3.1425 \(\int \frac{5-x}{(3+2 x) (2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=73 \[ \frac{41 x+26}{210 \left (3 x^2+2\right )^{3/2}}+\frac{2137 x+312}{7350 \sqrt{3 x^2+2}}-\frac{104 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

[Out]

(26 + 41*x)/(210*(2 + 3*x^2)^(3/2)) + (312 + 2137*x)/(7350*Sqrt[2 + 3*x^2]) - (104*ArcTanh[(4 - 9*x)/(Sqrt[35]
*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

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Rubi [A]  time = 0.0425624, antiderivative size = 73, 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 = {823, 12, 725, 206} \[ \frac{41 x+26}{210 \left (3 x^2+2\right )^{3/2}}+\frac{2137 x+312}{7350 \sqrt{3 x^2+2}}-\frac{104 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(210*(2 + 3*x^2)^(3/2)) + (312 + 2137*x)/(7350*Sqrt[2 + 3*x^2]) - (104*ArcTanh[(4 - 9*x)/(Sqrt[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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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) \left (2+3 x^2\right )^{5/2}} \, dx &=\frac{26+41 x}{210 \left (2+3 x^2\right )^{3/2}}-\frac{1}{630} \int \frac{-1206-492 x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac{26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac{312+2137 x}{7350 \sqrt{2+3 x^2}}+\frac{\int \frac{11232}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{132300}\\ &=\frac{26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac{312+2137 x}{7350 \sqrt{2+3 x^2}}+\frac{104 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{1225}\\ &=\frac{26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac{312+2137 x}{7350 \sqrt{2+3 x^2}}-\frac{104 \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}{210 \left (2+3 x^2\right )^{3/2}}+\frac{312+2137 x}{7350 \sqrt{2+3 x^2}}-\frac{104 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{1225 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.0398818, size = 63, normalized size = 0.86 \[ \frac{\frac{35 \left (6411 x^3+936 x^2+5709 x+1534\right )}{\left (3 x^2+2\right )^{3/2}}-624 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{257250} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((35*(1534 + 5709*x + 936*x^2 + 6411*x^3))/(2 + 3*x^2)^(3/2) - 624*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2
+ 3*x^2])])/257250

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Maple [B]  time = 0.007, size = 122, normalized size = 1.7 \begin{align*} -{\frac{x}{12} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{x}{12}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{13}{105} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{39\,x}{140} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{1833\,x}{4900}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{52}{1225}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{104\,\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*x/(3*x^2+2)^(3/2)-1/12*x/(3*x^2+2)^(1/2)+13/105/(3*(x+3/2)^2-9*x-19/4)^(3/2)+39/140*x/(3*(x+3/2)^2-9*x-1
9/4)^(3/2)+1833/4900*x/(3*(x+3/2)^2-9*x-19/4)^(1/2)+52/1225/(3*(x+3/2)^2-9*x-19/4)^(1/2)-104/42875*35^(1/2)*ar
ctanh(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 = 1.50479, size = 109, normalized size = 1.49 \begin{align*} \frac{104}{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{2137 \, x}{7350 \, \sqrt{3 \, x^{2} + 2}} + \frac{52}{1225 \, \sqrt{3 \, x^{2} + 2}} + \frac{41 \, x}{210 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{13}{105 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}

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]

104/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 2137/7350*x/sqrt(3*x^2 + 2
) + 52/1225/sqrt(3*x^2 + 2) + 41/210*x/(3*x^2 + 2)^(3/2) + 13/105/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 1.5448, size = 281, normalized size = 3.85 \begin{align*} \frac{312 \, \sqrt{35}{\left (9 \, x^{4} + 12 \, x^{2} + 4\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 (6411 \, x^{3} + 936 \, x^{2} + 5709 \, x + 1534\right )} \sqrt{3 \, x^{2} + 2}}{257250 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \end{align*}

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]

1/257250*(312*sqrt(35)*(9*x^4 + 12*x^2 + 4)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*
x^2 + 12*x + 9)) + 35*(6411*x^3 + 936*x^2 + 5709*x + 1534)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x^2 + 4)

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

[Out]

Timed out

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Giac [A]  time = 1.25445, size = 126, normalized size = 1.73 \begin{align*} \frac{104}{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{3 \,{\left ({\left (2137 \, x + 312\right )} x + 1903\right )} x + 1534}{7350 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}

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]

104/42875*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))) + 1/7350*(3*((2137*x + 312)*x + 1903)*x + 1534)/(3*x^2 + 2)^(3/2)