3.1376 \(\int \frac{(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=106 \[ \frac{(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}-\frac{3 (111 x+385) \sqrt{3 x^2+2}}{280 (2 x+3)}+\frac{11727 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{560 \sqrt{35}}+\frac{33}{16} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(-3*(385 + 111*x)*Sqrt[2 + 3*x^2])/(280*(3 + 2*x)) + ((229 + 456*x)*(2 + 3*x^2)^(3/2))/(420*(3 + 2*x)^3) + (33
*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/16 + (11727*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(560*Sqrt[35])

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Rubi [A]  time = 0.058458, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {811, 813, 844, 215, 725, 206} \[ \frac{(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}-\frac{3 (111 x+385) \sqrt{3 x^2+2}}{280 (2 x+3)}+\frac{11727 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{560 \sqrt{35}}+\frac{33}{16} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(385 + 111*x)*Sqrt[2 + 3*x^2])/(280*(3 + 2*x)) + ((229 + 456*x)*(2 + 3*x^2)^(3/2))/(420*(3 + 2*x)^3) + (33
*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/16 + (11727*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(560*Sqrt[35])

Rule 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx &=\frac{(229+456 x) \left (2+3 x^2\right )^{3/2}}{420 (3+2 x)^3}-\frac{1}{560} \int \frac{(-624+1332 x) \sqrt{2+3 x^2}}{(3+2 x)^2} \, dx\\ &=-\frac{3 (385+111 x) \sqrt{2+3 x^2}}{280 (3+2 x)}+\frac{(229+456 x) \left (2+3 x^2\right )^{3/2}}{420 (3+2 x)^3}+\frac{\int \frac{-10656+55440 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{4480}\\ &=-\frac{3 (385+111 x) \sqrt{2+3 x^2}}{280 (3+2 x)}+\frac{(229+456 x) \left (2+3 x^2\right )^{3/2}}{420 (3+2 x)^3}+\frac{99}{16} \int \frac{1}{\sqrt{2+3 x^2}} \, dx-\frac{11727}{560} \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=-\frac{3 (385+111 x) \sqrt{2+3 x^2}}{280 (3+2 x)}+\frac{(229+456 x) \left (2+3 x^2\right )^{3/2}}{420 (3+2 x)^3}+\frac{33}{16} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )+\frac{11727}{560} \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )\\ &=-\frac{3 (385+111 x) \sqrt{2+3 x^2}}{280 (3+2 x)}+\frac{(229+456 x) \left (2+3 x^2\right )^{3/2}}{420 (3+2 x)^3}+\frac{33}{16} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )+\frac{11727 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{560 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.123625, size = 89, normalized size = 0.84 \[ -\frac{\sqrt{3 x^2+2} \left (1260 x^3+24474 x^2+48747 x+30269\right )}{840 (2 x+3)^3}+\frac{11727 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{560 \sqrt{35}}+\frac{33}{16} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2 + 3*x^2]*(30269 + 48747*x + 24474*x^2 + 1260*x^3))/(840*(3 + 2*x)^3) + (33*Sqrt[3]*ArcSinh[Sqrt[3/2]*
x])/16 + (11727*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(560*Sqrt[35])

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Maple [B]  time = 0.01, size = 173, normalized size = 1.6 \begin{align*} -{\frac{13}{840} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1}{2450} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{446}{42875} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{3909}{85750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{3933\,x}{9800}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{33\,\sqrt{3}}{16}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{11727}{19600}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}+{\frac{11727\,\sqrt{35}}{19600}{\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{1338\,x}{42875} \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*x^2+2)^(3/2)/(3+2*x)^4,x)

[Out]

-13/840/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(5/2)-1/2450/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(5/2)-446/42875/(x+3/2)
*(3*(x+3/2)^2-9*x-19/4)^(5/2)-3909/85750*(3*(x+3/2)^2-9*x-19/4)^(3/2)+3933/9800*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)
+33/16*arcsinh(1/2*x*6^(1/2))*3^(1/2)-11727/19600*(12*(x+3/2)^2-36*x-19)^(1/2)+11727/19600*35^(1/2)*arctanh(2/
35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))+1338/42875*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)

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Maxima [A]  time = 1.56083, size = 203, normalized size = 1.92 \begin{align*} \frac{3}{2450} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{105 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{1225 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{3933}{9800} \, \sqrt{3 \, x^{2} + 2} x + \frac{33}{16} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{11727}{19600} \, \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{11727}{9800} \, \sqrt{3 \, x^{2} + 2} - \frac{223 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{1225 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^4,x, algorithm="maxima")

[Out]

3/2450*(3*x^2 + 2)^(3/2) - 13/105*(3*x^2 + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 2/1225*(3*x^2 + 2)^(5/2)/(4
*x^2 + 12*x + 9) + 3933/9800*sqrt(3*x^2 + 2)*x + 33/16*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 11727/19600*sqrt(35)*a
rcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 11727/9800*sqrt(3*x^2 + 2) - 223/1225*(3*x^2 +
2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 2.32958, size = 432, normalized size = 4.08 \begin{align*} \frac{121275 \, \sqrt{3}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 35181 \, \sqrt{35}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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 ) - 140 \,{\left (1260 \, x^{3} + 24474 \, x^{2} + 48747 \, x + 30269\right )} \sqrt{3 \, x^{2} + 2}}{117600 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^4,x, algorithm="fricas")

[Out]

1/117600*(121275*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 35181*sqrt
(35)*(8*x^3 + 36*x^2 + 54*x + 27)*log((sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) - 93*x^2 + 36*x - 43)/(4*x^2 + 12*x
+ 9)) - 140*(1260*x^3 + 24474*x^2 + 48747*x + 30269)*sqrt(3*x^2 + 2))/(8*x^3 + 36*x^2 + 54*x + 27)

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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*x**2+2)**(3/2)/(3+2*x)**4,x)

[Out]

Timed out

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Giac [B]  time = 1.32061, size = 352, normalized size = 3.32 \begin{align*} -\frac{33}{16} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{11727}{19600} \, \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}{16} \, \sqrt{3 \, x^{2} + 2} - \frac{44376 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 189285 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 423090 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 561630 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 499440 \, \sqrt{3} x - 50144 \, \sqrt{3} - 499440 \, \sqrt{3 \, x^{2} + 2}}{1120 \,{\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*x^2+2)^(3/2)/(3+2*x)^4,x, algorithm="giac")

[Out]

-33/16*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 11727/19600*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*s
qrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 3/16*sqrt(3*x^2 + 2) -
1/1120*(44376*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 189285*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 423090*(sqrt
(3)*x - sqrt(3*x^2 + 2))^3 - 561630*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 499440*sqrt(3)*x - 50144*sqrt(3)
- 499440*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