3.13.50 \(\int \frac {5-x}{(2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=37 \[ \frac {5 x}{6 \sqrt {3 x^2+2}}+\frac {15 x+2}{18 \left (3 x^2+2\right )^{3/2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {639, 191} \begin {gather*} \frac {5 x}{6 \sqrt {3 x^2+2}}+\frac {15 x+2}{18 \left (3 x^2+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - x)/(2 + 3*x^2)^(5/2),x]

[Out]

(2 + 15*x)/(18*(2 + 3*x^2)^(3/2)) + (5*x)/(6*Sqrt[2 + 3*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 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rubi steps

\begin {align*} \int \frac {5-x}{\left (2+3 x^2\right )^{5/2}} \, dx &=\frac {2+15 x}{18 \left (2+3 x^2\right )^{3/2}}+\frac {5}{3} \int \frac {1}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac {2+15 x}{18 \left (2+3 x^2\right )^{3/2}}+\frac {5 x}{6 \sqrt {2+3 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.68 \begin {gather*} \frac {45 x^3+45 x+2}{18 \left (3 x^2+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/(2 + 3*x^2)^(5/2),x]

[Out]

(2 + 45*x + 45*x^3)/(18*(2 + 3*x^2)^(3/2))

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IntegrateAlgebraic [A]  time = 0.28, size = 25, normalized size = 0.68 \begin {gather*} \frac {45 x^3+45 x+2}{18 \left (3 x^2+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(5 - x)/(2 + 3*x^2)^(5/2),x]

[Out]

(2 + 45*x + 45*x^3)/(18*(2 + 3*x^2)^(3/2))

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fricas [A]  time = 0.43, size = 35, normalized size = 0.95 \begin {gather*} \frac {{\left (45 \, x^{3} + 45 \, x + 2\right )} \sqrt {3 \, x^{2} + 2}}{18 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

1/18*(45*x^3 + 45*x + 2)*sqrt(3*x^2 + 2)/(9*x^4 + 12*x^2 + 4)

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giac [A]  time = 0.18, size = 21, normalized size = 0.57 \begin {gather*} \frac {45 \, {\left (x^{2} + 1\right )} x + 2}{18 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

1/18*(45*(x^2 + 1)*x + 2)/(3*x^2 + 2)^(3/2)

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maple [A]  time = 0.04, size = 22, normalized size = 0.59 \begin {gather*} \frac {45 x^{3}+45 x +2}{18 \left (3 x^{2}+2\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(3*x^2+2)^(5/2),x)

[Out]

1/18*(45*x^3+45*x+2)/(3*x^2+2)^(3/2)

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maxima [A]  time = 0.45, size = 36, normalized size = 0.97 \begin {gather*} \frac {5 \, x}{6 \, \sqrt {3 \, x^{2} + 2}} + \frac {5 \, x}{6 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {1}{9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

5/6*x/sqrt(3*x^2 + 2) + 5/6*x/(3*x^2 + 2)^(3/2) + 1/9/(3*x^2 + 2)^(3/2)

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mupad [B]  time = 1.69, size = 161, normalized size = 4.35 \begin {gather*} \frac {5\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{48\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {5\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{48\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {15}{16}+\frac {\sqrt {6}\,1{}\mathrm {i}}{16}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (-\frac {5}{8}+\frac {\sqrt {6}\,1{}\mathrm {i}}{24}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {15}{16}+\frac {\sqrt {6}\,1{}\mathrm {i}}{16}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (\frac {5}{8}+\frac {\sqrt {6}\,1{}\mathrm {i}}{24}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 5)/(3*x^2 + 2)^(5/2),x)

[Out]

(5*3^(1/2)*(x^2 + 2/3)^(1/2))/(48*(x - (6^(1/2)*1i)/3)) + (5*3^(1/2)*(x^2 + 2/3)^(1/2))/(48*(x + (6^(1/2)*1i)/
3)) - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*1i)/16 - 15/16)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2)*1i)/24 -
5/8)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 + (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*1i)/16 + 15/16)/(x + (6^(1/2
)*1i)/3) + (6^(1/2)*((6^(1/2)*1i)/24 + 5/8)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27

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sympy [B]  time = 43.08, size = 90, normalized size = 2.43 \begin {gather*} \frac {5 x^{3}}{6 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}} + \frac {5 x}{6 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}} + \frac {1}{27 x^{2} \sqrt {3 x^{2} + 2} + 18 \sqrt {3 x^{2} + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3*x**2+2)**(5/2),x)

[Out]

5*x**3/(6*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)) + 5*x/(6*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)) + 1
/(27*x**2*sqrt(3*x**2 + 2) + 18*sqrt(3*x**2 + 2))

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