3.17.14 \(\int (d+e x) (9+12 x+4 x^2)^{5/2} \, dx\) [1614]

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

[Out]

1/24*(2*d-3*e)*(3+2*x)*(4*x^2+12*x+9)^(5/2)+1/28*e*(4*x^2+12*x+9)^(7/2)

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Rubi [A]
time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {654, 623} \begin {gather*} \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} \end {gather*}

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 623

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 654

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 )^{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*}

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Mathematica [A]
time = 0.02, size = 81, normalized size = 1.62 \begin {gather*} \frac {x \sqrt {(3+2 x)^2} \left (14 d \left (729+1215 x+1080 x^2+540 x^3+144 x^4+16 x^5\right )+3 e x \left (1701+3780 x+3780 x^2+2016 x^3+560 x^4+64 x^5\right )\right )}{42 (3+2 x)} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs. \(2(42)=84\).
time = 0.56, size = 86, normalized size = 1.72

method	result	size
gosper	\(\frac {x \left (192 x^{6} e +224 x^{5} d +1680 x^{5} e +2016 d \,x^{4}+6048 x^{4} e +7560 d \,x^{3}+11340 e \,x^{3}+15120 d \,x^{2}+11340 e \,x^{2}+17010 d x +5103 e x +10206 d \right ) \left (\left (2 x +3\right )^{2}\right )^{\frac {5}{2}}}{42 \left (2 x +3\right )^{5}}\)	\(86\)
default	\(\frac {x \left (192 x^{6} e +224 x^{5} d +1680 x^{5} e +2016 d \,x^{4}+6048 x^{4} e +7560 d \,x^{3}+11340 e \,x^{3}+15120 d \,x^{2}+11340 e \,x^{2}+17010 d x +5103 e x +10206 d \right ) \left (\left (2 x +3\right )^{2}\right )^{\frac {5}{2}}}{42 \left (2 x +3\right )^{5}}\)	\(86\)
risch	\(\frac {32 \sqrt {\left (2 x +3\right )^{2}}\, e \,x^{7}}{7 \left (2 x +3\right )}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (32 d +240 e \right ) x^{6}}{12 x +18}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (240 d +720 e \right ) x^{5}}{10 x +15}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (720 d +1080 e \right ) x^{4}}{8 x +12}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (1080 d +810 e \right ) x^{3}}{6 x +9}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (810 d +243 e \right ) x^{2}}{4 x +6}+\frac {243 \sqrt {\left (2 x +3\right )^{2}}\, d x}{2 x +3}\)	\(184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)*((2*x+3)^2)^(5/2)/(2*x+3)^5

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Maxima [A]
time = 0.50, size = 81, normalized size = 1.62 \begin {gather*} \frac {1}{6} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}} d x + \frac {1}{28} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {7}{2}} e - \frac {1}{4} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}} x e + \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 {gather*}

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]

1/6*(4*x^2 + 12*x + 9)^(5/2)*d*x + 1/28*(4*x^2 + 12*x + 9)^(7/2)*e - 1/4*(4*x^2 + 12*x + 9)^(5/2)*x*e + 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 = 4.18, size = 70, normalized size = 1.40 \begin {gather*} \frac {16}{3} \, d x^{6} + 48 \, d x^{5} + 180 \, d x^{4} + 360 \, d x^{3} + 405 \, d x^{2} + 243 \, d x + \frac {1}{14} \, {\left (64 \, x^{7} + 560 \, x^{6} + 2016 \, x^{5} + 3780 \, x^{4} + 3780 \, x^{3} + 1701 \, x^{2}\right )} e \end {gather*}

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]

16/3*d*x^6 + 48*d*x^5 + 180*d*x^4 + 360*d*x^3 + 405*d*x^2 + 243*d*x + 1/14*(64*x^7 + 560*x^6 + 2016*x^5 + 3780
*x^4 + 3780*x^3 + 1701*x^2)*e

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

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] Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (44) = 88\).
time = 0.94, size = 165, normalized size = 3.30 \begin {gather*} \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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \left (d+e\,x\right )\,{\left (4\,x^2+12\,x+9\right )}^{5/2} \,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)^(5/2),x)

[Out]

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

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