### 3.1614 $$\int (d+e x) (9+12 x+4 x^2)^{5/2} \, dx$$

Optimal. Leaf size=50 $\frac{1}{24} (2 x+3) \left (4 x^2+12 x+9\right )^{5/2} (2 d-3 e)+\frac{1}{28} e \left (4 x^2+12 x+9\right )^{7/2}$

[Out]

((2*d - 3*e)*(3 + 2*x)*(9 + 12*x + 4*x^2)^(5/2))/24 + (e*(9 + 12*x + 4*x^2)^(7/2))/28

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Rubi [A]  time = 0.0127645, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, Rules used = {640, 609} $\frac{1}{24} (2 x+3) \left (4 x^2+12 x+9\right )^{5/2} (2 d-3 e)+\frac{1}{28} e \left (4 x^2+12 x+9\right )^{7/2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*(9 + 12*x + 4*x^2)^(5/2),x]

[Out]

((2*d - 3*e)*(3 + 2*x)*(9 + 12*x + 4*x^2)^(5/2))/24 + (e*(9 + 12*x + 4*x^2)^(7/2))/28

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]

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)]

Rubi steps

\begin{align*} \int (d+e x) \left (9+12 x+4 x^2\right )^{5/2} \, dx &=\frac{1}{28} e \left (9+12 x+4 x^2\right )^{7/2}+\frac{1}{2} (2 d-3 e) \int \left (9+12 x+4 x^2\right )^{5/2} \, dx\\ &=\frac{1}{24} (2 d-3 e) (3+2 x) \left (9+12 x+4 x^2\right )^{5/2}+\frac{1}{28} e \left (9+12 x+4 x^2\right )^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0311574, size = 81, normalized size = 1.62 $\frac{x \sqrt{(2 x+3)^2} \left (14 d \left (16 x^5+144 x^4+540 x^3+1080 x^2+1215 x+729\right )+3 e x \left (64 x^5+560 x^4+2016 x^3+3780 x^2+3780 x+1701\right )\right )}{42 (2 x+3)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)*(9 + 12*x + 4*x^2)^(5/2),x]

[Out]

(x*Sqrt[(3 + 2*x)^2]*(14*d*(729 + 1215*x + 1080*x^2 + 540*x^3 + 144*x^4 + 16*x^5) + 3*e*x*(1701 + 3780*x + 378
0*x^2 + 2016*x^3 + 560*x^4 + 64*x^5)))/(42*(3 + 2*x))

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Maple [B]  time = 0.081, size = 86, normalized size = 1.7 \begin{align*}{\frac{x \left ( 192\,e{x}^{6}+224\,{x}^{5}d+1680\,{x}^{5}e+2016\,d{x}^{4}+6048\,e{x}^{4}+7560\,d{x}^{3}+11340\,{x}^{3}e+15120\,d{x}^{2}+11340\,e{x}^{2}+17010\,dx+5103\,ex+10206\,d \right ) }{42\, \left ( 3+2\,x \right ) ^{5}} \left ( \left ( 3+2\,x \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(4*x^2+12*x+9)^(5/2),x)

[Out]

1/42*x*(192*e*x^6+224*d*x^5+1680*e*x^5+2016*d*x^4+6048*e*x^4+7560*d*x^3+11340*e*x^3+15120*d*x^2+11340*e*x^2+17
010*d*x+5103*e*x+10206*d)*((3+2*x)^2)^(5/2)/(3+2*x)^5

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Maxima [A]  time = 1.60213, size = 105, normalized size = 2.1 \begin{align*} \frac{1}{28} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{7}{2}} e + \frac{1}{6} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} d x - \frac{1}{4} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} e x + \frac{1}{4} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} d - \frac{3}{8} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} e \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x^2+12*x+9)^(5/2),x, algorithm="maxima")

[Out]

1/28*(4*x^2 + 12*x + 9)^(7/2)*e + 1/6*(4*x^2 + 12*x + 9)^(5/2)*d*x - 1/4*(4*x^2 + 12*x + 9)^(5/2)*e*x + 1/4*(4
*x^2 + 12*x + 9)^(5/2)*d - 3/8*(4*x^2 + 12*x + 9)^(5/2)*e

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Fricas [A]  time = 1.47588, size = 176, normalized size = 3.52 \begin{align*} \frac{32}{7} \, e x^{7} + \frac{8}{3} \,{\left (2 \, d + 15 \, e\right )} x^{6} + 48 \,{\left (d + 3 \, e\right )} x^{5} + 90 \,{\left (2 \, d + 3 \, e\right )} x^{4} + 90 \,{\left (4 \, d + 3 \, e\right )} x^{3} + \frac{81}{2} \,{\left (10 \, d + 3 \, e\right )} x^{2} + 243 \, d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x^2+12*x+9)^(5/2),x, algorithm="fricas")

[Out]

32/7*e*x^7 + 8/3*(2*d + 15*e)*x^6 + 48*(d + 3*e)*x^5 + 90*(2*d + 3*e)*x^4 + 90*(4*d + 3*e)*x^3 + 81/2*(10*d +
3*e)*x^2 + 243*d*x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right ) \left (\left (2 x + 3\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x**2+12*x+9)**(5/2),x)

[Out]

Integral((d + e*x)*((2*x + 3)**2)**(5/2), x)

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Giac [B]  time = 1.15301, size = 223, normalized size = 4.46 \begin{align*} \frac{32}{7} \, x^{7} e \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{16}{3} \, d x^{6} \mathrm{sgn}\left (2 \, x + 3\right ) + 40 \, x^{6} e \mathrm{sgn}\left (2 \, x + 3\right ) + 48 \, d x^{5} \mathrm{sgn}\left (2 \, x + 3\right ) + 144 \, x^{5} e \mathrm{sgn}\left (2 \, x + 3\right ) + 180 \, d x^{4} \mathrm{sgn}\left (2 \, x + 3\right ) + 270 \, x^{4} e \mathrm{sgn}\left (2 \, x + 3\right ) + 360 \, d x^{3} \mathrm{sgn}\left (2 \, x + 3\right ) + 270 \, x^{3} e \mathrm{sgn}\left (2 \, x + 3\right ) + 405 \, d x^{2} \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{243}{2} \, x^{2} e \mathrm{sgn}\left (2 \, x + 3\right ) + 243 \, d x \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{243}{56} \,{\left (14 \, d - 3 \, e\right )} \mathrm{sgn}\left (2 \, x + 3\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x^2+12*x+9)^(5/2),x, algorithm="giac")

[Out]

32/7*x^7*e*sgn(2*x + 3) + 16/3*d*x^6*sgn(2*x + 3) + 40*x^6*e*sgn(2*x + 3) + 48*d*x^5*sgn(2*x + 3) + 144*x^5*e*
sgn(2*x + 3) + 180*d*x^4*sgn(2*x + 3) + 270*x^4*e*sgn(2*x + 3) + 360*d*x^3*sgn(2*x + 3) + 270*x^3*e*sgn(2*x +
3) + 405*d*x^2*sgn(2*x + 3) + 243/2*x^2*e*sgn(2*x + 3) + 243*d*x*sgn(2*x + 3) + 243/56*(14*d - 3*e)*sgn(2*x +
3)