3.1427 \(\int \frac{5-x}{(3+2 x)^3 (2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=131 \[ \frac{41 x+26}{210 (2 x+3)^2 \left (3 x^2+2\right )^{3/2}}+\frac{857 \sqrt{3 x^2+2}}{128625 (2 x+3)}+\frac{83 \sqrt{3 x^2+2}}{1225 (2 x+3)^2}+\frac{419 x+4}{1050 (2 x+3)^2 \sqrt{3 x^2+2}}-\frac{3072 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42875 \sqrt{35}} \]

[Out]

(26 + 41*x)/(210*(3 + 2*x)^2*(2 + 3*x^2)^(3/2)) + (4 + 419*x)/(1050*(3 + 2*x)^2*Sqrt[2 + 3*x^2]) + (83*Sqrt[2
+ 3*x^2])/(1225*(3 + 2*x)^2) + (857*Sqrt[2 + 3*x^2])/(128625*(3 + 2*x)) - (3072*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sq
rt[2 + 3*x^2])])/(42875*Sqrt[35])

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Rubi [A]  time = 0.0767948, antiderivative size = 131, 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} \[ \frac{41 x+26}{210 (2 x+3)^2 \left (3 x^2+2\right )^{3/2}}+\frac{857 \sqrt{3 x^2+2}}{128625 (2 x+3)}+\frac{83 \sqrt{3 x^2+2}}{1225 (2 x+3)^2}+\frac{419 x+4}{1050 (2 x+3)^2 \sqrt{3 x^2+2}}-\frac{3072 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42875 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(210*(3 + 2*x)^2*(2 + 3*x^2)^(3/2)) + (4 + 419*x)/(1050*(3 + 2*x)^2*Sqrt[2 + 3*x^2]) + (83*Sqrt[2
+ 3*x^2])/(1225*(3 + 2*x)^2) + (857*Sqrt[2 + 3*x^2])/(128625*(3 + 2*x)) - (3072*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sq
rt[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)^3 \left (2+3 x^2\right )^{5/2}} \, dx &=\frac{26+41 x}{210 (3+2 x)^2 \left (2+3 x^2\right )^{3/2}}-\frac{1}{630} \int \frac{-1518-984 x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac{26+41 x}{210 (3+2 x)^2 \left (2+3 x^2\right )^{3/2}}+\frac{4+419 x}{1050 (3+2 x)^2 \sqrt{2+3 x^2}}+\frac{\int \frac{3024+211176 x}{(3+2 x)^3 \sqrt{2+3 x^2}} \, dx}{132300}\\ &=\frac{26+41 x}{210 (3+2 x)^2 \left (2+3 x^2\right )^{3/2}}+\frac{4+419 x}{1050 (3+2 x)^2 \sqrt{2+3 x^2}}+\frac{83 \sqrt{2+3 x^2}}{1225 (3+2 x)^2}-\frac{\int \frac{-1743840-1882440 x}{(3+2 x)^2 \sqrt{2+3 x^2}} \, dx}{9261000}\\ &=\frac{26+41 x}{210 (3+2 x)^2 \left (2+3 x^2\right )^{3/2}}+\frac{4+419 x}{1050 (3+2 x)^2 \sqrt{2+3 x^2}}+\frac{83 \sqrt{2+3 x^2}}{1225 (3+2 x)^2}+\frac{857 \sqrt{2+3 x^2}}{128625 (3+2 x)}+\frac{3072 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{42875}\\ &=\frac{26+41 x}{210 (3+2 x)^2 \left (2+3 x^2\right )^{3/2}}+\frac{4+419 x}{1050 (3+2 x)^2 \sqrt{2+3 x^2}}+\frac{83 \sqrt{2+3 x^2}}{1225 (3+2 x)^2}+\frac{857 \sqrt{2+3 x^2}}{128625 (3+2 x)}-\frac{3072 \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}{210 (3+2 x)^2 \left (2+3 x^2\right )^{3/2}}+\frac{4+419 x}{1050 (3+2 x)^2 \sqrt{2+3 x^2}}+\frac{83 \sqrt{2+3 x^2}}{1225 (3+2 x)^2}+\frac{857 \sqrt{2+3 x^2}}{128625 (3+2 x)}-\frac{3072 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{42875 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.111094, size = 80, normalized size = 0.61 \[ \frac{\frac{35 \left (10284 x^5+67716 x^4+116367 x^3+91268 x^2+89749 x+41366\right )}{(2 x+3)^2 \left (3 x^2+2\right )^{3/2}}-6144 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{3001250} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((35*(41366 + 89749*x + 91268*x^2 + 116367*x^3 + 67716*x^4 + 10284*x^5))/((3 + 2*x)^2*(2 + 3*x^2)^(3/2)) - 614
4*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/3001250

