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

Optimal. Leaf size=60 \[ \frac {(2 x+3) (2 d-3 e) \left (4 x^2+12 x+9\right )^p}{4 (2 p+1)}+\frac {e \left (4 x^2+12 x+9\right )^{p+1}}{8 (p+1)} \]

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Rubi [A]  time = 0.01, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {640, 609} \begin {gather*} \frac {(2 x+3) (2 d-3 e) \left (4 x^2+12 x+9\right )^p}{4 (2 p+1)}+\frac {e \left (4 x^2+12 x+9\right )^{p+1}}{8 (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(9 + 12*x + 4*x^2)^p,x]

[Out]

((2*d - 3*e)*(3 + 2*x)*(9 + 12*x + 4*x^2)^p)/(4*(1 + 2*p)) + (e*(9 + 12*x + 4*x^2)^(1 + p))/(8*(1 + p))

Rule 609

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int (d+e x) \left (9+12 x+4 x^2\right )^p \, dx &=\frac {e \left (9+12 x+4 x^2\right )^{1+p}}{8 (1+p)}+\frac {1}{2} (2 d-3 e) \int \left (9+12 x+4 x^2\right )^p \, dx\\ &=\frac {(2 d-3 e) (3+2 x) \left (9+12 x+4 x^2\right )^p}{4 (1+2 p)}+\frac {e \left (9+12 x+4 x^2\right )^{1+p}}{8 (1+p)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 48, normalized size = 0.80 \begin {gather*} \frac {(2 x+3) \left ((2 x+3)^2\right )^p (4 d (p+1)+e ((4 p+2) x-3))}{8 (p+1) (2 p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3 + 2*x)*((3 + 2*x)^2)^p*(4*d*(1 + p) + e*(-3 + (2 + 4*p)*x)))/(8*(1 + p)*(1 + 2*p))

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IntegrateAlgebraic [F]  time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x) \left (9+12 x+4 x^2\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 64, normalized size = 1.07 \begin {gather*} \frac {{\left (4 \, {\left (2 \, e p + e\right )} x^{2} + 12 \, d p + 4 \, {\left ({\left (2 \, d + 3 \, e\right )} p + 2 \, d\right )} x + 12 \, d - 9 \, e\right )} {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p}}{8 \, {\left (2 \, p^{2} + 3 \, p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*(2*e*p + e)*x^2 + 12*d*p + 4*((2*d + 3*e)*p + 2*d)*x + 12*d - 9*e)*(4*x^2 + 12*x + 9)^p/(2*p^2 + 3*p +
1)

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giac [B]  time = 0.18, size = 152, normalized size = 2.53 \begin {gather*} \frac {8 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} p x^{2} e + 8 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d p x + 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} p x e + 4 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} x^{2} e + 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d p + 8 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d x + 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d - 9 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} e}{8 \, {\left (2 \, p^{2} + 3 \, p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(8*(4*x^2 + 12*x + 9)^p*p*x^2*e + 8*(4*x^2 + 12*x + 9)^p*d*p*x + 12*(4*x^2 + 12*x + 9)^p*p*x*e + 4*(4*x^2
+ 12*x + 9)^p*x^2*e + 12*(4*x^2 + 12*x + 9)^p*d*p + 8*(4*x^2 + 12*x + 9)^p*d*x + 12*(4*x^2 + 12*x + 9)^p*d - 9
*(4*x^2 + 12*x + 9)^p*e)/(2*p^2 + 3*p + 1)

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maple [A]  time = 0.05, size = 52, normalized size = 0.87 \begin {gather*} \frac {\left (4 e p x +4 d p +2 e x +4 d -3 e \right ) \left (2 x +3\right ) \left (4 x^{2}+12 x +9\right )^{p}}{16 p^{2}+24 p +8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*x^2+12*x+9)^p*(4*e*p*x+4*d*p+2*e*x+4*d-3*e)*(2*x+3)/(2*p^2+3*p+1)

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maxima [A]  time = 1.06, size = 65, normalized size = 1.08 \begin {gather*} \frac {{\left (4 \, {\left (2 \, p + 1\right )} x^{2} + 12 \, p x - 9\right )} e {\left (2 \, x + 3\right )}^{2 \, p}}{8 \, {\left (2 \, p^{2} + 3 \, p + 1\right )}} + \frac {d {\left (2 \, x + 3\right )}^{2 \, p} {\left (2 \, x + 3\right )}}{2 \, {\left (2 \, p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*(2*p + 1)*x^2 + 12*p*x - 9)*e*(2*x + 3)^(2*p)/(2*p^2 + 3*p + 1) + 1/2*d*(2*x + 3)^(2*p)*(2*x + 3)/(2*p
+ 1)

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mupad [B]  time = 0.11, size = 87, normalized size = 1.45 \begin {gather*} \left (\frac {12\,d-9\,e+12\,d\,p}{16\,p^2+24\,p+8}+\frac {x\,\left (8\,d+8\,d\,p+12\,e\,p\right )}{16\,p^2+24\,p+8}+\frac {4\,e\,x^2\,\left (2\,p+1\right )}{16\,p^2+24\,p+8}\right )\,{\left (4\,x^2+12\,x+9\right )}^p \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)*(12*x + 4*x^2 + 9)^p,x)

[Out]

((12*d - 9*e + 12*d*p)/(24*p + 16*p^2 + 8) + (x*(8*d + 8*d*p + 12*e*p))/(24*p + 16*p^2 + 8) + (4*e*x^2*(2*p +
1))/(24*p + 16*p^2 + 8))*(12*x + 4*x^2 + 9)^p

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} - \frac {2 d}{8 x + 12} + \frac {2 e x \log {\left (x + \frac {3}{2} \right )}}{8 x + 12} + \frac {3 e \log {\left (x + \frac {3}{2} \right )}}{8 x + 12} + \frac {3 e}{8 x + 12} & \text {for}\: p = -1 \\\int \frac {d + e x}{\sqrt {\left (2 x + 3\right )^{2}}}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {8 d p x \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} + \frac {12 d p \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} + \frac {8 d x \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} + \frac {12 d \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} + \frac {8 e p x^{2} \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} + \frac {12 e p x \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} + \frac {4 e x^{2} \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} - \frac {9 e \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*d/(8*x + 12) + 2*e*x*log(x + 3/2)/(8*x + 12) + 3*e*log(x + 3/2)/(8*x + 12) + 3*e/(8*x + 12), Eq(
p, -1)), (Integral((d + e*x)/sqrt((2*x + 3)**2), x), Eq(p, -1/2)), (8*d*p*x*(4*x**2 + 12*x + 9)**p/(16*p**2 +
24*p + 8) + 12*d*p*(4*x**2 + 12*x + 9)**p/(16*p**2 + 24*p + 8) + 8*d*x*(4*x**2 + 12*x + 9)**p/(16*p**2 + 24*p
+ 8) + 12*d*(4*x**2 + 12*x + 9)**p/(16*p**2 + 24*p + 8) + 8*e*p*x**2*(4*x**2 + 12*x + 9)**p/(16*p**2 + 24*p +
8) + 12*e*p*x*(4*x**2 + 12*x + 9)**p/(16*p**2 + 24*p + 8) + 4*e*x**2*(4*x**2 + 12*x + 9)**p/(16*p**2 + 24*p +
8) - 9*e*(4*x**2 + 12*x + 9)**p/(16*p**2 + 24*p + 8), True))

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