3.1375 \(\int \frac{(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^3} \, dx\)

Optimal. Leaf size=104 \[ -\frac{(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}+\frac{3 (12 x+37) \sqrt{3 x^2+2}}{4 (2 x+3)}-\frac{1143 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8 \sqrt{35}}-\frac{111}{8} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(3*(37 + 12*x)*Sqrt[2 + 3*x^2])/(4*(3 + 2*x)) - ((8 + x)*(2 + 3*x^2)^(3/2))/(4*(3 + 2*x)^2) - (111*Sqrt[3]*Arc
Sinh[Sqrt[3/2]*x])/8 - (1143*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(8*Sqrt[35])

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Rubi [A]  time = 0.0608168, antiderivative size = 104, 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 = {813, 844, 215, 725, 206} \[ -\frac{(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}+\frac{3 (12 x+37) \sqrt{3 x^2+2}}{4 (2 x+3)}-\frac{1143 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8 \sqrt{35}}-\frac{111}{8} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(37 + 12*x)*Sqrt[2 + 3*x^2])/(4*(3 + 2*x)) - ((8 + x)*(2 + 3*x^2)^(3/2))/(4*(3 + 2*x)^2) - (111*Sqrt[3]*Arc
Sinh[Sqrt[3/2]*x])/8 - (1143*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(8*Sqrt[35])

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)^3} \, dx &=-\frac{(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac{3}{32} \int \frac{(16-192 x) \sqrt{2+3 x^2}}{(3+2 x)^2} \, dx\\ &=\frac{3 (37+12 x) \sqrt{2+3 x^2}}{4 (3+2 x)}-\frac{(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}+\frac{3}{256} \int \frac{1536-7104 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=\frac{3 (37+12 x) \sqrt{2+3 x^2}}{4 (3+2 x)}-\frac{(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac{333}{8} \int \frac{1}{\sqrt{2+3 x^2}} \, dx+\frac{1143}{8} \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=\frac{3 (37+12 x) \sqrt{2+3 x^2}}{4 (3+2 x)}-\frac{(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac{111}{8} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{1143}{8} \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )\\ &=\frac{3 (37+12 x) \sqrt{2+3 x^2}}{4 (3+2 x)}-\frac{(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac{111}{8} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{1143 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{8 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.119111, size = 89, normalized size = 0.86 \[ -\frac{\sqrt{3 x^2+2} \left (3 x^3-48 x^2-328 x-317\right )}{4 (2 x+3)^2}-\frac{1143 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8 \sqrt{35}}-\frac{111}{8} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2 + 3*x^2]*(-317 - 328*x - 48*x^2 + 3*x^3))/(4*(3 + 2*x)^2) - (111*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/8 - (1
143*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(8*Sqrt[35])

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Maple [A]  time = 0.009, size = 152, normalized size = 1.5 \begin{align*}{\frac{187}{4900} \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{381}{1225} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{171\,x}{70}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{111\,\sqrt{3}}{8}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{1143}{280}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{1143\,\sqrt{35}}{280}{\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{561\,x}{4900} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{13}{280} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-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)^3,x)

[Out]

187/4900/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(5/2)+381/1225*(3*(x+3/2)^2-9*x-19/4)^(3/2)-171/70*x*(3*(x+3/2)^2-9*x-
19/4)^(1/2)-111/8*arcsinh(1/2*x*6^(1/2))*3^(1/2)+1143/280*(12*(x+3/2)^2-36*x-19)^(1/2)-1143/280*35^(1/2)*arcta
nh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))-561/4900*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)-13/280/(x+3/2)^
2*(3*(x+3/2)^2-9*x-19/4)^(5/2)

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Maxima [A]  time = 1.55148, size = 165, normalized size = 1.59 \begin{align*} \frac{39}{280} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{70 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{171}{70} \, \sqrt{3 \, x^{2} + 2} x - \frac{111}{8} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{1143}{280} \, \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{1143}{140} \, \sqrt{3 \, x^{2} + 2} + \frac{187 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{280 \,{\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)^3,x, algorithm="maxima")

[Out]

39/280*(3*x^2 + 2)^(3/2) - 13/70*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) - 171/70*sqrt(3*x^2 + 2)*x - 111/8*sqrt(
3)*arcsinh(1/2*sqrt(6)*x) + 1143/280*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) +
1143/140*sqrt(3*x^2 + 2) + 187/280*(3*x^2 + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 2.25316, size = 370, normalized size = 3.56 \begin{align*} \frac{3885 \, \sqrt{3}{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 1143 \, \sqrt{35}{\left (4 \, x^{2} + 12 \, x + 9\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 (3 \, x^{3} - 48 \, x^{2} - 328 \, x - 317\right )} \sqrt{3 \, x^{2} + 2}}{560 \,{\left (4 \, x^{2} + 12 \, x + 9\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)^3,x, algorithm="fricas")

[Out]

1/560*(3885*sqrt(3)*(4*x^2 + 12*x + 9)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 1143*sqrt(35)*(4*x^2 + 12*
x + 9)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 140*(3*x^3 - 48*x^
2 - 328*x - 317)*sqrt(3*x^2 + 2))/(4*x^2 + 12*x + 9)

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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)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.25375, size = 296, normalized size = 2.85 \begin{align*} -\frac{3}{16} \, \sqrt{3 \, x^{2} + 2}{\left (x - 19\right )} + \frac{111}{8} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{1143}{280} \, \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{5 \,{\left (1452 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 3013 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 6528 \, \sqrt{3} x + 1048 \, \sqrt{3} + 6528 \, \sqrt{3 \, x^{2} + 2}\right )}}{64 \,{\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*x^2+2)^(3/2)/(3+2*x)^3,x, algorithm="giac")

[Out]

-3/16*sqrt(3*x^2 + 2)*(x - 19) + 111/8*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 1143/280*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))) + 5/64*(1452*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 3013*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 6528*sqrt(3
)*x + 1048*sqrt(3) + 6528*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