3.2448 \(\int (5-x) (3+2 x)^3 (2+5 x+3 x^2)^{7/2} \, dx\)

Optimal. Leaf size=204 \[ -\frac{1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}+\frac{34}{99} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{(390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}}{320760}+\frac{91087 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{311040}-\frac{637609 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{22394880}+\frac{637609 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{214990848}-\frac{637609 (6 x+5) \sqrt{3 x^2+5 x+2}}{1719926784}+\frac{637609 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{3439853568 \sqrt{3}} \]

[Out]

(-637609*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/1719926784 + (637609*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/214990848 -
(637609*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/22394880 + (91087*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/311040 + (34*(
3 + 2*x)^2*(2 + 5*x + 3*x^2)^(9/2))/99 - ((3 + 2*x)^3*(2 + 5*x + 3*x^2)^(9/2))/36 + ((863825 + 390798*x)*(2 +
5*x + 3*x^2)^(9/2))/320760 + (637609*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(3439853568*Sqrt[3]
)

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Rubi [A]  time = 0.106524, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {832, 779, 612, 621, 206} \[ -\frac{1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}+\frac{34}{99} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{(390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}}{320760}+\frac{91087 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{311040}-\frac{637609 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{22394880}+\frac{637609 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{214990848}-\frac{637609 (6 x+5) \sqrt{3 x^2+5 x+2}}{1719926784}+\frac{637609 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{3439853568 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(7/2),x]

[Out]

(-637609*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/1719926784 + (637609*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/214990848 -
(637609*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/22394880 + (91087*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/311040 + (34*(
3 + 2*x)^2*(2 + 5*x + 3*x^2)^(9/2))/99 - ((3 + 2*x)^3*(2 + 5*x + 3*x^2)^(9/2))/36 + ((863825 + 390798*x)*(2 +
5*x + 3*x^2)^(9/2))/320760 + (637609*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(3439853568*Sqrt[3]
)

Rule 832

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

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)^3 \left (2+5 x+3 x^2\right )^{7/2} \, dx &=-\frac{1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac{1}{36} \int (3+2 x)^2 \left (\frac{1239}{2}+408 x\right ) \left (2+5 x+3 x^2\right )^{7/2} \, dx\\ &=\frac{34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac{1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac{\int (3+2 x) \left (\frac{61053}{2}+21711 x\right ) \left (2+5 x+3 x^2\right )^{7/2} \, dx}{1188}\\ &=\frac{34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac{1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}+\frac{91087 \int \left (2+5 x+3 x^2\right )^{7/2} \, dx}{6480}\\ &=\frac{91087 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{311040}+\frac{34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac{1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}-\frac{637609 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{622080}\\ &=-\frac{637609 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{22394880}+\frac{91087 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{311040}+\frac{34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac{1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}+\frac{637609 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{8957952}\\ &=\frac{637609 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{214990848}-\frac{637609 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{22394880}+\frac{91087 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{311040}+\frac{34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac{1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}-\frac{637609 \int \sqrt{2+5 x+3 x^2} \, dx}{143327232}\\ &=-\frac{637609 (5+6 x) \sqrt{2+5 x+3 x^2}}{1719926784}+\frac{637609 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{214990848}-\frac{637609 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{22394880}+\frac{91087 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{311040}+\frac{34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac{1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}+\frac{637609 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{3439853568}\\ &=-\frac{637609 (5+6 x) \sqrt{2+5 x+3 x^2}}{1719926784}+\frac{637609 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{214990848}-\frac{637609 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{22394880}+\frac{91087 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{311040}+\frac{34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac{1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}+\frac{637609 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{1719926784}\\ &=-\frac{637609 (5+6 x) \sqrt{2+5 x+3 x^2}}{1719926784}+\frac{637609 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{214990848}-\frac{637609 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{22394880}+\frac{91087 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{311040}+\frac{34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac{1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}+\frac{637609 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{3439853568 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.160077, size = 163, normalized size = 0.8 \[ \frac{1}{36} \left (-(2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}+\frac{136}{11} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{(390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}}{8910}+\frac{91087 \left (6 \sqrt{3 x^2+5 x+2} \left (4478976 x^7+26127360 x^6+64800000 x^5+88560000 x^4+72023472 x^3+34858680 x^2+9298342 x+1054785\right )+35 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right )}{1433272320}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(7/2),x]

[Out]

((136*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(9/2))/11 - (3 + 2*x)^3*(2 + 5*x + 3*x^2)^(9/2) + ((863825 + 390798*x)*(2
+ 5*x + 3*x^2)^(9/2))/8910 + (91087*(6*Sqrt[2 + 5*x + 3*x^2]*(1054785 + 9298342*x + 34858680*x^2 + 72023472*x^
3 + 88560000*x^4 + 64800000*x^5 + 26127360*x^6 + 4478976*x^7) + 35*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x
+ 9*x^2])]))/1433272320)/36

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 170, normalized size = 0.8 \begin{align*} -{\frac{2\,{x}^{3}}{9} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{37\,{x}^{2}}{99} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{22807\,x}{5940} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{455435+546522\,x}{311040} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{3188045+3825654\,x}{22394880} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{3188045+3825654\,x}{214990848} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{637609\,\sqrt{3}}{10319560704}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }-{\frac{3188045+3825654\,x}{1719926784}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{322939}{64152} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3+2*x)^3*(3*x^2+5*x+2)^(7/2),x)

