3.1363 \(\int \frac{(5-x) \sqrt{2+3 x^2}}{(3+2 x)^3} \, dx\)

Optimal. Leaf size=79 \[ \frac{\sqrt{3 x^2+2} (187 x+53)}{140 (2 x+3)^2}-\frac{471 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{280 \sqrt{35}}-\frac{1}{8} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

((53 + 187*x)*Sqrt[2 + 3*x^2])/(140*(3 + 2*x)^2) - (Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/8 - (471*ArcTanh[(4 - 9*x)/(
Sqrt[35]*Sqrt[2 + 3*x^2])])/(280*Sqrt[35])

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Rubi [A]  time = 0.0421219, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {811, 844, 215, 725, 206} \[ \frac{\sqrt{3 x^2+2} (187 x+53)}{140 (2 x+3)^2}-\frac{471 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{280 \sqrt{35}}-\frac{1}{8} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((53 + 187*x)*Sqrt[2 + 3*x^2])/(140*(3 + 2*x)^2) - (Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/8 - (471*ArcTanh[(4 - 9*x)/(
Sqrt[35]*Sqrt[2 + 3*x^2])])/(280*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 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) \sqrt{2+3 x^2}}{(3+2 x)^3} \, dx &=\frac{(53+187 x) \sqrt{2+3 x^2}}{140 (3+2 x)^2}-\frac{1}{560} \int \frac{-312+420 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=\frac{(53+187 x) \sqrt{2+3 x^2}}{140 (3+2 x)^2}-\frac{3}{8} \int \frac{1}{\sqrt{2+3 x^2}} \, dx+\frac{471}{280} \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=\frac{(53+187 x) \sqrt{2+3 x^2}}{140 (3+2 x)^2}-\frac{1}{8} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{471}{280} \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )\\ &=\frac{(53+187 x) \sqrt{2+3 x^2}}{140 (3+2 x)^2}-\frac{1}{8} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{471 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{280 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.101637, size = 80, normalized size = 1.01 \[ \frac{\frac{70 (187 x+53) \sqrt{3 x^2+2}}{(2 x+3)^2}-471 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{9800}-\frac{1}{8} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/8 + ((70*(53 + 187*x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^2 - 471*Sqrt[35]*ArcTanh[(4 -
9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/9800

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Maple [A]  time = 0.009, size = 119, normalized size = 1.5 \begin{align*} -{\frac{47}{4900} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{471}{9800}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{\sqrt{3}}{8}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{471\,\sqrt{35}}{9800}{\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{141\,x}{4900}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{13}{280} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{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)^(1/2)/(3+2*x)^3,x)

[Out]

-47/4900/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(3/2)+471/9800*(12*(x+3/2)^2-36*x-19)^(1/2)-1/8*arcsinh(1/2*x*6^(1/2))
*3^(1/2)-471/9800*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))+141/4900*x*(3*(x+3/2)^2
-9*x-19/4)^(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 = 1.49043, size = 134, normalized size = 1.7 \begin{align*} -\frac{1}{8} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{471}{9800} \, \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{39}{280} \, \sqrt{3 \, x^{2} + 2} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{70 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{47 \, \sqrt{3 \, x^{2} + 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)^(1/2)/(3+2*x)^3,x, algorithm="maxima")

[Out]

-1/8*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 471/9800*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2
*x + 3)) + 39/280*sqrt(3*x^2 + 2) - 13/70*(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9) - 47/280*sqrt(3*x^2 + 2)/(2*x +
3)

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Fricas [B]  time = 2.89206, size = 347, normalized size = 4.39 \begin{align*} \frac{1225 \, \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 ) + 471 \, \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 \, \sqrt{3 \, x^{2} + 2}{\left (187 \, x + 53\right )}}{19600 \,{\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)^(1/2)/(3+2*x)^3,x, algorithm="fricas")

[Out]

1/19600*(1225*sqrt(3)*(4*x^2 + 12*x + 9)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 471*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*sqrt(3*x^2 +
2)*(187*x + 53))/(4*x^2 + 12*x + 9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{5 \sqrt{3 x^{2} + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx - \int \frac{x \sqrt{3 x^{2} + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+2)**(1/2)/(3+2*x)**3,x)

[Out]

-Integral(-5*sqrt(3*x**2 + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(x*sqrt(3*x**2 + 2)/(8*x**3 + 36*x*
*2 + 54*x + 27), x)

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Giac [B]  time = 1.32107, size = 277, normalized size = 3.51 \begin{align*} \frac{1}{8} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{471}{9800} \, \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{3048 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 4301 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 7368 \, \sqrt{3} x + 1496 \, \sqrt{3} + 7368 \, \sqrt{3 \, x^{2} + 2}}{560 \,{\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)^(1/2)/(3+2*x)^3,x, algorithm="giac")

[Out]

1/8*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 471/9800*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))) + 1/560*(3048*(sqrt(3)*x - sqr
t(3*x^2 + 2))^3 + 4301*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 7368*sqrt(3)*x + 1496*sqrt(3) + 7368*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