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Maple [A]  time = 0.01, size = 140, normalized size = 1.1 \begin{align*} -{\frac{107}{700} \left ( x+{\frac{3}{2}} \right ) ^{-1} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{128}{1225} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{173\,x}{2450} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{857\,x}{85750}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{1536}{42875}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{3072\,\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 ) }-{\frac{13}{280} \left ( x+{\frac{3}{2}} \right ) ^{-2} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-107/700/(x+3/2)/(3*(x+3/2)^2-9*x-19/4)^(3/2)+128/1225/(3*(x+3/2)^2-9*x-19/4)^(3/2)-173/2450*x/(3*(x+3/2)^2-9*
x-19/4)^(3/2)+857/85750*x/(3*(x+3/2)^2-9*x-19/4)^(1/2)+1536/42875/(3*(x+3/2)^2-9*x-19/4)^(1/2)-3072/1500625*35
^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))-13/280/(x+3/2)^2/(3*(x+3/2)^2-9*x-19/4)^(3/
2)

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Maxima [A]  time = 1.51806, size = 204, normalized size = 1.56 \begin{align*} \frac{3072}{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{857 \, x}{85750 \, \sqrt{3 \, x^{2} + 2}} + \frac{1536}{42875 \, \sqrt{3 \, x^{2} + 2}} - \frac{173 \, x}{2450 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{13}{70 \,{\left (4 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + 9 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}\right )}} - \frac{107}{350 \,{\left (2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + 3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}\right )}} + \frac{128}{1225 \,{\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/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

3072/1500625*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 857/85750*x/sqrt(3*x^2
+ 2) + 1536/42875/sqrt(3*x^2 + 2) - 173/2450*x/(3*x^2 + 2)^(3/2) - 13/70/(4*(3*x^2 + 2)^(3/2)*x^2 + 12*(3*x^2
+ 2)^(3/2)*x + 9*(3*x^2 + 2)^(3/2)) - 107/350/(2*(3*x^2 + 2)^(3/2)*x + 3*(3*x^2 + 2)^(3/2)) + 128/1225/(3*x^2
+ 2)^(3/2)

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Fricas [A]  time = 1.55391, size = 432, normalized size = 3.3 \begin{align*} \frac{3072 \, \sqrt{35}{\left (36 \, x^{6} + 108 \, x^{5} + 129 \, x^{4} + 144 \, x^{3} + 124 \, x^{2} + 48 \, x + 36\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 (10284 \, x^{5} + 67716 \, x^{4} + 116367 \, x^{3} + 91268 \, x^{2} + 89749 \, x + 41366\right )} \sqrt{3 \, x^{2} + 2}}{3001250 \,{\left (36 \, x^{6} + 108 \, x^{5} + 129 \, x^{4} + 144 \, x^{3} + 124 \, x^{2} + 48 \, x + 36\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3001250*(3072*sqrt(35)*(36*x^6 + 108*x^5 + 129*x^4 + 144*x^3 + 124*x^2 + 48*x + 36)*log(-(sqrt(35)*sqrt(3*x^
2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 35*(10284*x^5 + 67716*x^4 + 116367*x^3 + 91268*x^
2 + 89749*x + 41366)*sqrt(3*x^2 + 2))/(36*x^6 + 108*x^5 + 129*x^4 + 144*x^3 + 124*x^2 + 48*x + 36)

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

[Out]

Timed out

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Giac [A]  time = 1.3215, size = 281, normalized size = 2.15 \begin{align*} \frac{3072}{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 ({\left (59203 \, x + 69168\right )} x + 37637\right )} x + 190066}{3001250 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{4 \,{\left (9588 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 27991 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 68448 \, \sqrt{3} x + 9736 \, \sqrt{3} + 68448 \, \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 )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3072/1500625*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))) + 1/3001250*(3*((59203*x + 69168)*x + 37637)*x + 190066)/(3*x^2 + 2)^(3/2)
- 4/1500625*(9588*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 27991*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 68448*sqr
t(3)*x + 9736*sqrt(3) + 68448*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)^2