3.1416 \(\int \frac{5-x}{(3+2 x)^4 (2+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac{41 x+26}{70 (2 x+3)^3 \sqrt{3 x^2+2}}-\frac{1051 \sqrt{3 x^2+2}}{42875 (2 x+3)}-\frac{27 \sqrt{3 x^2+2}}{1225 (2 x+3)^2}+\frac{23 \sqrt{3 x^2+2}}{525 (2 x+3)^3}-\frac{3312 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42875 \sqrt{35}} \]

[Out]

(26 + 41*x)/(70*(3 + 2*x)^3*Sqrt[2 + 3*x^2]) + (23*Sqrt[2 + 3*x^2])/(525*(3 + 2*x)^3) - (27*Sqrt[2 + 3*x^2])/(
1225*(3 + 2*x)^2) - (1051*Sqrt[2 + 3*x^2])/(42875*(3 + 2*x)) - (3312*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^
2])])/(42875*Sqrt[35])

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Rubi [A]  time = 0.075735, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {823, 835, 807, 725, 206} \[ \frac{41 x+26}{70 (2 x+3)^3 \sqrt{3 x^2+2}}-\frac{1051 \sqrt{3 x^2+2}}{42875 (2 x+3)}-\frac{27 \sqrt{3 x^2+2}}{1225 (2 x+3)^2}+\frac{23 \sqrt{3 x^2+2}}{525 (2 x+3)^3}-\frac{3312 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42875 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(70*(3 + 2*x)^3*Sqrt[2 + 3*x^2]) + (23*Sqrt[2 + 3*x^2])/(525*(3 + 2*x)^3) - (27*Sqrt[2 + 3*x^2])/(
1225*(3 + 2*x)^2) - (1051*Sqrt[2 + 3*x^2])/(42875*(3 + 2*x)) - (3312*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^
2])])/(42875*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 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 \left (2+3 x^2\right )^{3/2}} \, dx &=\frac{26+41 x}{70 (3+2 x)^3 \sqrt{2+3 x^2}}-\frac{1}{210} \int \frac{-624-738 x}{(3+2 x)^4 \sqrt{2+3 x^2}} \, dx\\ &=\frac{26+41 x}{70 (3+2 x)^3 \sqrt{2+3 x^2}}+\frac{23 \sqrt{2+3 x^2}}{525 (3+2 x)^3}+\frac{\int \frac{25704+5796 x}{(3+2 x)^3 \sqrt{2+3 x^2}} \, dx}{22050}\\ &=\frac{26+41 x}{70 (3+2 x)^3 \sqrt{2+3 x^2}}+\frac{23 \sqrt{2+3 x^2}}{525 (3+2 x)^3}-\frac{27 \sqrt{2+3 x^2}}{1225 (3+2 x)^2}-\frac{\int \frac{-509040+102060 x}{(3+2 x)^2 \sqrt{2+3 x^2}} \, dx}{1543500}\\ &=\frac{26+41 x}{70 (3+2 x)^3 \sqrt{2+3 x^2}}+\frac{23 \sqrt{2+3 x^2}}{525 (3+2 x)^3}-\frac{27 \sqrt{2+3 x^2}}{1225 (3+2 x)^2}-\frac{1051 \sqrt{2+3 x^2}}{42875 (3+2 x)}+\frac{3312 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{42875}\\ &=\frac{26+41 x}{70 (3+2 x)^3 \sqrt{2+3 x^2}}+\frac{23 \sqrt{2+3 x^2}}{525 (3+2 x)^3}-\frac{27 \sqrt{2+3 x^2}}{1225 (3+2 x)^2}-\frac{1051 \sqrt{2+3 x^2}}{42875 (3+2 x)}-\frac{3312 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{42875}\\ &=\frac{26+41 x}{70 (3+2 x)^3 \sqrt{2+3 x^2}}+\frac{23 \sqrt{2+3 x^2}}{525 (3+2 x)^3}-\frac{27 \sqrt{2+3 x^2}}{1225 (3+2 x)^2}-\frac{1051 \sqrt{2+3 x^2}}{42875 (3+2 x)}-\frac{3312 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{42875 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.0862066, size = 75, normalized size = 0.6 \[ \frac{-\frac{35 \left (75672 x^4+261036 x^3+237930 x^2+23349 x+29438\right )}{(2 x+3)^3 \sqrt{3 x^2+2}}-19872 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{9003750} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-35*(29438 + 23349*x + 237930*x^2 + 261036*x^3 + 75672*x^4))/((3 + 2*x)^3*Sqrt[2 + 3*x^2]) - 19872*Sqrt[35]*
ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/9003750

