3.284 \(\int \frac{1}{(4+2 x+x^2)^{7/2}} \, dx\)

Optimal. Leaf size=58 \[ \frac{8 (x+1)}{405 \sqrt{x^2+2 x+4}}+\frac{4 (x+1)}{135 \left (x^2+2 x+4\right )^{3/2}}+\frac{x+1}{15 \left (x^2+2 x+4\right )^{5/2}} \]

[Out]

(1 + x)/(15*(4 + 2*x + x^2)^(5/2)) + (4*(1 + x))/(135*(4 + 2*x + x^2)^(3/2)) + (8*(1 + x))/(405*Sqrt[4 + 2*x +
x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0107579, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {614, 613} \[ \frac{8 (x+1)}{405 \sqrt{x^2+2 x+4}}+\frac{4 (x+1)}{135 \left (x^2+2 x+4\right )^{3/2}}+\frac{x+1}{15 \left (x^2+2 x+4\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(4 + 2*x + x^2)^(-7/2),x]

[Out]

(1 + x)/(15*(4 + 2*x + x^2)^(5/2)) + (4*(1 + x))/(135*(4 + 2*x + x^2)^(3/2)) + (8*(1 + x))/(405*Sqrt[4 + 2*x +
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 (4+2 x+x^2\right )^{7/2}} \, dx &=\frac{1+x}{15 \left (4+2 x+x^2\right )^{5/2}}+\frac{4}{15} \int \frac{1}{\left (4+2 x+x^2\right )^{5/2}} \, dx\\ &=\frac{1+x}{15 \left (4+2 x+x^2\right )^{5/2}}+\frac{4 (1+x)}{135 \left (4+2 x+x^2\right )^{3/2}}+\frac{8}{135} \int \frac{1}{\left (4+2 x+x^2\right )^{3/2}} \, dx\\ &=\frac{1+x}{15 \left (4+2 x+x^2\right )^{5/2}}+\frac{4 (1+x)}{135 \left (4+2 x+x^2\right )^{3/2}}+\frac{8 (1+x)}{405 \sqrt{4+2 x+x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0111127, size = 39, normalized size = 0.67 \[ \frac{(x+1) \left (8 x^4+32 x^3+108 x^2+152 x+203\right )}{405 \left (x^2+2 x+4\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(4 + 2*x + x^2)^(-7/2),x]

[Out]

((1 + x)*(203 + 152*x + 108*x^2 + 32*x^3 + 8*x^4))/(405*(4 + 2*x + x^2)^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 38, normalized size = 0.7 \begin{align*}{\frac{8\,{x}^{5}+40\,{x}^{4}+140\,{x}^{3}+260\,{x}^{2}+355\,x+203}{405} \left ({x}^{2}+2\,x+4 \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+2*x+4)^(7/2),x)

[Out]

1/405*(8*x^5+40*x^4+140*x^3+260*x^2+355*x+203)/(x^2+2*x+4)^(5/2)

________________________________________________________________________________________

Maxima [A]  time = 0.963592, size = 103, normalized size = 1.78 \begin{align*} \frac{8 \, x}{405 \, \sqrt{x^{2} + 2 \, x + 4}} + \frac{8}{405 \, \sqrt{x^{2} + 2 \, x + 4}} + \frac{4 \, x}{135 \,{\left (x^{2} + 2 \, x + 4\right )}^{\frac{3}{2}}} + \frac{4}{135 \,{\left (x^{2} + 2 \, x + 4\right )}^{\frac{3}{2}}} + \frac{x}{15 \,{\left (x^{2} + 2 \, x + 4\right )}^{\frac{5}{2}}} + \frac{1}{15 \,{\left (x^{2} + 2 \, x + 4\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+2*x+4)^(7/2),x, algorithm="maxima")

[Out]

8/405*x/sqrt(x^2 + 2*x + 4) + 8/405/sqrt(x^2 + 2*x + 4) + 4/135*x/(x^2 + 2*x + 4)^(3/2) + 4/135/(x^2 + 2*x + 4
)^(3/2) + 1/15*x/(x^2 + 2*x + 4)^(5/2) + 1/15/(x^2 + 2*x + 4)^(5/2)

________________________________________________________________________________________

Fricas [B]  time = 1.78644, size = 262, normalized size = 4.52 \begin{align*} \frac{8 \, x^{6} + 48 \, x^{5} + 192 \, x^{4} + 448 \, x^{3} + 768 \, x^{2} +{\left (8 \, x^{5} + 40 \, x^{4} + 140 \, x^{3} + 260 \, x^{2} + 355 \, x + 203\right )} \sqrt{x^{2} + 2 \, x + 4} + 768 \, x + 512}{405 \,{\left (x^{6} + 6 \, x^{5} + 24 \, x^{4} + 56 \, x^{3} + 96 \, x^{2} + 96 \, x + 64\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+2*x+4)^(7/2),x, algorithm="fricas")

[Out]

1/405*(8*x^6 + 48*x^5 + 192*x^4 + 448*x^3 + 768*x^2 + (8*x^5 + 40*x^4 + 140*x^3 + 260*x^2 + 355*x + 203)*sqrt(
x^2 + 2*x + 4) + 768*x + 512)/(x^6 + 6*x^5 + 24*x^4 + 56*x^3 + 96*x^2 + 96*x + 64)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x^{2} + 2 x + 4\right )^{\frac{7}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+2*x+4)**(7/2),x)

[Out]

Integral((x**2 + 2*x + 4)**(-7/2), x)

________________________________________________________________________________________

Giac [A]  time = 1.09836, size = 45, normalized size = 0.78 \begin{align*} \frac{{\left (4 \,{\left ({\left (2 \,{\left (x + 5\right )} x + 35\right )} x + 65\right )} x + 355\right )} x + 203}{405 \,{\left (x^{2} + 2 \, x + 4\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+2*x+4)^(7/2),x, algorithm="giac")

[Out]

1/405*((4*((2*(x + 5)*x + 35)*x + 65)*x + 355)*x + 203)/(x^2 + 2*x + 4)^(5/2)