3.2435 \(\int (5-x) (3+2 x) (2+5 x+3 x^2)^{5/2} \, dx\)

Optimal. Leaf size=131 \[ \frac{1}{168} (71-14 x) \left (3 x^2+5 x+2\right )^{7/2}+\frac{373 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{1728}-\frac{1865 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{82944}+\frac{1865 (6 x+5) \sqrt{3 x^2+5 x+2}}{663552}-\frac{1865 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1327104 \sqrt{3}} \]

[Out]

(1865*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/663552 - (1865*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/82944 + (373*(5 + 6*x
)*(2 + 5*x + 3*x^2)^(5/2))/1728 + ((71 - 14*x)*(2 + 5*x + 3*x^2)^(7/2))/168 - (1865*ArcTanh[(5 + 6*x)/(2*Sqrt[
3]*Sqrt[2 + 5*x + 3*x^2])])/(1327104*Sqrt[3])

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Rubi [A]  time = 0.0455891, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {779, 612, 621, 206} \[ \frac{1}{168} (71-14 x) \left (3 x^2+5 x+2\right )^{7/2}+\frac{373 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{1728}-\frac{1865 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{82944}+\frac{1865 (6 x+5) \sqrt{3 x^2+5 x+2}}{663552}-\frac{1865 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1327104 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1865*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/663552 - (1865*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/82944 + (373*(5 + 6*x
)*(2 + 5*x + 3*x^2)^(5/2))/1728 + ((71 - 14*x)*(2 + 5*x + 3*x^2)^(7/2))/168 - (1865*ArcTanh[(5 + 6*x)/(2*Sqrt[
3]*Sqrt[2 + 5*x + 3*x^2])])/(1327104*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 )^{5/2} \, dx &=\frac{1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}+\frac{373}{48} \int \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac{373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac{1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{1865 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{3456}\\ &=-\frac{1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac{373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac{1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}+\frac{1865 \int \sqrt{2+5 x+3 x^2} \, dx}{55296}\\ &=\frac{1865 (5+6 x) \sqrt{2+5 x+3 x^2}}{663552}-\frac{1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac{373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac{1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{1865 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{1327104}\\ &=\frac{1865 (5+6 x) \sqrt{2+5 x+3 x^2}}{663552}-\frac{1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac{373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac{1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{1865 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{663552}\\ &=\frac{1865 (5+6 x) \sqrt{2+5 x+3 x^2}}{663552}-\frac{1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac{373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac{1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{1865 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{1327104 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0855963, size = 101, normalized size = 0.77 \[ \frac{373 \left (6 \sqrt{3 x^2+5 x+2} \left (20736 x^5+86400 x^4+142128 x^3+115320 x^2+46166 x+7305\right )-5 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right )}{3981312}-\frac{1}{168} (14 x-71) \left (3 x^2+5 x+2\right )^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-71 + 14*x)*(2 + 5*x + 3*x^2)^(7/2))/168 + (373*(6*Sqrt[2 + 5*x + 3*x^2]*(7305 + 46166*x + 115320*x^2 + 142
128*x^3 + 86400*x^4 + 20736*x^5) - 5*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])]))/3981312

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Maple [A]  time = 0.006, size = 117, normalized size = 0.9 \begin{align*} -{\frac{x}{12} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{71}{168} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{1865+2238\,x}{1728} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{9325+11190\,x}{82944} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{9325+11190\,x}{663552}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{1865\,\sqrt{3}}{3981312}\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)^(5/2),x)

[Out]

-1/12*x*(3*x^2+5*x+2)^(7/2)+71/168*(3*x^2+5*x+2)^(7/2)+373/1728*(5+6*x)*(3*x^2+5*x+2)^(5/2)-1865/82944*(5+6*x)
*(3*x^2+5*x+2)^(3/2)+1865/663552*(5+6*x)*(3*x^2+5*x+2)^(1/2)-1865/3981312*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 = 1.51696, size = 196, normalized size = 1.5 \begin{align*} -\frac{1}{12} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{71}{168} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} + \frac{373}{288} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{1865}{1728} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{1865}{13824} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{9325}{82944} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{1865}{110592} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{1865}{3981312} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{9325}{663552} \, \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)^(5/2),x, algorithm="maxima")

[Out]

-1/12*(3*x^2 + 5*x + 2)^(7/2)*x + 71/168*(3*x^2 + 5*x + 2)^(7/2) + 373/288*(3*x^2 + 5*x + 2)^(5/2)*x + 1865/17
28*(3*x^2 + 5*x + 2)^(5/2) - 1865/13824*(3*x^2 + 5*x + 2)^(3/2)*x - 9325/82944*(3*x^2 + 5*x + 2)^(3/2) + 1865/
110592*sqrt(3*x^2 + 5*x + 2)*x - 1865/3981312*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 9325/66
3552*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.41356, size = 329, normalized size = 2.51 \begin{align*} -\frac{1}{4644864} \,{\left (10450944 \, x^{7} - 746496 \, x^{6} - 211154688 \, x^{5} - 655212672 \, x^{4} - 897818256 \, x^{3} - 642995688 \, x^{2} - 235223330 \, x - 34777419\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{1865}{7962624} \, \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)^(5/2),x, algorithm="fricas")

[Out]

-1/4644864*(10450944*x^7 - 746496*x^6 - 211154688*x^5 - 655212672*x^4 - 897818256*x^3 - 642995688*x^2 - 235223
330*x - 34777419)*sqrt(3*x^2 + 5*x + 2) + 1865/7962624*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 - 328 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 687 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 669 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 271 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 3 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 18 x^{6} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 60 \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)**(5/2),x)

[Out]

-Integral(-328*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-687*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-669*x*
*3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-271*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3*x**5*sqrt(3*x**2 +
5*x + 2), x) - Integral(18*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(-60*sqrt(3*x**2 + 5*x + 2), x)

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Giac [A]  time = 1.1177, size = 113, normalized size = 0.86 \begin{align*} -\frac{1}{4644864} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (14 \, x - 1\right )} x - 10183\right )} x - 189587\right )} x - 2078283\right )} x - 26791487\right )} x - 117611665\right )} x - 34777419\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{1865}{3981312} \, \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)^(5/2),x, algorithm="giac")

[Out]

-1/4644864*(2*(12*(18*(8*(6*(36*(14*x - 1)*x - 10183)*x - 189587)*x - 2078283)*x - 26791487)*x - 117611665)*x
- 34777419)*sqrt(3*x^2 + 5*x + 2) + 1865/3981312*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)
) - 5))