\(\int \frac {d+e x}{(9+12 x+4 x^2)^{7/2}} \, dx\) [1620]

Optimal result

Integrand size = 20, antiderivative size = 52 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx=-\frac {e}{20 \left (9+12 x+4 x^2\right )^{5/2}}-\frac {2 d-3 e}{24 (3+2 x) \left (9+12 x+4 x^2\right )^{5/2}} \]

[Out]

-1/20*e/(4*x^2+12*x+9)^(5/2)+1/24*(-2*d+3*e)/(3+2*x)/(4*x^2+12*x+9)^(5/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {654, 621} \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx=-\frac {2 d-3 e}{24 (2 x+3) \left (4 x^2+12 x+9\right )^{5/2}}-\frac {e}{20 \left (4 x^2+12 x+9\right )^{5/2}} \]

[In]

Int[(d + e*x)/(9 + 12*x + 4*x^2)^(7/2),x]

[Out]

-1/20*e/(9 + 12*x + 4*x^2)^(5/2) - (2*d - 3*e)/(24*(3 + 2*x)*(9 + 12*x + 4*x^2)^(5/2))

Rule 621

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[2*((a + b*x + c*x^2)^(p + 1)/((2*p + 1)*(b + 2*
c*x))), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -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*} \text {integral}& = -\frac {e}{20 \left (9+12 x+4 x^2\right )^{5/2}}+\frac {1}{2} (2 d-3 e) \int \frac {1}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx \\ & = -\frac {e}{20 \left (9+12 x+4 x^2\right )^{5/2}}-\frac {2 d-3 e}{24 (3+2 x) \left (9+12 x+4 x^2\right )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.65 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx=\frac {-10 d-3 (e+4 e x)}{120 (3+2 x)^5 \sqrt {(3+2 x)^2}} \]

[In]

Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^(7/2),x]

[Out]

(-10*d - 3*(e + 4*e*x))/(120*(3 + 2*x)^5*Sqrt[(3 + 2*x)^2])

Maple [A] (verified)

Time = 2.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.54

method	result	size
gosper	\(-\frac {\left (2 x +3\right ) \left (12 e x +10 d +3 e \right )}{120 \left (\left (2 x +3\right )^{2}\right )^{\frac {7}{2}}}\)	\(28\)
default	\(-\frac {\left (2 x +3\right ) \left (12 e x +10 d +3 e \right )}{120 \left (\left (2 x +3\right )^{2}\right )^{\frac {7}{2}}}\)	\(28\)
risch	\(\frac {64 \sqrt {\left (2 x +3\right )^{2}}\, \left (-\frac {1}{640} e x -\frac {1}{768} d -\frac {1}{2560} e \right )}{\left (2 x +3\right )^{7}}\)	\(30\)
meijerg	\(\frac {e \,x^{2} \left (\frac {16}{81} x^{4}+\frac {16}{9} x^{3}+\frac {20}{3} x^{2}+\frac {40}{3} x +15\right )}{65610 \left (1+\frac {2 x}{3}\right )^{6}}+\frac {d x \left (\frac {32}{243} x^{5}+\frac {32}{27} x^{4}+\frac {40}{9} x^{3}+\frac {80}{9} x^{2}+10 x +6\right )}{13122 \left (1+\frac {2 x}{3}\right )^{6}}\)	\(71\)

[In]

int((e*x+d)/(4*x^2+12*x+9)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/120*(2*x+3)*(12*e*x+10*d+3*e)/((2*x+3)^2)^(7/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.87 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx=-\frac {12 \, e x + 10 \, d + 3 \, e}{120 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]

[In]

integrate((e*x+d)/(4*x^2+12*x+9)^(7/2),x, algorithm="fricas")

[Out]

-1/120*(12*e*x + 10*d + 3*e)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)

Sympy [F]

\[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx=\int \frac {d + e x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate((e*x+d)/(4*x**2+12*x+9)**(7/2),x)

[Out]

Integral((d + e*x)/((2*x + 3)**2)**(7/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.69 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx=-\frac {e}{20 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}}} - \frac {d}{12 \, {\left (2 \, x + 3\right )}^{6}} + \frac {e}{8 \, {\left (2 \, x + 3\right )}^{6}} \]

[In]

integrate((e*x+d)/(4*x^2+12*x+9)^(7/2),x, algorithm="maxima")

[Out]

-1/20*e/(4*x^2 + 12*x + 9)^(5/2) - 1/12*d/(2*x + 3)^6 + 1/8*e/(2*x + 3)^6

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.54 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx=-\frac {12 \, e x + 10 \, d + 3 \, e}{120 \, {\left (2 \, x + 3\right )}^{6} \mathrm {sgn}\left (2 \, x + 3\right )} \]

[In]

integrate((e*x+d)/(4*x^2+12*x+9)^(7/2),x, algorithm="giac")

[Out]

-1/120*(12*e*x + 10*d + 3*e)/((2*x + 3)^6*sgn(2*x + 3))

Mupad [B] (verification not implemented)

Time = 9.54 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.62 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx=-\frac {\left (10\,d+3\,e+12\,e\,x\right )\,\sqrt {4\,x^2+12\,x+9}}{120\,{\left (2\,x+3\right )}^7} \]

[In]

int((d + e*x)/(12*x + 4*x^2 + 9)^(7/2),x)

[Out]

-((10*d + 3*e + 12*e*x)*(12*x + 4*x^2 + 9)^(1/2))/(120*(2*x + 3)^7)