3.15 \(\int (3-x+2 x^2) (2+3 x+5 x^2)^4 \, dx\)

Optimal. Leaf size=68 \[ \frac {1250 x^{11}}{11}+\frac {475 x^{10}}{2}+\frac {5075 x^9}{9}+\frac {3415 x^8}{4}+1176 x^7+\frac {2377 x^6}{2}+\frac {5099 x^5}{5}+656 x^4+\frac {1064 x^3}{3}+136 x^2+48 x \]

[Out]

48*x+136*x^2+1064/3*x^3+656*x^4+5099/5*x^5+2377/2*x^6+1176*x^7+3415/4*x^8+5075/9*x^9+475/2*x^10+1250/11*x^11

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Rubi [A]  time = 0.05, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1657} \[ \frac {1250 x^{11}}{11}+\frac {475 x^{10}}{2}+\frac {5075 x^9}{9}+\frac {3415 x^8}{4}+1176 x^7+\frac {2377 x^6}{2}+\frac {5099 x^5}{5}+656 x^4+\frac {1064 x^3}{3}+136 x^2+48 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

48*x + 136*x^2 + (1064*x^3)/3 + 656*x^4 + (5099*x^5)/5 + (2377*x^6)/2 + 1176*x^7 + (3415*x^8)/4 + (5075*x^9)/9
+ (475*x^10)/2 + (1250*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 ) \left (2+3 x+5 x^2\right )^4 \, dx &=\int \left (48+272 x+1064 x^2+2624 x^3+5099 x^4+7131 x^5+8232 x^6+6830 x^7+5075 x^8+2375 x^9+1250 x^{10}\right ) \, dx\\ &=48 x+136 x^2+\frac {1064 x^3}{3}+656 x^4+\frac {5099 x^5}{5}+\frac {2377 x^6}{2}+1176 x^7+\frac {3415 x^8}{4}+\frac {5075 x^9}{9}+\frac {475 x^{10}}{2}+\frac {1250 x^{11}}{11}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 68, normalized size = 1.00 \[ \frac {1250 x^{11}}{11}+\frac {475 x^{10}}{2}+\frac {5075 x^9}{9}+\frac {3415 x^8}{4}+1176 x^7+\frac {2377 x^6}{2}+\frac {5099 x^5}{5}+656 x^4+\frac {1064 x^3}{3}+136 x^2+48 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

48*x + 136*x^2 + (1064*x^3)/3 + 656*x^4 + (5099*x^5)/5 + (2377*x^6)/2 + 1176*x^7 + (3415*x^8)/4 + (5075*x^9)/9
+ (475*x^10)/2 + (1250*x^11)/11

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fricas [A]  time = 0.72, size = 54, normalized size = 0.79 \[ \frac {1250}{11} x^{11} + \frac {475}{2} x^{10} + \frac {5075}{9} x^{9} + \frac {3415}{4} x^{8} + 1176 x^{7} + \frac {2377}{2} x^{6} + \frac {5099}{5} x^{5} + 656 x^{4} + \frac {1064}{3} x^{3} + 136 x^{2} + 48 x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1250/11*x^11 + 475/2*x^10 + 5075/9*x^9 + 3415/4*x^8 + 1176*x^7 + 2377/2*x^6 + 5099/5*x^5 + 656*x^4 + 1064/3*x^
3 + 136*x^2 + 48*x

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giac [A]  time = 0.21, size = 54, normalized size = 0.79 \[ \frac {1250}{11} \, x^{11} + \frac {475}{2} \, x^{10} + \frac {5075}{9} \, x^{9} + \frac {3415}{4} \, x^{8} + 1176 \, x^{7} + \frac {2377}{2} \, x^{6} + \frac {5099}{5} \, x^{5} + 656 \, x^{4} + \frac {1064}{3} \, x^{3} + 136 \, x^{2} + 48 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1250/11*x^11 + 475/2*x^10 + 5075/9*x^9 + 3415/4*x^8 + 1176*x^7 + 2377/2*x^6 + 5099/5*x^5 + 656*x^4 + 1064/3*x^
3 + 136*x^2 + 48*x

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maple [A]  time = 0.00, size = 55, normalized size = 0.81 \[ \frac {1250}{11} x^{11}+\frac {475}{2} x^{10}+\frac {5075}{9} x^{9}+\frac {3415}{4} x^{8}+1176 x^{7}+\frac {2377}{2} x^{6}+\frac {5099}{5} x^{5}+656 x^{4}+\frac {1064}{3} x^{3}+136 x^{2}+48 x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

48*x+136*x^2+1064/3*x^3+656*x^4+5099/5*x^5+2377/2*x^6+1176*x^7+3415/4*x^8+5075/9*x^9+475/2*x^10+1250/11*x^11

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maxima [A]  time = 0.46, size = 54, normalized size = 0.79 \[ \frac {1250}{11} \, x^{11} + \frac {475}{2} \, x^{10} + \frac {5075}{9} \, x^{9} + \frac {3415}{4} \, x^{8} + 1176 \, x^{7} + \frac {2377}{2} \, x^{6} + \frac {5099}{5} \, x^{5} + 656 \, x^{4} + \frac {1064}{3} \, x^{3} + 136 \, x^{2} + 48 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1250/11*x^11 + 475/2*x^10 + 5075/9*x^9 + 3415/4*x^8 + 1176*x^7 + 2377/2*x^6 + 5099/5*x^5 + 656*x^4 + 1064/3*x^
3 + 136*x^2 + 48*x

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mupad [B]  time = 0.06, size = 54, normalized size = 0.79 \[ \frac {1250\,x^{11}}{11}+\frac {475\,x^{10}}{2}+\frac {5075\,x^9}{9}+\frac {3415\,x^8}{4}+1176\,x^7+\frac {2377\,x^6}{2}+\frac {5099\,x^5}{5}+656\,x^4+\frac {1064\,x^3}{3}+136\,x^2+48\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

48*x + 136*x^2 + (1064*x^3)/3 + 656*x^4 + (5099*x^5)/5 + (2377*x^6)/2 + 1176*x^7 + (3415*x^8)/4 + (5075*x^9)/9
+ (475*x^10)/2 + (1250*x^11)/11

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sympy [A]  time = 0.09, size = 65, normalized size = 0.96 \[ \frac {1250 x^{11}}{11} + \frac {475 x^{10}}{2} + \frac {5075 x^{9}}{9} + \frac {3415 x^{8}}{4} + 1176 x^{7} + \frac {2377 x^{6}}{2} + \frac {5099 x^{5}}{5} + 656 x^{4} + \frac {1064 x^{3}}{3} + 136 x^{2} + 48 x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1250*x**11/11 + 475*x**10/2 + 5075*x**9/9 + 3415*x**8/4 + 1176*x**7 + 2377*x**6/2 + 5099*x**5/5 + 656*x**4 + 1
064*x**3/3 + 136*x**2 + 48*x

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