} \frac{5}{2} \, x^{8} e + \frac{20}{7} \, d x^{7} - \frac{17}{7} \, x^{7} e - \frac{17}{6} \, d x^{6} + \frac{17}{6} \, x^{6} e + \frac{17}{5} \, d x^{5} - \frac{4}{5} \, x^{5} e - d x^{4} + \frac{21}{4} \, x^{4} e + 7 \, d x^{3} + \frac{7}{3} \, x^{3} e + \frac{7}{2} \, d x^{2} + 3 \, x^{2} e + 6 \, 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)*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="giac")

[Out]

5/2*x^8*e + 20/7*d*x^7 - 17/7*x^7*e - 17/6*d*x^6 + 17/6*x^6*e + 17/5*d*x^5 - 4/5*x^5*e - d*x^4 + 21/4*x^4*e +
7*d*x^3 + 7/3*x^3*e + 7/2*d*x^2 + 3*x^2*e + 6*d*x