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

Optimal. Leaf size=82 \[ \frac{1000 x^{13}}{13}+25 x^{12}+\frac{4830 x^{11}}{11}+\frac{3061 x^{10}}{10}+\frac{3316 x^9}{3}+\frac{7869 x^8}{8}+\frac{12016 x^7}{7}+\frac{2873 x^6}{2}+\frac{8292 x^5}{5}+\frac{4483 x^4}{4}+870 x^3+378 x^2+216 x \]

[Out]

216*x + 378*x^2 + 870*x^3 + (4483*x^4)/4 + (8292*x^5)/5 + (2873*x^6)/2 + (12016*x^7)/7 + (7869*x^8)/8 + (3316*
x^9)/3 + (3061*x^10)/10 + (4830*x^11)/11 + 25*x^12 + (1000*x^13)/13

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Rubi [A]  time = 0.0555126, antiderivative size = 82, 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} \[ \frac{1000 x^{13}}{13}+25 x^{12}+\frac{4830 x^{11}}{11}+\frac{3061 x^{10}}{10}+\frac{3316 x^9}{3}+\frac{7869 x^8}{8}+\frac{12016 x^7}{7}+\frac{2873 x^6}{2}+\frac{8292 x^5}{5}+\frac{4483 x^4}{4}+870 x^3+378 x^2+216 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

216*x + 378*x^2 + 870*x^3 + (4483*x^4)/4 + (8292*x^5)/5 + (2873*x^6)/2 + (12016*x^7)/7 + (7869*x^8)/8 + (3316*
x^9)/3 + (3061*x^10)/10 + (4830*x^11)/11 + 25*x^12 + (1000*x^13)/13

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 )^3 \left (2+3 x+5 x^2\right )^3 \, dx &=\int \left (216+756 x+2610 x^2+4483 x^3+8292 x^4+8619 x^5+12016 x^6+7869 x^7+9948 x^8+3061 x^9+4830 x^{10}+300 x^{11}+1000 x^{12}\right ) \, dx\\ &=216 x+378 x^2+870 x^3+\frac{4483 x^4}{4}+\frac{8292 x^5}{5}+\frac{2873 x^6}{2}+\frac{12016 x^7}{7}+\frac{7869 x^8}{8}+\frac{3316 x^9}{3}+\frac{3061 x^{10}}{10}+\frac{4830 x^{11}}{11}+25 x^{12}+\frac{1000 x^{13}}{13}\\ \end{align*}

Mathematica [A]  time = 0.0018334, size = 82, normalized size = 1. \[ \frac{1000 x^{13}}{13}+25 x^{12}+\frac{4830 x^{11}}{11}+\frac{3061 x^{10}}{10}+\frac{3316 x^9}{3}+\frac{7869 x^8}{8}+\frac{12016 x^7}{7}+\frac{2873 x^6}{2}+\frac{8292 x^5}{5}+\frac{4483 x^4}{4}+870 x^3+378 x^2+216 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

216*x + 378*x^2 + 870*x^3 + (4483*x^4)/4 + (8292*x^5)/5 + (2873*x^6)/2 + (12016*x^7)/7 + (7869*x^8)/8 + (3316*
x^9)/3 + (3061*x^10)/10 + (4830*x^11)/11 + 25*x^12 + (1000*x^13)/13

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Maple [A]  time = 0.045, size = 65, normalized size = 0.8 \begin{align*} 216\,x+378\,{x}^{2}+870\,{x}^{3}+{\frac{4483\,{x}^{4}}{4}}+{\frac{8292\,{x}^{5}}{5}}+{\frac{2873\,{x}^{6}}{2}}+{\frac{12016\,{x}^{7}}{7}}+{\frac{7869\,{x}^{8}}{8}}+{\frac{3316\,{x}^{9}}{3}}+{\frac{3061\,{x}^{10}}{10}}+{\frac{4830\,{x}^{11}}{11}}+25\,{x}^{12}+{\frac{1000\,{x}^{13}}{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

