\(\int \frac {1}{(4+2 x+x^2)^{7/2}} \, dx\) [284]

Optimal result

Integrand size = 12, antiderivative size = 58 \[ \int \frac {1}{\left (4+2 x+x^2\right )^{7/2}} \, dx=\frac {1+x}{15 \left (4+2 x+x^2\right )^{5/2}}+\frac {4 (1+x)}{135 \left (4+2 x+x^2\right )^{3/2}}+\frac {8 (1+x)}{405 \sqrt {4+2 x+x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {628, 627} \[ \int \frac {1}{\left (4+2 x+x^2\right )^{7/2}} \, dx=\frac {8 (x+1)}{405 \sqrt {x^2+2 x+4}}+\frac {4 (x+1)}{135 \left (x^2+2 x+4\right )^{3/2}}+\frac {x+1}{15 \left (x^2+2 x+4\right )^{5/2}} \]

[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 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 628

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (4+2 x+x^2\right )^{7/2}} \, dx=\frac {(1+x) \left (203+152 x+108 x^2+32 x^3+8 x^4\right )}{405 \left (4+2 x+x^2\right )^{5/2}} \]

[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] (verified)

Time = 0.44 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.66

method	result	size
gosper	\(\frac {8 x^{5}+40 x^{4}+140 x^{3}+260 x^{2}+355 x +203}{405 \left (x^{2}+2 x +4\right )^{\frac {5}{2}}}\)	\(38\)
trager	\(\frac {8 x^{5}+40 x^{4}+140 x^{3}+260 x^{2}+355 x +203}{405 \left (x^{2}+2 x +4\right )^{\frac {5}{2}}}\)	\(38\)
risch	\(\frac {8 x^{5}+40 x^{4}+140 x^{3}+260 x^{2}+355 x +203}{405 \left (x^{2}+2 x +4\right )^{\frac {5}{2}}}\)	\(38\)
default	\(\frac {2 x +2}{30 \left (x^{2}+2 x +4\right )^{\frac {5}{2}}}+\frac {\frac {4}{135}+\frac {4 x}{135}}{\left (x^{2}+2 x +4\right )^{\frac {3}{2}}}+\frac {\frac {8}{405}+\frac {8 x}{405}}{\sqrt {x^{2}+2 x +4}}\)	\(53\)

[In]

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

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (46) = 92\).

Time = 0.24 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.69 \[ \int \frac {1}{\left (4+2 x+x^2\right )^{7/2}} \, dx=\frac {8 \, x^{6} + 48 \, x^{5} + 192 \, x^{4} + 448 \, x^{3} + 768 \, x^{2} + {\left (8 \, x^{5} + 40 \, x^{4} + 140 \, x^{3} + 260 \, x^{2} + 355 \, x + 203\right )} \sqrt {x^{2} + 2 \, x + 4} + 768 \, x + 512}{405 \, {\left (x^{6} + 6 \, x^{5} + 24 \, x^{4} + 56 \, x^{3} + 96 \, x^{2} + 96 \, x + 64\right )}} \]

[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]

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

[In]

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

[Out]

Integral((x**2 + 2*x + 4)**(-7/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.31 \[ \int \frac {1}{\left (4+2 x+x^2\right )^{7/2}} \, dx=\frac {8 \, x}{405 \, \sqrt {x^{2} + 2 \, x + 4}} + \frac {8}{405 \, \sqrt {x^{2} + 2 \, x + 4}} + \frac {4 \, x}{135 \, {\left (x^{2} + 2 \, x + 4\right )}^{\frac {3}{2}}} + \frac {4}{135 \, {\left (x^{2} + 2 \, x + 4\right )}^{\frac {3}{2}}} + \frac {x}{15 \, {\left (x^{2} + 2 \, x + 4\right )}^{\frac {5}{2}}} + \frac {1}{15 \, {\left (x^{2} + 2 \, x + 4\right )}^{\frac {5}{2}}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\left (4+2 x+x^2\right )^{7/2}} \, dx=\frac {{\left (4 \, {\left ({\left (2 \, {\left (x + 5\right )} x + 35\right )} x + 65\right )} x + 355\right )} x + 203}{405 \, {\left (x^{2} + 2 \, x + 4\right )}^{\frac {5}{2}}} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (4+2 x+x^2\right )^{7/2}} \, dx=\frac {51\,x+8\,x\,{\left (x^2+2\,x+4\right )}^2+8\,{\left (x^2+2\,x+4\right )}^2+12\,x^2+12\,x\,\left (x^2+2\,x+4\right )+75}{{\left (x^2+2\,x+4\right )}^{3/2}\,\left (405\,x^2+810\,x+1620\right )} \]

[In]

int(1/(2*x + x^2 + 4)^(7/2),x)

[Out]

(51*x + 8*x*(2*x + x^2 + 4)^2 + 8*(2*x + x^2 + 4)^2 + 12*x^2 + 12*x*(2*x + x^2 + 4) + 75)/((2*x + x^2 + 4)^(3/
2)*(810*x + 405*x^2 + 1620))