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

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 = 0.0239542, 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 \[ \frac{8 (x+1)}{405 \sqrt{x^2+2 x+4}}+\frac{4 (x+1)}{135 \left (x^2+2 x+4\right )^{3/2}}+\frac{x+1}{15 \left (x^2+2 x+4\right )^{5/2}} \]

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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.0255, size = 54, normalized size = 0.93 \[ \frac{2 \left (2 x + 2\right )}{135 \left (x^{2} + 2 x + 4\right )^{\frac{3}{2}}} + \frac{2 x + 2}{30 \left (x^{2} + 2 x + 4\right )^{\frac{5}{2}}} + \frac{2 \left (4 x + 4\right )}{405 \sqrt{x^{2} + 2 x + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2+2*x+4)**(7/2),x)

[Out]

2*(2*x + 2)/(135*(x**2 + 2*x + 4)**(3/2)) + (2*x + 2)/(30*(x**2 + 2*x + 4)**(5/2
)) + 2*(4*x + 4)/(405*sqrt(x**2 + 2*x + 4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.021684, 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 = 0.005, size = 38, normalized size = 0.7 \[{\frac{8\,{x}^{5}+40\,{x}^{4}+140\,{x}^{3}+260\,{x}^{2}+355\,x+203}{405} \left ({x}^{2}+2\,x+4 \right ) ^{-{\frac{5}{2}}}} \]

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 = 1.36689, size = 103, normalized size = 1.78 \[ \frac{8 \, x}{405 \, \sqrt{x^{2} + 2 \, x + 4}} + \frac{8}{405 \, \sqrt{x^{2} + 2 \, x + 4}} + \frac{4 \, x}{135 \,{\left (x^{2} + 2 \, x + 4\right )}^{\frac{3}{2}}} + \frac{4}{135 \,{\left (x^{2} + 2 \, x + 4\right )}^{\frac{3}{2}}} + \frac{x}{15 \,{\left (x^{2} + 2 \, x + 4\right )}^{\frac{5}{2}}} + \frac{1}{15 \,{\left (x^{2} + 2 \, x + 4\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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 [A]  time = 0.213773, size = 213, normalized size = 3.67 \[ \frac{40 \, x^{4} + 160 \, x^{3} + 375 \, x^{2} - 5 \,{\left (8 \, x^{3} + 24 \, x^{2} + 39 \, x + 23\right )} \sqrt{x^{2} + 2 \, x + 4} + 430 \, x + 247}{15 \,{\left (16 \, x^{10} + 160 \, x^{9} + 900 \, x^{8} + 3360 \, x^{7} + 9165 \, x^{6} + 18702 \, x^{5} + 28920 \, x^{4} + 33240 \, x^{3} + 27360 \, x^{2} -{\left (16 \, x^{9} + 144 \, x^{8} + 732 \, x^{7} + 2436 \, x^{6} + 5841 \, x^{5} + 10221 \, x^{4} + 13104 \, x^{3} + 11772 \, x^{2} + 6816 \, x + 1936\right )} \sqrt{x^{2} + 2 \, x + 4} + 14560 \, x + 3904\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 4)^(-7/2),x, algorithm="fricas")

[Out]

1/15*(40*x^4 + 160*x^3 + 375*x^2 - 5*(8*x^3 + 24*x^2 + 39*x + 23)*sqrt(x^2 + 2*x
+ 4) + 430*x + 247)/(16*x^10 + 160*x^9 + 900*x^8 + 3360*x^7 + 9165*x^6 + 18702*
x^5 + 28920*x^4 + 33240*x^3 + 27360*x^2 - (16*x^9 + 144*x^8 + 732*x^7 + 2436*x^6
+ 5841*x^5 + 10221*x^4 + 13104*x^3 + 11772*x^2 + 6816*x + 1936)*sqrt(x^2 + 2*x
+ 4) + 14560*x + 3904)

_______________________________________________________________________________________

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

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/XCAS [A]  time = 0.202849, size = 45, normalized size = 0.78 \[ \frac{{\left (4 \,{\left ({\left (2 \,{\left (x + 5\right )} x + 35\right )} x + 65\right )} x + 355\right )} x + 203}{405 \,{\left (x^{2} + 2 \, x + 4\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)