3.1423 \(\int \frac {(5-x) (3+2 x)}{(2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=37 \[ \frac {43 x}{18 \sqrt {3 x^2+2}}-\frac {7 (2-7 x)}{18 \left (3 x^2+2\right )^{3/2}} \]

[Out]

-7/18*(2-7*x)/(3*x^2+2)^(3/2)+43/18*x/(3*x^2+2)^(1/2)

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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {778, 191} \[ \frac {43 x}{18 \sqrt {3 x^2+2}}-\frac {7 (2-7 x)}{18 \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 - 7*x))/(18*(2 + 3*x^2)^(3/2)) + (43*x)/(18*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 778

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

Rubi steps

\begin {align*} \int \frac {(5-x) (3+2 x)}{\left (2+3 x^2\right )^{5/2}} \, dx &=-\frac {7 (2-7 x)}{18 \left (2+3 x^2\right )^{3/2}}+\frac {43}{9} \int \frac {1}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {7 (2-7 x)}{18 \left (2+3 x^2\right )^{3/2}}+\frac {43 x}{18 \sqrt {2+3 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 0.68 \[ -\frac {-129 x^3-135 x+14}{18 \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/18*(14 - 135*x - 129*x^3)/(2 + 3*x^2)^(3/2)

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fricas [A]  time = 0.73, size = 35, normalized size = 0.95 \[ \frac {{\left (129 \, x^{3} + 135 \, x - 14\right )} \sqrt {3 \, x^{2} + 2}}{18 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(129*x^3 + 135*x - 14)*sqrt(3*x^2 + 2)/(9*x^4 + 12*x^2 + 4)

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giac [A]  time = 0.18, size = 23, normalized size = 0.62 \[ \frac {3 \, {\left (43 \, x^{2} + 45\right )} x - 14}{18 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(3*(43*x^2 + 45)*x - 14)/(3*x^2 + 2)^(3/2)

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maple [A]  time = 0.04, size = 22, normalized size = 0.59 \[ \frac {129 x^{3}+135 x -14}{18 \left (3 x^{2}+2\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(129*x^3+135*x-14)/(3*x^2+2)^(3/2)

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maxima [A]  time = 0.50, size = 36, normalized size = 0.97 \[ \frac {43 \, x}{18 \, \sqrt {3 \, x^{2} + 2}} + \frac {49 \, x}{18 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {7}{9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

43/18*x/sqrt(3*x^2 + 2) + 49/18*x/(3*x^2 + 2)^(3/2) - 7/9/(3*x^2 + 2)^(3/2)

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mupad [B]  time = 0.04, size = 161, normalized size = 4.35 \[ \frac {41\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{144\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {41\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{144\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {49}{16}+\frac {\sqrt {6}\,7{}\mathrm {i}}{16}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {49}{24}+\frac {\sqrt {6}\,7{}\mathrm {i}}{24}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {49}{16}+\frac {\sqrt {6}\,7{}\mathrm {i}}{16}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {49}{24}+\frac {\sqrt {6}\,7{}\mathrm {i}}{24}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(41*3^(1/2)*(x^2 + 2/3)^(1/2))/(144*(x - (6^(1/2)*1i)/3)) + (41*3^(1/2)*(x^2 + 2/3)^(1/2))/(144*(x + (6^(1/2)*
1i)/3)) - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*7i)/16 - 49/16)/(x + (6^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*7i)/
24 - 49/24)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 + (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*7i)/16 + 49/16)/(x - (
6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2)*7i)/24 + 49/24)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27

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sympy [B]  time = 58.40, size = 122, normalized size = 3.30 \[ - \frac {2 x^{3}}{18 x^{2} \sqrt {3 x^{2} + 2} + 12 \sqrt {3 x^{2} + 2}} + \frac {15 x^{3}}{6 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}} + \frac {15 x}{6 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}} - \frac {7}{27 x^{2} \sqrt {3 x^{2} + 2} + 18 \sqrt {3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x**3/(18*x**2*sqrt(3*x**2 + 2) + 12*sqrt(3*x**2 + 2)) + 15*x**3/(6*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 +
2)) + 15*x/(6*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)) - 7/(27*x**2*sqrt(3*x**2 + 2) + 18*sqrt(3*x**2 + 2))

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