3.2421 \(\int (5-x) (3+2 x) (2+5 x+3 x^2)^{3/2} \, dx\)

Optimal. Leaf size=108 \[ \frac{1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac{839 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac{839 (6 x+5) \sqrt{3 x^2+5 x+2}}{20736}+\frac{839 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{41472 \sqrt{3}} \]

[Out]

(-839*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/20736 + (839*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/2592 + ((161 - 30*x)*(2
+ 5*x + 3*x^2)^(5/2))/270 + (839*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(41472*Sqrt[3])

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Rubi [A]  time = 0.0391732, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {779, 612, 621, 206} \[ \frac{1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac{839 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac{839 (6 x+5) \sqrt{3 x^2+5 x+2}}{20736}+\frac{839 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{41472 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-839*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/20736 + (839*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/2592 + ((161 - 30*x)*(2
+ 5*x + 3*x^2)^(5/2))/270 + (839*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(41472*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) \left (2+5 x+3 x^2\right )^{3/2} \, dx &=\frac{1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac{839}{108} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac{839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac{1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac{839 \int \sqrt{2+5 x+3 x^2} \, dx}{1728}\\ &=-\frac{839 (5+6 x) \sqrt{2+5 x+3 x^2}}{20736}+\frac{839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac{1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac{839 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{41472}\\ &=-\frac{839 (5+6 x) \sqrt{2+5 x+3 x^2}}{20736}+\frac{839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac{1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac{839 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{20736}\\ &=-\frac{839 (5+6 x) \sqrt{2+5 x+3 x^2}}{20736}+\frac{839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac{1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac{839 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{41472 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0400528, size = 77, normalized size = 0.71 \[ \frac{4195 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )-6 \sqrt{3 x^2+5 x+2} \left (103680 x^5-210816 x^4-2032560 x^3-3567288 x^2-2406950 x-561921\right )}{622080} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-561921 - 2406950*x - 3567288*x^2 - 2032560*x^3 - 210816*x^4 + 103680*x^5) + 4195*S
qrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/622080

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Maple [A]  time = 0.006, size = 98, normalized size = 0.9 \begin{align*} -{\frac{x}{9} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{161}{270} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{4195+5034\,x}{2592} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{4195+5034\,x}{20736}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{839\,\sqrt{3}}{124416}\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)^(3/2),x)

[Out]

-1/9*x*(3*x^2+5*x+2)^(5/2)+161/270*(3*x^2+5*x+2)^(5/2)+839/2592*(5+6*x)*(3*x^2+5*x+2)^(3/2)-839/20736*(5+6*x)*
(3*x^2+5*x+2)^(1/2)+839/124416*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 = 1.47674, size = 157, normalized size = 1.45 \begin{align*} -\frac{1}{9} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{161}{270} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{839}{432} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{4195}{2592} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{839}{3456} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{839}{124416} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{4195}{20736} \, \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)^(3/2),x, algorithm="maxima")

[Out]

-1/9*(3*x^2 + 5*x + 2)^(5/2)*x + 161/270*(3*x^2 + 5*x + 2)^(5/2) + 839/432*(3*x^2 + 5*x + 2)^(3/2)*x + 4195/25
92*(3*x^2 + 5*x + 2)^(3/2) - 839/3456*sqrt(3*x^2 + 5*x + 2)*x + 839/124416*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 +
5*x + 2) + 6*x + 5) - 4195/20736*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.36267, size = 267, normalized size = 2.47 \begin{align*} -\frac{1}{103680} \,{\left (103680 \, x^{5} - 210816 \, x^{4} - 2032560 \, x^{3} - 3567288 \, x^{2} - 2406950 \, x - 561921\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{839}{248832} \, \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)^(3/2),x, algorithm="fricas")

[Out]

-1/103680*(103680*x^5 - 210816*x^4 - 2032560*x^3 - 3567288*x^2 - 2406950*x - 561921)*sqrt(3*x^2 + 5*x + 2) + 8
39/248832*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 - 89 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 76 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 11 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 6 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 30 \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)**(3/2),x)

[Out]

-Integral(-89*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-76*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-11*x**3*
sqrt(3*x**2 + 5*x + 2), x) - Integral(6*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-30*sqrt(3*x**2 + 5*x + 2),
x)

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Giac [A]  time = 1.11094, size = 100, normalized size = 0.93 \begin{align*} -\frac{1}{103680} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \,{\left (30 \, x - 61\right )} x - 4705\right )} x - 148637\right )} x - 1203475\right )} x - 561921\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{839}{124416} \, \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)^(3/2),x, algorithm="giac")

[Out]

-1/103680*(2*(12*(18*(8*(30*x - 61)*x - 4705)*x - 148637)*x - 1203475)*x - 561921)*sqrt(3*x^2 + 5*x + 2) - 839
/124416*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))