\(\int \frac {5-x}{(2+5 x+3 x^2)^{5/2}} \, dx\) [2520]

Optimal result

Integrand size = 20, antiderivative size = 47 \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {280 (5+6 x)}{3 \sqrt {2+5 x+3 x^2}} \]

[Out]

-2/3*(29+35*x)/(3*x^2+5*x+2)^(3/2)+280/3*(5+6*x)/(3*x^2+5*x+2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, 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 = {652, 627} \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {280 (6 x+5)}{3 \sqrt {3 x^2+5 x+2}}-\frac {2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]

[In]

Int[(5 - x)/(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(-2*(29 + 35*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (280*(5 + 6*x))/(3*Sqrt[2 + 5*x + 3*x^2])

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 652

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), x] - Dist[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 -
4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {140}{3} \int \frac {1}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx \\ & = -\frac {2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {280 (5+6 x)}{3 \sqrt {2+5 x+3 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {2+5 x+3 x^2} \left (457+1715 x+2100 x^2+840 x^3\right )}{(1+x)^2 (2+3 x)^2} \]

[In]

Integrate[(5 - x)/(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(2*Sqrt[2 + 5*x + 3*x^2]*(457 + 1715*x + 2100*x^2 + 840*x^3))/((1 + x)^2*(2 + 3*x)^2)

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.64

method	result	size
trager	\(\frac {1680 x^{3}+4200 x^{2}+3430 x +914}{\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}\)	\(30\)
risch	\(\frac {1680 x^{3}+4200 x^{2}+3430 x +914}{\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}\)	\(30\)
gosper	\(\frac {2 \left (840 x^{3}+2100 x^{2}+1715 x +457\right ) \left (1+x \right ) \left (2+3 x \right )}{\left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}\)	\(38\)
default	\(-\frac {35 \left (5+6 x \right )}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {\frac {1400}{3}+560 x}{\sqrt {3 x^{2}+5 x +2}}+\frac {1}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}\)	\(54\)

[In]

int((5-x)/(3*x^2+5*x+2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2*(840*x^3+2100*x^2+1715*x+457)/(3*x^2+5*x+2)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.09 \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (840 \, x^{3} + 2100 \, x^{2} + 1715 \, x + 457\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4} \]

[In]

integrate((5-x)/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")

[Out]

2*(840*x^3 + 2100*x^2 + 1715*x + 457)*sqrt(3*x^2 + 5*x + 2)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)

Sympy [F]

\[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \frac {x}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]

[In]

integrate((5-x)/(3*x**2+5*x+2)**(5/2),x)

[Out]

-Integral(x/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) +
20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30
*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 +
5*x + 2)), x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.26 \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {560 \, x}{\sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {1400}{3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {70 \, x}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {58}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]

[In]

integrate((5-x)/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")

[Out]

560*x/sqrt(3*x^2 + 5*x + 2) + 1400/3/sqrt(3*x^2 + 5*x + 2) - 70/3*x/(3*x^2 + 5*x + 2)^(3/2) - 58/3/(3*x^2 + 5*
x + 2)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.62 \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (35 \, {\left (12 \, {\left (2 \, x + 5\right )} x + 49\right )} x + 457\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]

[In]

integrate((5-x)/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")

[Out]

2*(35*(12*(2*x + 5)*x + 49)*x + 457)/(3*x^2 + 5*x + 2)^(3/2)

Mupad [B] (verification not implemented)

Time = 11.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77 \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2310\,x+560\,x\,\left (3\,x^2+5\,x+2\right )+1400\,x^2+914}{{\left (3\,x^2+5\,x+2\right )}^{3/2}} \]

[In]

int(-(x - 5)/(5*x + 3*x^2 + 2)^(5/2),x)

[Out]

(2310*x + 560*x*(5*x + 3*x^2 + 2) + 1400*x^2 + 914)/(5*x + 3*x^2 + 2)^(3/2)