3.285 \(\int \frac{1}{(1+8 x+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=47 \[ \frac{2 (3 x+4)}{169 \sqrt{3 x^2+8 x+1}}-\frac{3 x+4}{39 \left (3 x^2+8 x+1\right )^{3/2}} \]

[Out]

-(4 + 3*x)/(39*(1 + 8*x + 3*x^2)^(3/2)) + (2*(4 + 3*x))/(169*Sqrt[1 + 8*x + 3*x^2])

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Rubi [A]  time = 0.0072238, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {614, 613} \[ \frac{2 (3 x+4)}{169 \sqrt{3 x^2+8 x+1}}-\frac{3 x+4}{39 \left (3 x^2+8 x+1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 8*x + 3*x^2)^(-5/2),x]

[Out]

-(4 + 3*x)/(39*(1 + 8*x + 3*x^2)^(3/2)) + (2*(4 + 3*x))/(169*Sqrt[1 + 8*x + 3*x^2])

Rule 614

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

Rule 613

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]

Rubi steps

\begin{align*} \int \frac{1}{\left (1+8 x+3 x^2\right )^{5/2}} \, dx &=-\frac{4+3 x}{39 \left (1+8 x+3 x^2\right )^{3/2}}-\frac{2}{13} \int \frac{1}{\left (1+8 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{4+3 x}{39 \left (1+8 x+3 x^2\right )^{3/2}}+\frac{2 (4+3 x)}{169 \sqrt{1+8 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0163809, size = 33, normalized size = 0.7 \[ \frac{(3 x+4) \left (18 x^2+48 x-7\right )}{507 \left (3 x^2+8 x+1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 8*x + 3*x^2)^(-5/2),x]

[Out]

((4 + 3*x)*(-7 + 48*x + 18*x^2))/(507*(1 + 8*x + 3*x^2)^(3/2))

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Maple [A]  time = 0.002, size = 30, normalized size = 0.6 \begin{align*}{\frac{54\,{x}^{3}+216\,{x}^{2}+171\,x-28}{507} \left ( 3\,{x}^{2}+8\,x+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^2+8*x+1)^(5/2),x)

[Out]

1/507*(54*x^3+216*x^2+171*x-28)/(3*x^2+8*x+1)^(3/2)

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Maxima [A]  time = 0.961582, size = 80, normalized size = 1.7 \begin{align*} \frac{6 \, x}{169 \, \sqrt{3 \, x^{2} + 8 \, x + 1}} + \frac{8}{169 \, \sqrt{3 \, x^{2} + 8 \, x + 1}} - \frac{x}{13 \,{\left (3 \, x^{2} + 8 \, x + 1\right )}^{\frac{3}{2}}} - \frac{4}{39 \,{\left (3 \, x^{2} + 8 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+8*x+1)^(5/2),x, algorithm="maxima")

[Out]

6/169*x/sqrt(3*x^2 + 8*x + 1) + 8/169/sqrt(3*x^2 + 8*x + 1) - 1/13*x/(3*x^2 + 8*x + 1)^(3/2) - 4/39/(3*x^2 + 8
*x + 1)^(3/2)

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Fricas [A]  time = 1.64053, size = 197, normalized size = 4.19 \begin{align*} -\frac{252 \, x^{4} + 1344 \, x^{3} + 1960 \, x^{2} -{\left (54 \, x^{3} + 216 \, x^{2} + 171 \, x - 28\right )} \sqrt{3 \, x^{2} + 8 \, x + 1} + 448 \, x + 28}{507 \,{\left (9 \, x^{4} + 48 \, x^{3} + 70 \, x^{2} + 16 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+8*x+1)^(5/2),x, algorithm="fricas")

[Out]

-1/507*(252*x^4 + 1344*x^3 + 1960*x^2 - (54*x^3 + 216*x^2 + 171*x - 28)*sqrt(3*x^2 + 8*x + 1) + 448*x + 28)/(9
*x^4 + 48*x^3 + 70*x^2 + 16*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (3 x^{2} + 8 x + 1\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x**2+8*x+1)**(5/2),x)

[Out]

Integral((3*x**2 + 8*x + 1)**(-5/2), x)

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Giac [A]  time = 1.07882, size = 36, normalized size = 0.77 \begin{align*} \frac{9 \,{\left (6 \,{\left (x + 4\right )} x + 19\right )} x - 28}{507 \,{\left (3 \, x^{2} + 8 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+8*x+1)^(5/2),x, algorithm="giac")

[Out]

1/507*(9*(6*(x + 4)*x + 19)*x - 28)/(3*x^2 + 8*x + 1)^(3/2)