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

Optimal. Leaf size=44 \[ \frac {50 x^7}{7}+\frac {35 x^6}{6}+\frac {103 x^5}{5}+\frac {85 x^4}{4}+\frac {83 x^3}{3}+16 x^2+12 x \]

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Rubi [A]  time = 0.03, antiderivative size = 44, 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} \begin {gather*} \frac {50 x^7}{7}+\frac {35 x^6}{6}+\frac {103 x^5}{5}+\frac {85 x^4}{4}+\frac {83 x^3}{3}+16 x^2+12 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

12*x + 16*x^2 + (83*x^3)/3 + (85*x^4)/4 + (103*x^5)/5 + (35*x^6)/6 + (50*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-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2 \, dx &=\int \left (12+32 x+83 x^2+85 x^3+103 x^4+35 x^5+50 x^6\right ) \, dx\\ &=12 x+16 x^2+\frac {83 x^3}{3}+\frac {85 x^4}{4}+\frac {103 x^5}{5}+\frac {35 x^6}{6}+\frac {50 x^7}{7}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 44, normalized size = 1.00 \begin {gather*} \frac {50 x^7}{7}+\frac {35 x^6}{6}+\frac {103 x^5}{5}+\frac {85 x^4}{4}+\frac {83 x^3}{3}+16 x^2+12 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

12*x + 16*x^2 + (83*x^3)/3 + (85*x^4)/4 + (103*x^5)/5 + (35*x^6)/6 + (50*x^7)/7

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.34, size = 34, normalized size = 0.77 \begin {gather*} \frac {50}{7} x^{7} + \frac {35}{6} x^{6} + \frac {103}{5} x^{5} + \frac {85}{4} x^{4} + \frac {83}{3} x^{3} + 16 x^{2} + 12 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50/7*x^7 + 35/6*x^6 + 103/5*x^5 + 85/4*x^4 + 83/3*x^3 + 16*x^2 + 12*x

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giac [A]  time = 0.21, size = 34, normalized size = 0.77 \begin {gather*} \frac {50}{7} \, x^{7} + \frac {35}{6} \, x^{6} + \frac {103}{5} \, x^{5} + \frac {85}{4} \, x^{4} + \frac {83}{3} \, x^{3} + 16 \, x^{2} + 12 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50/7*x^7 + 35/6*x^6 + 103/5*x^5 + 85/4*x^4 + 83/3*x^3 + 16*x^2 + 12*x

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maple [A]  time = 0.00, size = 35, normalized size = 0.80 \begin {gather*} \frac {50}{7} x^{7}+\frac {35}{6} x^{6}+\frac {103}{5} x^{5}+\frac {85}{4} x^{4}+\frac {83}{3} x^{3}+16 x^{2}+12 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12*x+16*x^2+83/3*x^3+85/4*x^4+103/5*x^5+35/6*x^6+50/7*x^7

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maxima [A]  time = 0.43, size = 34, normalized size = 0.77 \begin {gather*} \frac {50}{7} \, x^{7} + \frac {35}{6} \, x^{6} + \frac {103}{5} \, x^{5} + \frac {85}{4} \, x^{4} + \frac {83}{3} \, x^{3} + 16 \, x^{2} + 12 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50/7*x^7 + 35/6*x^6 + 103/5*x^5 + 85/4*x^4 + 83/3*x^3 + 16*x^2 + 12*x

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mupad [B]  time = 0.03, size = 34, normalized size = 0.77 \begin {gather*} \frac {50\,x^7}{7}+\frac {35\,x^6}{6}+\frac {103\,x^5}{5}+\frac {85\,x^4}{4}+\frac {83\,x^3}{3}+16\,x^2+12\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12*x + 16*x^2 + (83*x^3)/3 + (85*x^4)/4 + (103*x^5)/5 + (35*x^6)/6 + (50*x^7)/7

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sympy [A]  time = 0.08, size = 41, normalized size = 0.93 \begin {gather*} \frac {50 x^{7}}{7} + \frac {35 x^{6}}{6} + \frac {103 x^{5}}{5} + \frac {85 x^{4}}{4} + \frac {83 x^{3}}{3} + 16 x^{2} + 12 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50*x**7/7 + 35*x**6/6 + 103*x**5/5 + 85*x**4/4 + 83*x**3/3 + 16*x**2 + 12*x

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