### 3.292 $$\int (3+2 x+5 x^2) (2+x+3 x^2-5 x^3+4 x^4) \, dx$$

Optimal. Leaf size=42 $\frac{20 x^7}{7}-\frac{17 x^6}{6}+\frac{17 x^5}{5}-x^4+7 x^3+\frac{7 x^2}{2}+6 x$

[Out]

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

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Rubi [A]  time = 0.0248029, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.034, Rules used = {1657} $\frac{20 x^7}{7}-\frac{17 x^6}{6}+\frac{17 x^5}{5}-x^4+7 x^3+\frac{7 x^2}{2}+6 x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int \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+7 x+21 x^2-4 x^3+17 x^4-17 x^5+20 x^6\right ) \, dx\\ &=6 x+\frac{7 x^2}{2}+7 x^3-x^4+\frac{17 x^5}{5}-\frac{17 x^6}{6}+\frac{20 x^7}{7}\\ \end{align*}

Mathematica [A]  time = 0.0014784, size = 42, normalized size = 1. $\frac{20 x^7}{7}-\frac{17 x^6}{6}+\frac{17 x^5}{5}-x^4+7 x^3+\frac{7 x^2}{2}+6 x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 35, normalized size = 0.8 \begin{align*} 6\,x+{\frac{7\,{x}^{2}}{2}}+7\,{x}^{3}-{x}^{4}+{\frac{17\,{x}^{5}}{5}}-{\frac{17\,{x}^{6}}{6}}+{\frac{20\,{x}^{7}}{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*x+7/2*x^2+7*x^3-x^4+17/5*x^5-17/6*x^6+20/7*x^7

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Maxima [A]  time = 0.969959, size = 46, normalized size = 1.1 \begin{align*} \frac{20}{7} \, x^{7} - \frac{17}{6} \, x^{6} + \frac{17}{5} \, x^{5} - x^{4} + 7 \, x^{3} + \frac{7}{2} \, x^{2} + 6 \, x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20/7*x^7 - 17/6*x^6 + 17/5*x^5 - x^4 + 7*x^3 + 7/2*x^2 + 6*x

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Fricas [A]  time = 0.868683, size = 84, normalized size = 2. \begin{align*} \frac{20}{7} x^{7} - \frac{17}{6} x^{6} + \frac{17}{5} x^{5} - x^{4} + 7 x^{3} + \frac{7}{2} x^{2} + 6 x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20/7*x^7 - 17/6*x^6 + 17/5*x^5 - x^4 + 7*x^3 + 7/2*x^2 + 6*x

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Sympy [A]  time = 0.063344, size = 37, normalized size = 0.88 \begin{align*} \frac{20 x^{7}}{7} - \frac{17 x^{6}}{6} + \frac{17 x^{5}}{5} - x^{4} + 7 x^{3} + \frac{7 x^{2}}{2} + 6 x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20*x**7/7 - 17*x**6/6 + 17*x**5/5 - x**4 + 7*x**3 + 7*x**2/2 + 6*x

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Giac [A]  time = 1.14635, size = 46, normalized size = 1.1 \begin{align*} \frac{20}{7} \, x^{7} - \frac{17}{6} \, x^{6} + \frac{17}{5} \, x^{5} - x^{4} + 7 \, x^{3} + \frac{7}{2} \, x^{2} + 6 \, x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20/7*x^7 - 17/6*x^6 + 17/5*x^5 - x^4 + 7*x^3 + 7/2*x^2 + 6*x