3.2411 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{3+2 x} \, dx\)

Optimal. Leaf size=100 \[ \frac{1}{24} \sqrt{3 x^2+5 x+2} (73-6 x)-\frac{311 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{48 \sqrt{3}}+\frac{13}{8} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

((73 - 6*x)*Sqrt[2 + 5*x + 3*x^2])/24 - (311*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(48*Sqrt[3]
) + (13*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/8

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Rubi [A]  time = 0.0619984, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {814, 843, 621, 206, 724} \[ \frac{1}{24} \sqrt{3 x^2+5 x+2} (73-6 x)-\frac{311 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{48 \sqrt{3}}+\frac{13}{8} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((73 - 6*x)*Sqrt[2 + 5*x + 3*x^2])/24 - (311*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(48*Sqrt[3]
) + (13*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/8

Rule 814

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

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

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])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{3+2 x} \, dx &=\frac{1}{24} (73-6 x) \sqrt{2+5 x+3 x^2}-\frac{1}{48} \int \frac{543+622 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{1}{24} (73-6 x) \sqrt{2+5 x+3 x^2}-\frac{311}{48} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{65}{8} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{1}{24} (73-6 x) \sqrt{2+5 x+3 x^2}-\frac{311}{24} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )-\frac{65}{4} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{1}{24} (73-6 x) \sqrt{2+5 x+3 x^2}-\frac{311 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{48 \sqrt{3}}+\frac{13}{8} \sqrt{5} \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0443231, size = 93, normalized size = 0.93 \[ \frac{1}{144} \left (-6 \sqrt{3 x^2+5 x+2} (6 x-73)-234 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )-311 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(-73 + 6*x)*Sqrt[2 + 5*x + 3*x^2] - 234*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] - 31
1*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/144

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Maple [A]  time = 0.009, size = 127, normalized size = 1.3 \begin{align*} -{\frac{5+6\,x}{24}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{\sqrt{3}}{144}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{13}{8}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{13\,\sqrt{3}}{6}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }-{\frac{13\,\sqrt{5}}{8}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(5+6*x)*(3*x^2+5*x+2)^(1/2)+1/144*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+13/8*(12*(x+3/2)
^2-16*x-19)^(1/2)-13/6*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)-13/8*5^(1/2)*arctanh(2/5
*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))

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Maxima [A]  time = 1.52806, size = 134, normalized size = 1.34 \begin{align*} -\frac{1}{4} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{311}{144} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{13}{8} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{73}{24} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x),x, algorithm="maxima")

[Out]

-1/4*sqrt(3*x^2 + 5*x + 2)*x - 311/144*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 13/8*sqrt(5)*l
og(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 73/24*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.48285, size = 317, normalized size = 3.17 \begin{align*} -\frac{1}{24} \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x - 73\right )} + \frac{311}{288} \, \sqrt{3} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + \frac{13}{16} \, \sqrt{5} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{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+5*x+2)^(1/2)/(3+2*x),x, algorithm="fricas")

[Out]

-1/24*sqrt(3*x^2 + 5*x + 2)*(6*x - 73) + 311/288*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x
^2 + 120*x + 49) + 13/16*sqrt(5)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(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} + 5 x + 2}}{2 x + 3}\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(x*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x)

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Giac [A]  time = 1.21363, size = 170, normalized size = 1.7 \begin{align*} -\frac{1}{24} \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x - 73\right )} + \frac{13}{8} \, \sqrt{5} \log \left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{311}{144} \, \sqrt{3} \log \left ({\left | -6 \, \sqrt{3} x - 5 \, \sqrt{3} + 6 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x),x, algorithm="giac")

[Out]

-1/24*sqrt(3*x^2 + 5*x + 2)*(6*x - 73) + 13/8*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*
x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) + 311/144*sqrt(3)*log(abs
(-6*sqrt(3)*x - 5*sqrt(3) + 6*sqrt(3*x^2 + 5*x + 2)))