216*x+378*x^2+870*x^3+4483/4*x^4+8292/5*x^5+2873/2*x^6+12016/7*x^7+7869/8*x^8+3316/3*x^9+3061/10*x^10+4830/11*
x^11+25*x^12+1000/13*x^13

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Maxima [A]  time = 0.969923, size = 86, normalized size = 1.05 \begin{align*} \frac{1000}{13} \, x^{13} + 25 \, x^{12} + \frac{4830}{11} \, x^{11} + \frac{3061}{10} \, x^{10} + \frac{3316}{3} \, x^{9} + \frac{7869}{8} \, x^{8} + \frac{12016}{7} \, x^{7} + \frac{2873}{2} \, x^{6} + \frac{8292}{5} \, x^{5} + \frac{4483}{4} \, x^{4} + 870 \, x^{3} + 378 \, x^{2} + 216 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1000/13*x^13 + 25*x^12 + 4830/11*x^11 + 3061/10*x^10 + 3316/3*x^9 + 7869/8*x^8 + 12016/7*x^7 + 2873/2*x^6 + 82
92/5*x^5 + 4483/4*x^4 + 870*x^3 + 378*x^2 + 216*x

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Fricas [A]  time = 0.83181, size = 217, normalized size = 2.65 \begin{align*} \frac{1000}{13} x^{13} + 25 x^{12} + \frac{4830}{11} x^{11} + \frac{3061}{10} x^{10} + \frac{3316}{3} x^{9} + \frac{7869}{8} x^{8} + \frac{12016}{7} x^{7} + \frac{2873}{2} x^{6} + \frac{8292}{5} x^{5} + \frac{4483}{4} x^{4} + 870 x^{3} + 378 x^{2} + 216 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1000/13*x^13 + 25*x^12 + 4830/11*x^11 + 3061/10*x^10 + 3316/3*x^9 + 7869/8*x^8 + 12016/7*x^7 + 2873/2*x^6 + 82
92/5*x^5 + 4483/4*x^4 + 870*x^3 + 378*x^2 + 216*x

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Sympy [A]  time = 0.084503, size = 78, normalized size = 0.95 \begin{align*} \frac{1000 x^{13}}{13} + 25 x^{12} + \frac{4830 x^{11}}{11} + \frac{3061 x^{10}}{10} + \frac{3316 x^{9}}{3} + \frac{7869 x^{8}}{8} + \frac{12016 x^{7}}{7} + \frac{2873 x^{6}}{2} + \frac{8292 x^{5}}{5} + \frac{4483 x^{4}}{4} + 870 x^{3} + 378 x^{2} + 216 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1000*x**13/13 + 25*x**12 + 4830*x**11/11 + 3061*x**10/10 + 3316*x**9/3 + 7869*x**8/8 + 12016*x**7/7 + 2873*x**
6/2 + 8292*x**5/5 + 4483*x**4/4 + 870*x**3 + 378*x**2 + 216*x

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Giac [A]  time = 1.13975, size = 86, normalized size = 1.05 \begin{align*} \frac{1000}{13} \, x^{13} + 25 \, x^{12} + \frac{4830}{11} \, x^{11} + \frac{3061}{10} \, x^{10} + \frac{3316}{3} \, x^{9} + \frac{7869}{8} \, x^{8} + \frac{12016}{7} \, x^{7} + \frac{2873}{2} \, x^{6} + \frac{8292}{5} \, x^{5} + \frac{4483}{4} \, x^{4} + 870 \, x^{3} + 378 \, x^{2} + 216 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1000/13*x^13 + 25*x^12 + 4830/11*x^11 + 3061/10*x^10 + 3316/3*x^9 + 7869/8*x^8 + 12016/7*x^7 + 2873/2*x^6 + 82
92/5*x^5 + 4483/4*x^4 + 870*x^3 + 378*x^2 + 216*x