3.1405 \(\int \frac{5-x}{(3+2 x)^4 \sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=99 \[ -\frac{10 \sqrt{3 x^2+2}}{343 (2 x+3)}-\frac{16 \sqrt{3 x^2+2}}{245 (2 x+3)^2}-\frac{13 \sqrt{3 x^2+2}}{105 (2 x+3)^3}-\frac{57 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1715 \sqrt{35}} \]

[Out]

(-13*Sqrt[2 + 3*x^2])/(105*(3 + 2*x)^3) - (16*Sqrt[2 + 3*x^2])/(245*(3 + 2*x)^2) - (10*Sqrt[2 + 3*x^2])/(343*(
3 + 2*x)) - (57*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1715*Sqrt[35])

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Rubi [A]  time = 0.0572682, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {835, 807, 725, 206} \[ -\frac{10 \sqrt{3 x^2+2}}{343 (2 x+3)}-\frac{16 \sqrt{3 x^2+2}}{245 (2 x+3)^2}-\frac{13 \sqrt{3 x^2+2}}{105 (2 x+3)^3}-\frac{57 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1715 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*Sqrt[2 + 3*x^2])/(105*(3 + 2*x)^3) - (16*Sqrt[2 + 3*x^2])/(245*(3 + 2*x)^2) - (10*Sqrt[2 + 3*x^2])/(343*(
3 + 2*x)) - (57*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1715*Sqrt[35])

Rule 835

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))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

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 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}{(3+2 x)^4 \sqrt{2+3 x^2}} \, dx &=-\frac{13 \sqrt{2+3 x^2}}{105 (3+2 x)^3}-\frac{1}{105} \int \frac{-123+78 x}{(3+2 x)^3 \sqrt{2+3 x^2}} \, dx\\ &=-\frac{13 \sqrt{2+3 x^2}}{105 (3+2 x)^3}-\frac{16 \sqrt{2+3 x^2}}{245 (3+2 x)^2}+\frac{\int \frac{1590-1440 x}{(3+2 x)^2 \sqrt{2+3 x^2}} \, dx}{7350}\\ &=-\frac{13 \sqrt{2+3 x^2}}{105 (3+2 x)^3}-\frac{16 \sqrt{2+3 x^2}}{245 (3+2 x)^2}-\frac{10 \sqrt{2+3 x^2}}{343 (3+2 x)}+\frac{57 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{1715}\\ &=-\frac{13 \sqrt{2+3 x^2}}{105 (3+2 x)^3}-\frac{16 \sqrt{2+3 x^2}}{245 (3+2 x)^2}-\frac{10 \sqrt{2+3 x^2}}{343 (3+2 x)}-\frac{57 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{1715}\\ &=-\frac{13 \sqrt{2+3 x^2}}{105 (3+2 x)^3}-\frac{16 \sqrt{2+3 x^2}}{245 (3+2 x)^2}-\frac{10 \sqrt{2+3 x^2}}{343 (3+2 x)}-\frac{57 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{1715 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.0725998, size = 65, normalized size = 0.66 \[ -\frac{\sqrt{3 x^2+2} \left (600 x^2+2472 x+2995\right )}{5145 (2 x+3)^3}-\frac{57 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1715 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2 + 3*x^2]*(2995 + 2472*x + 600*x^2))/(5145*(3 + 2*x)^3) - (57*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x
^2])])/(1715*Sqrt[35])

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Maple [A]  time = 0.01, size = 95, normalized size = 1. \begin{align*} -{\frac{13}{840}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{4}{245}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{5}{343}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{57\,\sqrt{35}}{60025}{\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/840/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(1/2)-4/245/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(1/2)-5/343/(x+3/2)*(3*(
x+3/2)^2-9*x-19/4)^(1/2)-57/60025*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))

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Maxima [A]  time = 1.5171, size = 140, normalized size = 1.41 \begin{align*} \frac{57}{60025} \, \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{13 \, \sqrt{3 \, x^{2} + 2}}{105 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{16 \, \sqrt{3 \, x^{2} + 2}}{245 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{10 \, \sqrt{3 \, x^{2} + 2}}{343 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

57/60025*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 13/105*sqrt(3*x^2 + 2)/(8*x
^3 + 36*x^2 + 54*x + 27) - 16/245*sqrt(3*x^2 + 2)/(4*x^2 + 12*x + 9) - 10/343*sqrt(3*x^2 + 2)/(2*x + 3)

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Fricas [A]  time = 1.86666, size = 288, normalized size = 2.91 \begin{align*} \frac{171 \, \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 ) - 70 \,{\left (600 \, x^{2} + 2472 \, x + 2995\right )} \sqrt{3 \, x^{2} + 2}}{360150 \,{\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+2*x)^4/(3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

1/360150*(171*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)) - 70*(600*x^2 + 2472*x + 2995)*sqrt(3*x^2 + 2))/(8*x^3 + 36*x^2 + 54*x + 27)

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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+2*x)**4/(3*x**2+2)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.27993, size = 308, normalized size = 3.11 \begin{align*} \frac{57}{60025} \, \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{114 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 855 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 6750 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 13290 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 10344 \, \sqrt{3} x - 800 \, \sqrt{3} - 10344 \, \sqrt{3 \, x^{2} + 2}}{3430 \,{\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+2*x)^4/(3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

57/60025*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))) - 1/3430*(114*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 855*sqrt(3)*(sqrt(3)*x - sqrt(
3*x^2 + 2))^4 + 6750*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 13290*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 10344*s
qrt(3)*x - 800*sqrt(3) - 10344*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