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Maple [A]  time = 0.01, size = 128, normalized size = 1. \begin{align*} -{\frac{13}{840} \left ( x+{\frac{3}{2}} \right ) ^{-3}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{17}{700} \left ( x+{\frac{3}{2}} \right ) ^{-2}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{101}{2450} \left ( x+{\frac{3}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{1656}{42875}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{3153\,x}{85750}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{3312\,\sqrt{35}}{1500625}{\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)^(3/2),x)

[Out]

-13/840/(x+3/2)^3/(3*(x+3/2)^2-9*x-19/4)^(1/2)-17/700/(x+3/2)^2/(3*(x+3/2)^2-9*x-19/4)^(1/2)-101/2450/(x+3/2)/
(3*(x+3/2)^2-9*x-19/4)^(1/2)+1656/42875/(3*(x+3/2)^2-9*x-19/4)^(1/2)-3153/85750*x/(3*(x+3/2)^2-9*x-19/4)^(1/2)
-3312/1500625*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 = 1.50208, size = 248, normalized size = 1.97 \begin{align*} \frac{3312}{1500625} \, \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{3153 \, x}{85750 \, \sqrt{3 \, x^{2} + 2}} + \frac{1656}{42875 \, \sqrt{3 \, x^{2} + 2}} - \frac{13}{105 \,{\left (8 \, \sqrt{3 \, x^{2} + 2} x^{3} + 36 \, \sqrt{3 \, x^{2} + 2} x^{2} + 54 \, \sqrt{3 \, x^{2} + 2} x + 27 \, \sqrt{3 \, x^{2} + 2}\right )}} - \frac{17}{175 \,{\left (4 \, \sqrt{3 \, x^{2} + 2} x^{2} + 12 \, \sqrt{3 \, x^{2} + 2} x + 9 \, \sqrt{3 \, x^{2} + 2}\right )}} - \frac{101}{1225 \,{\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)^4/(3*x^2+2)^(3/2),x, algorithm="maxima")

[Out]

3312/1500625*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 3153/85750*x/sqrt(3*x^2
+ 2) + 1656/42875/sqrt(3*x^2 + 2) - 13/105/(8*sqrt(3*x^2 + 2)*x^3 + 36*sqrt(3*x^2 + 2)*x^2 + 54*sqrt(3*x^2 +
2)*x + 27*sqrt(3*x^2 + 2)) - 17/175/(4*sqrt(3*x^2 + 2)*x^2 + 12*sqrt(3*x^2 + 2)*x + 9*sqrt(3*x^2 + 2)) - 101/1
225/(2*sqrt(3*x^2 + 2)*x + 3*sqrt(3*x^2 + 2))

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Fricas [A]  time = 1.57329, size = 393, normalized size = 3.12 \begin{align*} \frac{9936 \, \sqrt{35}{\left (24 \, x^{5} + 108 \, x^{4} + 178 \, x^{3} + 153 \, x^{2} + 108 \, x + 54\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 (75672 \, x^{4} + 261036 \, x^{3} + 237930 \, x^{2} + 23349 \, x + 29438\right )} \sqrt{3 \, x^{2} + 2}}{9003750 \,{\left (24 \, x^{5} + 108 \, x^{4} + 178 \, x^{3} + 153 \, x^{2} + 108 \, x + 54\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)^(3/2),x, algorithm="fricas")

[Out]

1/9003750*(9936*sqrt(35)*(24*x^5 + 108*x^4 + 178*x^3 + 153*x^2 + 108*x + 54)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9
*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(75672*x^4 + 261036*x^3 + 237930*x^2 + 23349*x + 29438)
*sqrt(3*x^2 + 2))/(24*x^5 + 108*x^4 + 178*x^3 + 153*x^2 + 108*x + 54)

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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)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.19474, size = 329, normalized size = 2.61 \begin{align*} \frac{3312}{1500625} \, \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 (10281 \, x - 12674\right )}}{3001250 \, \sqrt{3 \, x^{2} + 2}} - \frac{2 \,{\left (38949 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 253320 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 894510 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 1481160 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 1275420 \, \sqrt{3} x - 106016 \, \sqrt{3} - 1275420 \, \sqrt{3 \, x^{2} + 2}\right )}}{1500625 \,{\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)^(3/2),x, algorithm="giac")

[Out]

3312/1500625*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))) - 3/3001250*(10281*x - 12674)/sqrt(3*x^2 + 2) - 2/1500625*(38949*(sqrt(3)*
x - sqrt(3*x^2 + 2))^5 + 253320*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 894510*(sqrt(3)*x - sqrt(3*x^2 + 2))
^3 - 1481160*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 1275420*sqrt(3)*x - 106016*sqrt(3) - 1275420*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