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

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

[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])

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Rubi [A]  time = 0.0077619, antiderivative size = 49, normalized size of antiderivative = 1., 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 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]

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]

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 = 0.007732, 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))

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Maple [A]  time = 0.002, size = 25, normalized size = 0.5 \begin{align*}{\frac{x \left ( 32\,{x}^{4}+120\,{x}^{2}+135 \right ) }{405} \left ( 2\,{x}^{2}+3 \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.959111, size = 50, normalized size = 1.02 \begin{align*} \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}}} \end{align*}

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)

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Fricas [A]  time = 1.91383, size = 109, normalized size = 2.22 \begin{align*} \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 )}} \end{align*}

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)

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Sympy [B]  time = 75.9236, size = 139, normalized size = 2.84 \begin{align*} \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}} \end{align*}

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))

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Giac [A]  time = 1.09071, size = 35, normalized size = 0.71 \begin{align*} \frac{{\left (8 \,{\left (4 \, x^{2} + 15\right )} x^{2} + 135\right )} x}{405 \,{\left (2 \, x^{2} + 3\right )}^{\frac{5}{2}}} \end{align*}

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)