3.2409 \(\int (5-x) (3+2 x) \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=85 \[ \frac{1}{108} (109-18 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac{559}{864} (6 x+5) \sqrt{3 x^2+5 x+2}-\frac{559 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1728 \sqrt{3}} \]

[Out]

(559*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/864 + ((109 - 18*x)*(2 + 5*x + 3*x^2)^(3/2))/108 - (559*ArcTanh[(5 + 6*x
)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1728*Sqrt[3])

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Rubi [A]  time = 0.0268036, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {779, 612, 621, 206} \[ \frac{1}{108} (109-18 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac{559}{864} (6 x+5) \sqrt{3 x^2+5 x+2}-\frac{559 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1728 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(559*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/864 + ((109 - 18*x)*(2 + 5*x + 3*x^2)^(3/2))/108 - (559*ArcTanh[(5 + 6*x
)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1728*Sqrt[3])

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 612

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

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])

Rubi steps

\begin{align*} \int (5-x) (3+2 x) \sqrt{2+5 x+3 x^2} \, dx &=\frac{1}{108} (109-18 x) \left (2+5 x+3 x^2\right )^{3/2}+\frac{559}{72} \int \sqrt{2+5 x+3 x^2} \, dx\\ &=\frac{559}{864} (5+6 x) \sqrt{2+5 x+3 x^2}+\frac{1}{108} (109-18 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{559 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{1728}\\ &=\frac{559}{864} (5+6 x) \sqrt{2+5 x+3 x^2}+\frac{1}{108} (109-18 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{559}{864} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{559}{864} (5+6 x) \sqrt{2+5 x+3 x^2}+\frac{1}{108} (109-18 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{559 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{1728 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0290742, size = 67, normalized size = 0.79 \[ \frac{-6 \sqrt{3 x^2+5 x+2} \left (432 x^3-1896 x^2-7426 x-4539\right )-559 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{5184} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-4539 - 7426*x - 1896*x^2 + 432*x^3) - 559*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15
*x + 9*x^2])])/5184

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Maple [A]  time = 0.006, size = 79, normalized size = 0.9 \begin{align*} -{\frac{x}{6} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{109}{108} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{2795+3354\,x}{864}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{559\,\sqrt{3}}{5184}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*x*(3*x^2+5*x+2)^(3/2)+109/108*(3*x^2+5*x+2)^(3/2)+559/864*(5+6*x)*(3*x^2+5*x+2)^(1/2)-559/5184*ln(1/3*(5/
2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)

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Maxima [A]  time = 1.53841, size = 117, normalized size = 1.38 \begin{align*} -\frac{1}{6} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{109}{108} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{559}{144} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{559}{5184} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{2795}{864} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*x^2 + 5*x + 2)^(3/2)*x + 109/108*(3*x^2 + 5*x + 2)^(3/2) + 559/144*sqrt(3*x^2 + 5*x + 2)*x - 559/5184*
sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 2795/864*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.43788, size = 212, normalized size = 2.49 \begin{align*} -\frac{1}{864} \,{\left (432 \, x^{3} - 1896 \, x^{2} - 7426 \, x - 4539\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{559}{10368} \, \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/864*(432*x^3 - 1896*x^2 - 7426*x - 4539)*sqrt(3*x^2 + 5*x + 2) + 559/10368*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^
2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - 7 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 2 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 15 \sqrt{3 x^{2} + 5 x + 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-7*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(2*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-15*sqrt(3*x
**2 + 5*x + 2), x)

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Giac [A]  time = 1.18511, size = 86, normalized size = 1.01 \begin{align*} -\frac{1}{864} \,{\left (2 \,{\left (12 \,{\left (18 \, x - 79\right )} x - 3713\right )} x - 4539\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{559}{5184} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/864*(2*(12*(18*x - 79)*x - 3713)*x - 4539)*sqrt(3*x^2 + 5*x + 2) + 559/5184*sqrt(3)*log(abs(-2*sqrt(3)*(sqr
t(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))