3.1414 \(\int \frac{5-x}{(3+2 x)^2 (2+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=82 \[ \frac{41 x+26}{70 (2 x+3) \sqrt{3 x^2+2}}+\frac{19 \sqrt{3 x^2+2}}{1225 (2 x+3)}-\frac{632 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

[Out]

(26 + 41*x)/(70*(3 + 2*x)*Sqrt[2 + 3*x^2]) + (19*Sqrt[2 + 3*x^2])/(1225*(3 + 2*x)) - (632*ArcTanh[(4 - 9*x)/(S
qrt[35]*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

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Rubi [A]  time = 0.0420368, 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 = {823, 807, 725, 206} \[ \frac{41 x+26}{70 (2 x+3) \sqrt{3 x^2+2}}+\frac{19 \sqrt{3 x^2+2}}{1225 (2 x+3)}-\frac{632 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(70*(3 + 2*x)*Sqrt[2 + 3*x^2]) + (19*Sqrt[2 + 3*x^2])/(1225*(3 + 2*x)) - (632*ArcTanh[(4 - 9*x)/(S
qrt[35]*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[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)^2 \left (2+3 x^2\right )^{3/2}} \, dx &=\frac{26+41 x}{70 (3+2 x) \sqrt{2+3 x^2}}-\frac{1}{210} \int \frac{-312-246 x}{(3+2 x)^2 \sqrt{2+3 x^2}} \, dx\\ &=\frac{26+41 x}{70 (3+2 x) \sqrt{2+3 x^2}}+\frac{19 \sqrt{2+3 x^2}}{1225 (3+2 x)}+\frac{632 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{1225}\\ &=\frac{26+41 x}{70 (3+2 x) \sqrt{2+3 x^2}}+\frac{19 \sqrt{2+3 x^2}}{1225 (3+2 x)}-\frac{632 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{1225}\\ &=\frac{26+41 x}{70 (3+2 x) \sqrt{2+3 x^2}}+\frac{19 \sqrt{2+3 x^2}}{1225 (3+2 x)}-\frac{632 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{1225 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.0363447, size = 65, normalized size = 0.79 \[ \frac{\frac{35 \left (114 x^2+1435 x+986\right )}{(2 x+3) \sqrt{3 x^2+2}}-1264 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{85750} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((35*(986 + 1435*x + 114*x^2))/((3 + 2*x)*Sqrt[2 + 3*x^2]) - 1264*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2
+ 3*x^2])])/85750

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Maple [A]  time = 0.011, size = 86, normalized size = 1.1 \begin{align*}{\frac{316}{1225}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{57\,x}{2450}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{632\,\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{13}{70} \left ( x+{\frac{3}{2}} \right ) ^{-1}{\frac{1}{\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+2*x)^2/(3*x^2+2)^(3/2),x)

[Out]

316/1225/(3*(x+3/2)^2-9*x-19/4)^(1/2)+57/2450*x/(3*(x+3/2)^2-9*x-19/4)^(1/2)-632/42875*35^(1/2)*arctanh(2/35*(
4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))-13/70/(x+3/2)/(3*(x+3/2)^2-9*x-19/4)^(1/2)

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Maxima [A]  time = 1.51752, size = 116, normalized size = 1.41 \begin{align*} \frac{632}{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{57 \, x}{2450 \, \sqrt{3 \, x^{2} + 2}} + \frac{316}{1225 \, \sqrt{3 \, x^{2} + 2}} - \frac{13}{35 \,{\left (2 \, \sqrt{3 \, x^{2} + 2} x + 3 \, \sqrt{3 \, x^{2} + 2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

632/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 57/2450*x/sqrt(3*x^2 + 2)
+ 316/1225/sqrt(3*x^2 + 2) - 13/35/(2*sqrt(3*x^2 + 2)*x + 3*sqrt(3*x^2 + 2))

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Fricas [A]  time = 1.56076, size = 277, normalized size = 3.38 \begin{align*} \frac{632 \, \sqrt{35}{\left (6 \, x^{3} + 9 \, x^{2} + 4 \, x + 6\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 (114 \, x^{2} + 1435 \, x + 986\right )} \sqrt{3 \, x^{2} + 2}}{85750 \,{\left (6 \, x^{3} + 9 \, x^{2} + 4 \, x + 6\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/85750*(632*sqrt(35)*(6*x^3 + 9*x^2 + 4*x + 6)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)
/(4*x^2 + 12*x + 9)) + 35*(114*x^2 + 1435*x + 986)*sqrt(3*x^2 + 2))/(6*x^3 + 9*x^2 + 4*x + 6)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x - 5}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (2 \, x + 3\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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