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

Optimal. Leaf size=48 \[ \frac {(2 x+3)^2 (15 x+2)}{18 \left (3 x^2+2\right )^{3/2}}-\frac {41 (4-9 x)}{54 \sqrt {3 x^2+2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {805, 637} \begin {gather*} \frac {(2 x+3)^2 (15 x+2)}{18 \left (3 x^2+2\right )^{3/2}}-\frac {41 (4-9 x)}{54 \sqrt {3 x^2+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3 + 2*x)^2*(2 + 15*x))/(18*(2 + 3*x^2)^(3/2)) - (41*(4 - 9*x))/(54*Sqrt[2 + 3*x^2])

Rule 637

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

Rule 805

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*
(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] - Dist[(m*(c*d*f + a*e*g))/(2*a*c*(p + 1)), Int[(d + e*
x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplif
y[m + 2*p + 3], 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {(5-x) (3+2 x)^2}{\left (2+3 x^2\right )^{5/2}} \, dx &=\frac {(3+2 x)^2 (2+15 x)}{18 \left (2+3 x^2\right )^{3/2}}+\frac {41}{9} \int \frac {3+2 x}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac {(3+2 x)^2 (2+15 x)}{18 \left (2+3 x^2\right )^{3/2}}-\frac {41 (4-9 x)}{54 \sqrt {2+3 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 30, normalized size = 0.62 \begin {gather*} -\frac {-1287 x^3-72 x^2-1215 x+274}{54 \left (3 x^2+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/54*(274 - 1215*x - 72*x^2 - 1287*x^3)/(2 + 3*x^2)^(3/2)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.38, size = 30, normalized size = 0.62 \begin {gather*} \frac {1287 x^3+72 x^2+1215 x-274}{54 \left (3 x^2+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-274 + 1215*x + 72*x^2 + 1287*x^3)/(54*(2 + 3*x^2)^(3/2))

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 40, normalized size = 0.83 \begin {gather*} \frac {{\left (1287 \, x^{3} + 72 \, x^{2} + 1215 \, x - 274\right )} \sqrt {3 \, x^{2} + 2}}{54 \, {\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+2*x)^2/(3*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

1/54*(1287*x^3 + 72*x^2 + 1215*x - 274)*sqrt(3*x^2 + 2)/(9*x^4 + 12*x^2 + 4)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 25, normalized size = 0.52 \begin {gather*} \frac {9 \, {\left ({\left (143 \, x + 8\right )} x + 135\right )} x - 274}{54 \, {\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+2*x)^2/(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

1/54*(9*((143*x + 8)*x + 135)*x - 274)/(3*x^2 + 2)^(3/2)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 27, normalized size = 0.56 \begin {gather*} \frac {1287 x^{3}+72 x^{2}+1215 x -274}{54 \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)*(2*x+3)^2/(3*x^2+2)^(5/2),x)

[Out]

1/54*(1287*x^3+72*x^2+1215*x-274)/(3*x^2+2)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 0.53, size = 50, normalized size = 1.04 \begin {gather*} \frac {143 \, x}{18 \, \sqrt {3 \, x^{2} + 2}} + \frac {4 \, x^{2}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {119 \, x}{18 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {137}{27 \, {\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+2*x)^2/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

143/18*x/sqrt(3*x^2 + 2) + 4/3*x^2/(3*x^2 + 2)^(3/2) + 119/18*x/(3*x^2 + 2)^(3/2) - 137/27/(3*x^2 + 2)^(3/2)

________________________________________________________________________________________

mupad [B]  time = 1.70, size = 185, normalized size = 3.85 \begin {gather*} -\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {119}{16}+\frac {\sqrt {6}\,161{}\mathrm {i}}{48}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {119}{24}+\frac {\sqrt {6}\,161{}\mathrm {i}}{72}\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 {119}{16}+\frac {\sqrt {6}\,161{}\mathrm {i}}{48}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {119}{24}+\frac {\sqrt {6}\,161{}\mathrm {i}}{72}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-96+\sqrt {6}\,453{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (96+\sqrt {6}\,453{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*161i)/48 + 119/16)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2)*161i)/72 +
119/24)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*161i)/48 - 119/16)/(x + (6
^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*161i)/72 - 119/24)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*6^(1/2)*(
6^(1/2)*453i - 96)*(x^2 + 2/3)^(1/2)*1i)/(2592*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*453i + 96)*(x
^2 + 2/3)^(1/2)*1i)/(2592*(x - (6^(1/2)*1i)/3))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {51 x}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {8 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \frac {4 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {45}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-51*x/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-8*x*
*2/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(4*x**3/(9*x**4*sqr
t(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-45/(9*x**4*sqrt(3*x**2 + 2) + 1
2*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x)

________________________________________________________________________________________