3.252 \(\int \frac {1}{(3+2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=49 \[ \frac {8 x}{405 \sqrt {2 x^2+3}}+\frac {4 x}{135 \left (2 x^2+3\right )^{3/2}}+\frac {x}{15 \left (2 x^2+3\right )^{5/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {192, 191} \[ \frac {8 x}{405 \sqrt {2 x^2+3}}+\frac {4 x}{135 \left (2 x^2+3\right )^{3/2}}+\frac {x}{15 \left (2 x^2+3\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 2*x^2)^(-7/2),x]

[Out]

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 28, normalized size = 0.57 \[ \frac {x \left (32 x^4+120 x^2+135\right )}{405 \left (2 x^2+3\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x^2)^(-7/2),x]

[Out]

(x*(135 + 120*x^2 + 32*x^4))/(405*(3 + 2*x^2)^(5/2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.04, size = 28, normalized size = 0.57 \[ \frac {x \left (32 x^4+120 x^2+135\right )}{405 \left (2 x^2+3\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(3 + 2*x^2)^(-7/2),x]

[Out]

(x*(135 + 120*x^2 + 32*x^4))/(405*(3 + 2*x^2)^(5/2))

________________________________________________________________________________________

fricas [A]  time = 0.58, size = 44, normalized size = 0.90 \[ \frac {{\left (32 \, x^{5} + 120 \, x^{3} + 135 \, x\right )} \sqrt {2 \, x^{2} + 3}}{405 \, {\left (8 \, x^{6} + 36 \, x^{4} + 54 \, x^{2} + 27\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^2+3)^(7/2),x, algorithm="fricas")

[Out]

1/405*(32*x^5 + 120*x^3 + 135*x)*sqrt(2*x^2 + 3)/(8*x^6 + 36*x^4 + 54*x^2 + 27)

________________________________________________________________________________________

giac [A]  time = 0.64, size = 26, normalized size = 0.53 \[ \frac {{\left (8 \, {\left (4 \, x^{2} + 15\right )} x^{2} + 135\right )} x}{405 \, {\left (2 \, x^{2} + 3\right )}^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^2+3)^(7/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.28, size = 25, normalized size = 0.51

method	result	size
gosper	\(\frac {x \left (32 x^{4}+120 x^{2}+135\right )}{405 \left (2 x^{2}+3\right )^{\frac {5}{2}}}\)	\(25\)
trager	\(\frac {x \left (32 x^{4}+120 x^{2}+135\right )}{405 \left (2 x^{2}+3\right )^{\frac {5}{2}}}\)	\(25\)
risch	\(\frac {x \left (32 x^{4}+120 x^{2}+135\right )}{405 \left (2 x^{2}+3\right )^{\frac {5}{2}}}\)	\(25\)
meijerg	\(\frac {\sqrt {3}\, x \left (\frac {32}{9} x^{4}+\frac {40}{3} x^{2}+15\right )}{1215 \left (1+\frac {2 x^{2}}{3}\right )^{\frac {5}{2}}}\)	\(28\)
default	\(\frac {x}{15 \left (2 x^{2}+3\right )^{\frac {5}{2}}}+\frac {4 x}{135 \left (2 x^{2}+3\right )^{\frac {3}{2}}}+\frac {8 x}{405 \sqrt {2 x^{2}+3}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x^2+3)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/405*x*(32*x^4+120*x^2+135)/(2*x^2+3)^(5/2)

