3.1373 \(\int \frac{(5-x) (2+3 x^2)^{3/2}}{3+2 x} \, dx\)

Optimal. Leaf size=92 \[ \frac{1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}+\frac{1}{16} (455-123 x) \sqrt{3 x^2+2}-\frac{455}{32} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{1529}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

((455 - 123*x)*Sqrt[2 + 3*x^2])/16 + ((26 - 3*x)*(2 + 3*x^2)^(3/2))/24 - (1529*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/3
2 - (455*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/32

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Rubi [A]  time = 0.0570738, antiderivative size = 92, 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 = {815, 844, 215, 725, 206} \[ \frac{1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}+\frac{1}{16} (455-123 x) \sqrt{3 x^2+2}-\frac{455}{32} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{1529}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((455 - 123*x)*Sqrt[2 + 3*x^2])/16 + ((26 - 3*x)*(2 + 3*x^2)^(3/2))/24 - (1529*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/3
2 - (455*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/32

Rule 815

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m
+ 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 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} \, dx &=\frac{1}{24} (26-3 x) \left (2+3 x^2\right )^{3/2}+\frac{1}{48} \int \frac{(516-1476 x) \sqrt{2+3 x^2}}{3+2 x} \, dx\\ &=\frac{1}{16} (455-123 x) \sqrt{2+3 x^2}+\frac{1}{24} (26-3 x) \left (2+3 x^2\right )^{3/2}+\frac{\int \frac{77904-330264 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{1152}\\ &=\frac{1}{16} (455-123 x) \sqrt{2+3 x^2}+\frac{1}{24} (26-3 x) \left (2+3 x^2\right )^{3/2}-\frac{4587}{32} \int \frac{1}{\sqrt{2+3 x^2}} \, dx+\frac{15925}{32} \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=\frac{1}{16} (455-123 x) \sqrt{2+3 x^2}+\frac{1}{24} (26-3 x) \left (2+3 x^2\right )^{3/2}-\frac{1529}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{15925}{32} \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )\\ &=\frac{1}{16} (455-123 x) \sqrt{2+3 x^2}+\frac{1}{24} (26-3 x) \left (2+3 x^2\right )^{3/2}-\frac{1529}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{455}{32} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0429007, size = 80, normalized size = 0.87 \[ \frac{1}{96} \left (-2 \sqrt{3 x^2+2} \left (18 x^3-156 x^2+381 x-1469\right )-1365 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-4587 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2 + 3*x^2]*(-1469 + 381*x - 156*x^2 + 18*x^3) - 4587*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] - 1365*Sqrt[35]*Arc
Tanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/96

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Maple [A]  time = 0.006, size = 117, normalized size = 1.3 \begin{align*} -{\frac{x}{8} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{3\,x}{8}\sqrt{3\,{x}^{2}+2}}-{\frac{1529\,\sqrt{3}}{32}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{13}{12} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{117\,x}{16}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{455}{32}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{455\,\sqrt{35}}{32}{\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*x^2+2)^(3/2)/(3+2*x),x)

[Out]

-1/8*x*(3*x^2+2)^(3/2)-3/8*x*(3*x^2+2)^(1/2)-1529/32*arcsinh(1/2*x*6^(1/2))*3^(1/2)+13/12*(3*(x+3/2)^2-9*x-19/
4)^(3/2)-117/16*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)+455/32*(12*(x+3/2)^2-36*x-19)^(1/2)-455/32*35^(1/2)*arctanh(2/3
5*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))

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Maxima [A]  time = 1.49091, size = 126, normalized size = 1.37 \begin{align*} -\frac{1}{8} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{13}{12} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{123}{16} \, \sqrt{3 \, x^{2} + 2} x - \frac{1529}{32} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{455}{32} \, \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{455}{16} \, \sqrt{3 \, x^{2} + 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),x, algorithm="maxima")

[Out]

-1/8*(3*x^2 + 2)^(3/2)*x + 13/12*(3*x^2 + 2)^(3/2) - 123/16*sqrt(3*x^2 + 2)*x - 1529/32*sqrt(3)*arcsinh(1/2*sq
rt(6)*x) + 455/32*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 455/16*sqrt(3*x^2
+ 2)

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Fricas [A]  time = 2.23264, size = 296, normalized size = 3.22 \begin{align*} -\frac{1}{48} \,{\left (18 \, x^{3} - 156 \, x^{2} + 381 \, x - 1469\right )} \sqrt{3 \, x^{2} + 2} + \frac{1529}{64} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + \frac{455}{64} \, \sqrt{35} \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 ) \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),x, algorithm="fricas")

[Out]

-1/48*(18*x^3 - 156*x^2 + 381*x - 1469)*sqrt(3*x^2 + 2) + 1529/64*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^
2 - 1) + 455/64*sqrt(35)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{10 \sqrt{3 x^{2} + 2}}{2 x + 3}\, dx - \int \frac{2 x \sqrt{3 x^{2} + 2}}{2 x + 3}\, dx - \int - \frac{15 x^{2} \sqrt{3 x^{2} + 2}}{2 x + 3}\, dx - \int \frac{3 x^{3} \sqrt{3 x^{2} + 2}}{2 x + 3}\, dx \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),x)

[Out]

-Integral(-10*sqrt(3*x**2 + 2)/(2*x + 3), x) - Integral(2*x*sqrt(3*x**2 + 2)/(2*x + 3), x) - Integral(-15*x**2
*sqrt(3*x**2 + 2)/(2*x + 3), x) - Integral(3*x**3*sqrt(3*x**2 + 2)/(2*x + 3), x)

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Giac [A]  time = 1.22443, size = 157, normalized size = 1.71 \begin{align*} -\frac{1}{48} \,{\left (3 \,{\left (2 \,{\left (3 \, x - 26\right )} x + 127\right )} x - 1469\right )} \sqrt{3 \, x^{2} + 2} + \frac{1529}{32} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{455}{32} \, \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 ) \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),x, algorithm="giac")

[Out]

-1/48*(3*(2*(3*x - 26)*x + 127)*x - 1469)*sqrt(3*x^2 + 2) + 1529/32*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))
+ 455/32*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)))