3.13.40 \(\int (d+e x) (9+12 x+4 x^2)^2 \, dx\)

Optimal. Leaf size=31 \[ \frac {1}{20} (2 x+3)^5 (2 d-3 e)+\frac {1}{24} e (2 x+3)^6 \]

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 43} \begin {gather*} \frac {1}{20} (2 x+3)^5 (2 d-3 e)+\frac {1}{24} e (2 x+3)^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(9 + 12*x + 4*x^2)^2,x]

[Out]

((2*d - 3*e)*(3 + 2*x)^5)/20 + (e*(3 + 2*x)^6)/24

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (d+e x) \left (9+12 x+4 x^2\right )^2 \, dx &=\int (3+2 x)^4 (d+e x) \, dx\\ &=\int \left (\frac {1}{2} (2 d-3 e) (3+2 x)^4+\frac {1}{2} e (3+2 x)^5\right ) \, dx\\ &=\frac {1}{20} (2 d-3 e) (3+2 x)^5+\frac {1}{24} e (3+2 x)^6\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 59, normalized size = 1.90 \begin {gather*} \frac {16}{5} x^5 (d+6 e)+6 x^4 (4 d+9 e)+72 x^3 (d+e)+\frac {27}{2} x^2 (8 d+3 e)+81 d x+\frac {8 e x^6}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*(9 + 12*x + 4*x^2)^2,x]

[Out]

81*d*x + (27*(8*d + 3*e)*x^2)/2 + 72*(d + e)*x^3 + 6*(4*d + 9*e)*x^4 + (16*(d + 6*e)*x^5)/5 + (8*e*x^6)/3

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x) \left (9+12 x+4 x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)*(9 + 12*x + 4*x^2)^2,x]

[Out]

IntegrateAlgebraic[(d + e*x)*(9 + 12*x + 4*x^2)^2, x]

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fricas [B]  time = 0.35, size = 59, normalized size = 1.90 \begin {gather*} \frac {8}{3} x^{6} e + \frac {96}{5} x^{5} e + \frac {16}{5} x^{5} d + 54 x^{4} e + 24 x^{4} d + 72 x^{3} e + 72 x^{3} d + \frac {81}{2} x^{2} e + 108 x^{2} d + 81 x d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x^2+12*x+9)^2,x, algorithm="fricas")

[Out]

8/3*x^6*e + 96/5*x^5*e + 16/5*x^5*d + 54*x^4*e + 24*x^4*d + 72*x^3*e + 72*x^3*d + 81/2*x^2*e + 108*x^2*d + 81*
x*d

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giac [B]  time = 0.15, size = 64, normalized size = 2.06 \begin {gather*} \frac {8}{3} \, x^{6} e + \frac {16}{5} \, d x^{5} + \frac {96}{5} \, x^{5} e + 24 \, d x^{4} + 54 \, x^{4} e + 72 \, d x^{3} + 72 \, x^{3} e + 108 \, d x^{2} + \frac {81}{2} \, x^{2} e + 81 \, d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x^2+12*x+9)^2,x, algorithm="giac")

[Out]

8/3*x^6*e + 16/5*d*x^5 + 96/5*x^5*e + 24*d*x^4 + 54*x^4*e + 72*d*x^3 + 72*x^3*e + 108*d*x^2 + 81/2*x^2*e + 81*
d*x

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maple [B]  time = 0.04, size = 60, normalized size = 1.94 \begin {gather*} \frac {8 e \,x^{6}}{3}+\frac {\left (16 d +96 e \right ) x^{5}}{5}+\frac {\left (96 d +216 e \right ) x^{4}}{4}+\frac {\left (216 d +216 e \right ) x^{3}}{3}+81 d x +\frac {\left (216 d +81 e \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(4*x^2+12*x+9)^2,x)

[Out]

8/3*e*x^6+1/5*(16*d+96*e)*x^5+1/4*(96*d+216*e)*x^4+1/3*(216*d+216*e)*x^3+1/2*(216*d+81*e)*x^2+81*d*x

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maxima [A]  time = 1.33, size = 53, normalized size = 1.71 \begin {gather*} \frac {8}{3} \, e x^{6} + \frac {16}{5} \, {\left (d + 6 \, e\right )} x^{5} + 6 \, {\left (4 \, d + 9 \, e\right )} x^{4} + 72 \, {\left (d + e\right )} x^{3} + \frac {27}{2} \, {\left (8 \, d + 3 \, e\right )} x^{2} + 81 \, d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x^2+12*x+9)^2,x, algorithm="maxima")

[Out]

8/3*e*x^6 + 16/5*(d + 6*e)*x^5 + 6*(4*d + 9*e)*x^4 + 72*(d + e)*x^3 + 27/2*(8*d + 3*e)*x^2 + 81*d*x

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mupad [B]  time = 0.51, size = 55, normalized size = 1.77 \begin {gather*} \frac {8\,e\,x^6}{3}+\left (\frac {16\,d}{5}+\frac {96\,e}{5}\right )\,x^5+\left (24\,d+54\,e\right )\,x^4+\left (72\,d+72\,e\right )\,x^3+\left (108\,d+\frac {81\,e}{2}\right )\,x^2+81\,d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)*(12*x + 4*x^2 + 9)^2,x)

[Out]

x^4*(24*d + 54*e) + x^5*((16*d)/5 + (96*e)/5) + x^3*(72*d + 72*e) + x^2*(108*d + (81*e)/2) + 81*d*x + (8*e*x^6
)/3

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sympy [B]  time = 0.08, size = 58, normalized size = 1.87 \begin {gather*} 81 d x + \frac {8 e x^{6}}{3} + x^{5} \left (\frac {16 d}{5} + \frac {96 e}{5}\right ) + x^{4} \left (24 d + 54 e\right ) + x^{3} \left (72 d + 72 e\right ) + x^{2} \left (108 d + \frac {81 e}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x**2+12*x+9)**2,x)

[Out]

81*d*x + 8*e*x**6/3 + x**5*(16*d/5 + 96*e/5) + x**4*(24*d + 54*e) + x**3*(72*d + 72*e) + x**2*(108*d + 81*e/2)

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