________________________________________________________________________________________

maxima [A]  time = 0.54, size = 37, normalized size = 0.76 \[ \frac {8 \, x}{405 \, \sqrt {2 \, x^{2} + 3}} + \frac {4 \, x}{135 \, {\left (2 \, x^{2} + 3\right )}^{\frac {3}{2}}} + \frac {x}{15 \, {\left (2 \, x^{2} + 3\right )}^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^2+3)^(7/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 187, normalized size = 3.82 \[ \frac {2\,\sqrt {2}\,\sqrt {x^2+\frac {3}{2}}}{405\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{2}\right )}+\frac {2\,\sqrt {2}\,\sqrt {x^2+\frac {3}{2}}}{405\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{2}\right )}+\frac {\sqrt {2}\,\sqrt {x^2+\frac {3}{2}}}{1440\,\left (-x^3+\frac {3{}\mathrm {i}\,\sqrt {6}\,x^2}{2}+\frac {9\,x}{2}-\frac {\sqrt {6}\,3{}\mathrm {i}}{4}\right )}+\frac {\sqrt {2}\,\sqrt {x^2+\frac {3}{2}}}{1440\,\left (-x^3-\frac {3{}\mathrm {i}\,\sqrt {6}\,x^2}{2}+\frac {9\,x}{2}+\frac {\sqrt {6}\,3{}\mathrm {i}}{4}\right )}+\frac {\sqrt {2}\,\sqrt {6}\,\sqrt {x^2+\frac {3}{2}}\,19{}\mathrm {i}}{25920\,\left (x^2+1{}\mathrm {i}\,\sqrt {6}\,x-\frac {3}{2}\right )}+\frac {\sqrt {2}\,\sqrt {6}\,\sqrt {x^2+\frac {3}{2}}\,19{}\mathrm {i}}{25920\,\left (-x^2+1{}\mathrm {i}\,\sqrt {6}\,x+\frac {3}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x^2 + 3)^(7/2),x)

[Out]

(2*2^(1/2)*(x^2 + 3/2)^(1/2))/(405*(x - (6^(1/2)*1i)/2)) + (2*2^(1/2)*(x^2 + 3/2)^(1/2))/(405*(x + (6^(1/2)*1i
)/2)) + (2^(1/2)*(x^2 + 3/2)^(1/2))/(1440*((9*x)/2 - (6^(1/2)*3i)/4 + (6^(1/2)*x^2*3i)/2 - x^3)) + (2^(1/2)*(x
^2 + 3/2)^(1/2))/(1440*((9*x)/2 + (6^(1/2)*3i)/4 - (6^(1/2)*x^2*3i)/2 - x^3)) + (2^(1/2)*6^(1/2)*(x^2 + 3/2)^(
1/2)*19i)/(25920*(6^(1/2)*x*1i + x^2 - 3/2)) + (2^(1/2)*6^(1/2)*(x^2 + 3/2)^(1/2)*19i)/(25920*(6^(1/2)*x*1i -
x^2 + 3/2))

________________________________________________________________________________________

sympy [B]  time = 9.96, size = 139, normalized size = 2.84 \[ \frac {32 x^{5}}{1620 x^{4} \sqrt {2 x^{2} + 3} + 4860 x^{2} \sqrt {2 x^{2} + 3} + 3645 \sqrt {2 x^{2} + 3}} + \frac {120 x^{3}}{1620 x^{4} \sqrt {2 x^{2} + 3} + 4860 x^{2} \sqrt {2 x^{2} + 3} + 3645 \sqrt {2 x^{2} + 3}} + \frac {135 x}{1620 x^{4} \sqrt {2 x^{2} + 3} + 4860 x^{2} \sqrt {2 x^{2} + 3} + 3645 \sqrt {2 x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x**2+3)**(7/2),x)

[Out]

32*x**5/(1620*x**4*sqrt(2*x**2 + 3) + 4860*x**2*sqrt(2*x**2 + 3) + 3645*sqrt(2*x**2 + 3)) + 120*x**3/(1620*x**
4*sqrt(2*x**2 + 3) + 4860*x**2*sqrt(2*x**2 + 3) + 3645*sqrt(2*x**2 + 3)) + 135*x/(1620*x**4*sqrt(2*x**2 + 3) +
4860*x**2*sqrt(2*x**2 + 3) + 3645*sqrt(2*x**2 + 3))

________________________________________________________________________________________