3.1426 \(\int \frac{5-x}{(3+2 x)^2 (2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=109 \[ \frac{41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}+\frac{277 \sqrt{3 x^2+2}}{5145 (2 x+3)}+\frac{507 x+34}{1470 (2 x+3) \sqrt{3 x^2+2}}-\frac{176 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1715 \sqrt{35}} \]

[Out]

(26 + 41*x)/(210*(3 + 2*x)*(2 + 3*x^2)^(3/2)) + (34 + 507*x)/(1470*(3 + 2*x)*Sqrt[2 + 3*x^2]) + (277*Sqrt[2 +
3*x^2])/(5145*(3 + 2*x)) - (176*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1715*Sqrt[35])

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Rubi [A]  time = 0.0587916, antiderivative size = 109, 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, 807, 725, 206} \[ \frac{41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}+\frac{277 \sqrt{3 x^2+2}}{5145 (2 x+3)}+\frac{507 x+34}{1470 (2 x+3) \sqrt{3 x^2+2}}-\frac{176 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1715 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(210*(3 + 2*x)*(2 + 3*x^2)^(3/2)) + (34 + 507*x)/(1470*(3 + 2*x)*Sqrt[2 + 3*x^2]) + (277*Sqrt[2 +
3*x^2])/(5145*(3 + 2*x)) - (176*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1715*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 )^{5/2}} \, dx &=\frac{26+41 x}{210 (3+2 x) \left (2+3 x^2\right )^{3/2}}-\frac{1}{630} \int \frac{-1362-738 x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac{26+41 x}{210 (3+2 x) \left (2+3 x^2\right )^{3/2}}+\frac{34+507 x}{1470 (3+2 x) \sqrt{2+3 x^2}}+\frac{\int \frac{12240+91260 x}{(3+2 x)^2 \sqrt{2+3 x^2}} \, dx}{132300}\\ &=\frac{26+41 x}{210 (3+2 x) \left (2+3 x^2\right )^{3/2}}+\frac{34+507 x}{1470 (3+2 x) \sqrt{2+3 x^2}}+\frac{277 \sqrt{2+3 x^2}}{5145 (3+2 x)}+\frac{176 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{1715}\\ &=\frac{26+41 x}{210 (3+2 x) \left (2+3 x^2\right )^{3/2}}+\frac{34+507 x}{1470 (3+2 x) \sqrt{2+3 x^2}}+\frac{277 \sqrt{2+3 x^2}}{5145 (3+2 x)}-\frac{176 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{1715}\\ &=\frac{26+41 x}{210 (3+2 x) \left (2+3 x^2\right )^{3/2}}+\frac{34+507 x}{1470 (3+2 x) \sqrt{2+3 x^2}}+\frac{277 \sqrt{2+3 x^2}}{5145 (3+2 x)}-\frac{176 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{1715 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.047807, size = 101, normalized size = 0.93 \[ \frac{35 \left (4986 x^4+10647 x^3+7362 x^2+9107 x+3966\right )-1056 \sqrt{35} \sqrt{3 x^2+2} \left (6 x^3+9 x^2+4 x+6\right ) \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{360150 (2 x+3) \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*(3966 + 9107*x + 7362*x^2 + 10647*x^3 + 4986*x^4) - 1056*Sqrt[35]*Sqrt[2 + 3*x^2]*(6 + 4*x + 9*x^2 + 6*x^3
)*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(360150*(3 + 2*x)*(2 + 3*x^2)^(3/2))

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Maple [A]  time = 0.012, size = 119, normalized size = 1.1 \begin{align*}{\frac{22}{147} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{17\,x}{490} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{277\,x}{3430}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{88}{1715}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{176\,\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 ) }-{\frac{13}{70} \left ( x+{\frac{3}{2}} \right ) ^{-1} \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)^2/(3*x^2+2)^(5/2),x)

[Out]

22/147/(3*(x+3/2)^2-9*x-19/4)^(3/2)-17/490*x/(3*(x+3/2)^2-9*x-19/4)^(3/2)+277/3430*x/(3*(x+3/2)^2-9*x-19/4)^(1
/2)+88/1715/(3*(x+3/2)^2-9*x-19/4)^(1/2)-176/60025*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-1
9)^(1/2))-13/70/(x+3/2)/(3*(x+3/2)^2-9*x-19/4)^(3/2)

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Maxima [A]  time = 1.50225, size = 147, normalized size = 1.35 \begin{align*} \frac{176}{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{277 \, x}{3430 \, \sqrt{3 \, x^{2} + 2}} + \frac{88}{1715 \, \sqrt{3 \, x^{2} + 2}} - \frac{17 \, x}{490 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{13}{35 \,{\left (2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + 3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}\right )}} + \frac{22}{147 \,{\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)^2/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

176/60025*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 277/3430*x/sqrt(3*x^2 + 2)
+ 88/1715/sqrt(3*x^2 + 2) - 17/490*x/(3*x^2 + 2)^(3/2) - 13/35/(2*(3*x^2 + 2)^(3/2)*x + 3*(3*x^2 + 2)^(3/2))
+ 22/147/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 1.55089, size = 369, normalized size = 3.39 \begin{align*} \frac{528 \, \sqrt{35}{\left (18 \, x^{5} + 27 \, x^{4} + 24 \, x^{3} + 36 \, x^{2} + 8 \, x + 12\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 (4986 \, x^{4} + 10647 \, x^{3} + 7362 \, x^{2} + 9107 \, x + 3966\right )} \sqrt{3 \, x^{2} + 2}}{360150 \,{\left (18 \, x^{5} + 27 \, x^{4} + 24 \, x^{3} + 36 \, x^{2} + 8 \, x + 12\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)^(5/2),x, algorithm="fricas")

[Out]

1/360150*(528*sqrt(35)*(18*x^5 + 27*x^4 + 24*x^3 + 36*x^2 + 8*x + 12)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4)
+ 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 35*(4986*x^4 + 10647*x^3 + 7362*x^2 + 9107*x + 3966)*sqrt(3*x^2 +
2))/(18*x^5 + 27*x^4 + 24*x^3 + 36*x^2 + 8*x + 12)

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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)**(5/2),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{5}{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)^(5/2),x, algorithm="giac")

[Out]

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