3.23 \(\int (3-x+2 x^2)^2 (2+3 x+5 x^2)^3 \, dx\)

Optimal. Leaf size=66 \[ \frac{500 x^{11}}{11}+40 x^{10}+\frac{1865 x^9}{9}+\frac{1863 x^8}{8}+444 x^7+449 x^6+\frac{2693 x^5}{5}+\frac{1615 x^4}{4}+\frac{914 x^3}{3}+138 x^2+72 x \]

[Out]

72*x + 138*x^2 + (914*x^3)/3 + (1615*x^4)/4 + (2693*x^5)/5 + 449*x^6 + 444*x^7 + (1863*x^8)/8 + (1865*x^9)/9 +
40*x^10 + (500*x^11)/11

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Rubi [A]  time = 0.0481998, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1657} \[ \frac{500 x^{11}}{11}+40 x^{10}+\frac{1865 x^9}{9}+\frac{1863 x^8}{8}+444 x^7+449 x^6+\frac{2693 x^5}{5}+\frac{1615 x^4}{4}+\frac{914 x^3}{3}+138 x^2+72 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

72*x + 138*x^2 + (914*x^3)/3 + (1615*x^4)/4 + (2693*x^5)/5 + 449*x^6 + 444*x^7 + (1863*x^8)/8 + (1865*x^9)/9 +
40*x^10 + (500*x^11)/11

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-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^3 \, dx &=\int \left (72+276 x+914 x^2+1615 x^3+2693 x^4+2694 x^5+3108 x^6+1863 x^7+1865 x^8+400 x^9+500 x^{10}\right ) \, dx\\ &=72 x+138 x^2+\frac{914 x^3}{3}+\frac{1615 x^4}{4}+\frac{2693 x^5}{5}+449 x^6+444 x^7+\frac{1863 x^8}{8}+\frac{1865 x^9}{9}+40 x^{10}+\frac{500 x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.0021084, size = 66, normalized size = 1. \[ \frac{500 x^{11}}{11}+40 x^{10}+\frac{1865 x^9}{9}+\frac{1863 x^8}{8}+444 x^7+449 x^6+\frac{2693 x^5}{5}+\frac{1615 x^4}{4}+\frac{914 x^3}{3}+138 x^2+72 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

72*x + 138*x^2 + (914*x^3)/3 + (1615*x^4)/4 + (2693*x^5)/5 + 449*x^6 + 444*x^7 + (1863*x^8)/8 + (1865*x^9)/9 +
40*x^10 + (500*x^11)/11

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Maple [A]  time = 0.042, size = 55, normalized size = 0.8 \begin{align*} 72\,x+138\,{x}^{2}+{\frac{914\,{x}^{3}}{3}}+{\frac{1615\,{x}^{4}}{4}}+{\frac{2693\,{x}^{5}}{5}}+449\,{x}^{6}+444\,{x}^{7}+{\frac{1863\,{x}^{8}}{8}}+{\frac{1865\,{x}^{9}}{9}}+40\,{x}^{10}+{\frac{500\,{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

72*x+138*x^2+914/3*x^3+1615/4*x^4+2693/5*x^5+449*x^6+444*x^7+1863/8*x^8+1865/9*x^9+40*x^10+500/11*x^11

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Maxima [A]  time = 0.964319, size = 73, normalized size = 1.11 \begin{align*} \frac{500}{11} \, x^{11} + 40 \, x^{10} + \frac{1865}{9} \, x^{9} + \frac{1863}{8} \, x^{8} + 444 \, x^{7} + 449 \, x^{6} + \frac{2693}{5} \, x^{5} + \frac{1615}{4} \, x^{4} + \frac{914}{3} \, x^{3} + 138 \, x^{2} + 72 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500/11*x^11 + 40*x^10 + 1865/9*x^9 + 1863/8*x^8 + 444*x^7 + 449*x^6 + 2693/5*x^5 + 1615/4*x^4 + 914/3*x^3 + 13
8*x^2 + 72*x

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Fricas [A]  time = 0.734116, size = 167, normalized size = 2.53 \begin{align*} \frac{500}{11} x^{11} + 40 x^{10} + \frac{1865}{9} x^{9} + \frac{1863}{8} x^{8} + 444 x^{7} + 449 x^{6} + \frac{2693}{5} x^{5} + \frac{1615}{4} x^{4} + \frac{914}{3} x^{3} + 138 x^{2} + 72 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500/11*x^11 + 40*x^10 + 1865/9*x^9 + 1863/8*x^8 + 444*x^7 + 449*x^6 + 2693/5*x^5 + 1615/4*x^4 + 914/3*x^3 + 13
8*x^2 + 72*x

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Sympy [A]  time = 0.137197, size = 63, normalized size = 0.95 \begin{align*} \frac{500 x^{11}}{11} + 40 x^{10} + \frac{1865 x^{9}}{9} + \frac{1863 x^{8}}{8} + 444 x^{7} + 449 x^{6} + \frac{2693 x^{5}}{5} + \frac{1615 x^{4}}{4} + \frac{914 x^{3}}{3} + 138 x^{2} + 72 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500*x**11/11 + 40*x**10 + 1865*x**9/9 + 1863*x**8/8 + 444*x**7 + 449*x**6 + 2693*x**5/5 + 1615*x**4/4 + 914*x*
*3/3 + 138*x**2 + 72*x

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Giac [A]  time = 1.16031, size = 73, normalized size = 1.11 \begin{align*} \frac{500}{11} \, x^{11} + 40 \, x^{10} + \frac{1865}{9} \, x^{9} + \frac{1863}{8} \, x^{8} + 444 \, x^{7} + 449 \, x^{6} + \frac{2693}{5} \, x^{5} + \frac{1615}{4} \, x^{4} + \frac{914}{3} \, x^{3} + 138 \, x^{2} + 72 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500/11*x^11 + 40*x^10 + 1865/9*x^9 + 1863/8*x^8 + 444*x^7 + 449*x^6 + 2693/5*x^5 + 1615/4*x^4 + 914/3*x^3 + 13
8*x^2 + 72*x