[Out]

-2/9*x^3*(3*x^2+5*x+2)^(9/2)+37/99*x^2*(3*x^2+5*x+2)^(9/2)+22807/5940*x*(3*x^2+5*x+2)^(9/2)+91087/311040*(5+6*
x)*(3*x^2+5*x+2)^(7/2)-637609/22394880*(5+6*x)*(3*x^2+5*x+2)^(5/2)+637609/214990848*(5+6*x)*(3*x^2+5*x+2)^(3/2
)+637609/10319560704*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-637609/1719926784*(5+6*x)*(3*x^2+5*
x+2)^(1/2)+322939/64152*(3*x^2+5*x+2)^(9/2)

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Maxima [A]  time = 1.98858, size = 281, normalized size = 1.38 \begin{align*} -\frac{2}{9} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x^{3} + \frac{37}{99} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x^{2} + \frac{22807}{5940} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x + \frac{322939}{64152} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} + \frac{91087}{51840} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{91087}{62208} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{637609}{3732480} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{637609}{4478976} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{637609}{35831808} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{3188045}{214990848} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{637609}{286654464} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{637609}{10319560704} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{3188045}{1719926784} \, \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*(3*x^2+5*x+2)^(7/2),x, algorithm="maxima")

[Out]

-2/9*(3*x^2 + 5*x + 2)^(9/2)*x^3 + 37/99*(3*x^2 + 5*x + 2)^(9/2)*x^2 + 22807/5940*(3*x^2 + 5*x + 2)^(9/2)*x +
322939/64152*(3*x^2 + 5*x + 2)^(9/2) + 91087/51840*(3*x^2 + 5*x + 2)^(7/2)*x + 91087/62208*(3*x^2 + 5*x + 2)^(
7/2) - 637609/3732480*(3*x^2 + 5*x + 2)^(5/2)*x - 637609/4478976*(3*x^2 + 5*x + 2)^(5/2) + 637609/35831808*(3*
x^2 + 5*x + 2)^(3/2)*x + 3188045/214990848*(3*x^2 + 5*x + 2)^(3/2) - 637609/286654464*sqrt(3*x^2 + 5*x + 2)*x
+ 637609/10319560704*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 3188045/1719926784*sqrt(3*x^2 +
5*x + 2)

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Fricas [A]  time = 1.51146, size = 529, normalized size = 2.59 \begin{align*} -\frac{1}{94595973120} \,{\left (1702727516160 \, x^{11} + 8487838679040 \, x^{10} - 15591566278656 \, x^{9} - 235832896880640 \, x^{8} - 866110416795648 \, x^{7} - 1766184385305600 \, x^{6} - 2298912734198016 \, x^{5} - 1992318117275520 \, x^{4} - 1149328734822000 \, x^{3} - 425035984788120 \, x^{2} - 91318722047870 \, x - 8675936123685\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{637609}{20639121408} \, \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*(3*x^2+5*x+2)^(7/2),x, algorithm="fricas")

[Out]

-1/94595973120*(1702727516160*x^11 + 8487838679040*x^10 - 15591566278656*x^9 - 235832896880640*x^8 - 866110416
795648*x^7 - 1766184385305600*x^6 - 2298912734198016*x^5 - 1992318117275520*x^4 - 1149328734822000*x^3 - 42503
5984788120*x^2 - 91318722047870*x - 8675936123685)*sqrt(3*x^2 + 5*x + 2) + 637609/20639121408*sqrt(3)*log(4*sq
rt(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 - 10044 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 40698 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 93965 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 135392 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 124716 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 71336 x^{6} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 22247 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 1710 x^{8} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 972 x^{9} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 216 x^{10} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 1080 \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*(3*x**2+5*x+2)**(7/2),x)

[Out]

-Integral(-10044*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-40698*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-93
965*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-135392*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-124716*x**5
*sqrt(3*x**2 + 5*x + 2), x) - Integral(-71336*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(-22247*x**7*sqrt(3*x*
*2 + 5*x + 2), x) - Integral(-1710*x**8*sqrt(3*x**2 + 5*x + 2), x) - Integral(972*x**9*sqrt(3*x**2 + 5*x + 2),
x) - Integral(216*x**10*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1080*sqrt(3*x**2 + 5*x + 2), x)

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Giac [A]  time = 1.28865, size = 140, normalized size = 0.69 \begin{align*} -\frac{1}{94595973120} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (2 \,{\left (48 \,{\left (54 \,{\left (20 \,{\left (66 \, x + 329\right )} x - 12087\right )} x - 9872495\right )} x - 1740351757\right )} x - 7097898925\right )} x - 332597328443\right )} x - 1729442810135\right )} x - 7981449547375\right )} x - 17709832699505\right )} x - 45659361023935\right )} x - 8675936123685\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{637609}{10319560704} \, \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*(3*x^2+5*x+2)^(7/2),x, algorithm="giac")

[Out]

-1/94595973120*(2*(12*(6*(8*(6*(36*(2*(48*(54*(20*(66*x + 329)*x - 12087)*x - 9872495)*x - 1740351757)*x - 709
7898925)*x - 332597328443)*x - 1729442810135)*x - 7981449547375)*x - 17709832699505)*x - 45659361023935)*x - 8
675936123685)*sqrt(3*x^2 + 5*x + 2) - 637609/10319560704*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2)) - 5))