3.1364 \(\int \frac{(5-x) \sqrt{2+3 x^2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=82 \[ -\frac{13 \left (3 x^2+2\right )^{3/2}}{105 (2 x+3)^3}-\frac{41 (4-9 x) \sqrt{3 x^2+2}}{2450 (2 x+3)^2}-\frac{123 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

[Out]

(-41*(4 - 9*x)*Sqrt[2 + 3*x^2])/(2450*(3 + 2*x)^2) - (13*(2 + 3*x^2)^(3/2))/(105*(3 + 2*x)^3) - (123*ArcTanh[(
4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

________________________________________________________________________________________

Rubi [A]  time = 0.0339467, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {807, 721, 725, 206} \[ -\frac{13 \left (3 x^2+2\right )^{3/2}}{105 (2 x+3)^3}-\frac{41 (4-9 x) \sqrt{3 x^2+2}}{2450 (2 x+3)^2}-\frac{123 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^4,x]

[Out]

(-41*(4 - 9*x)*Sqrt[2 + 3*x^2])/(2450*(3 + 2*x)^2) - (13*(2 + 3*x^2)^(3/2))/(105*(3 + 2*x)^3) - (123*ArcTanh[(
4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

Rule 807

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

Rule 721

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*
d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*
x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2,
0] && GtQ[p, 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(5-x) \sqrt{2+3 x^2}}{(3+2 x)^4} \, dx &=-\frac{13 \left (2+3 x^2\right )^{3/2}}{105 (3+2 x)^3}+\frac{41}{35} \int \frac{\sqrt{2+3 x^2}}{(3+2 x)^3} \, dx\\ &=-\frac{41 (4-9 x) \sqrt{2+3 x^2}}{2450 (3+2 x)^2}-\frac{13 \left (2+3 x^2\right )^{3/2}}{105 (3+2 x)^3}+\frac{123 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{1225}\\ &=-\frac{41 (4-9 x) \sqrt{2+3 x^2}}{2450 (3+2 x)^2}-\frac{13 \left (2+3 x^2\right )^{3/2}}{105 (3+2 x)^3}-\frac{123 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{1225}\\ &=-\frac{41 (4-9 x) \sqrt{2+3 x^2}}{2450 (3+2 x)^2}-\frac{13 \left (2+3 x^2\right )^{3/2}}{105 (3+2 x)^3}-\frac{123 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{1225 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.0555901, size = 65, normalized size = 0.79 \[ \frac{-\frac{35 \sqrt{3 x^2+2} \left (516 x^2-2337 x+3296\right )}{(2 x+3)^3}-738 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{257250} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^4,x]

[Out]

((-35*Sqrt[2 + 3*x^2]*(3296 - 2337*x + 516*x^2))/(3 + 2*x)^3 - 738*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2
+ 3*x^2])])/257250

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 128, normalized size = 1.6 \begin{align*} -{\frac{13}{840} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{41}{4900} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{369}{85750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{123}{42875}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{123\,\sqrt{35}}{42875}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{1107\,x}{85750}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^4,x)

[Out]

-13/840/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(3/2)-41/4900/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(3/2)-369/85750/(x+3/2
)*(3*(x+3/2)^2-9*x-19/4)^(3/2)+123/42875*(12*(x+3/2)^2-36*x-19)^(1/2)-123/42875*35^(1/2)*arctanh(2/35*(4-9*x)*
35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))+1107/85750*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.4866, size = 155, normalized size = 1.89 \begin{align*} \frac{123}{42875} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{123}{4900} \, \sqrt{3 \, x^{2} + 2} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{105 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{41 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{1225 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{369 \, \sqrt{3 \, x^{2} + 2}}{4900 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^4,x, algorithm="maxima")

[Out]

123/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 123/4900*sqrt(3*x^2 + 2) -
13/105*(3*x^2 + 2)^(3/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 41/1225*(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9) - 369/49
00*sqrt(3*x^2 + 2)/(2*x + 3)

________________________________________________________________________________________

Fricas [A]  time = 2.44595, size = 288, normalized size = 3.51 \begin{align*} \frac{369 \, \sqrt{35}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac{\sqrt{35} \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \,{\left (516 \, x^{2} - 2337 \, x + 3296\right )} \sqrt{3 \, x^{2} + 2}}{257250 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^4,x, algorithm="fricas")

[Out]

1/257250*(369*sqrt(35)*(8*x^3 + 36*x^2 + 54*x + 27)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x +
43)/(4*x^2 + 12*x + 9)) - 35*(516*x^2 - 2337*x + 3296)*sqrt(3*x^2 + 2))/(8*x^3 + 36*x^2 + 54*x + 27)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+2)**(1/2)/(3+2*x)**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.25969, size = 308, normalized size = 3.76 \begin{align*} \frac{123}{42875} \, \sqrt{35} \log \left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{4659 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 30 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 11610 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 25740 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 60 \, \sqrt{3} x - 1376 \, \sqrt{3} + 60 \, \sqrt{3 \, x^{2} + 2}}{9800 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^4,x, algorithm="giac")

[Out]

123/42875*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) +
3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 1/9800*(4659*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 30*sqrt(3)*(sqrt(3)*x - sqrt
(3*x^2 + 2))^4 + 11610*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 25740*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 60*sq
rt(3)*x - 1376*sqrt(3) + 60*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*
x^2 + 2)) - 2)^3