3.2432 \(\int (5-x) (3+2 x)^4 (2+5 x+3 x^2)^{5/2} \, dx\)

Optimal. Leaf size=206 \[ -\frac{1}{33} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^4+\frac{41}{110} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^3+\frac{3298 \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^2}{4455}+\frac{(3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}}{1496880}+\frac{249299 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac{249299 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac{249299 (6 x+5) \sqrt{3 x^2+5 x+2}}{35831808}-\frac{249299 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{71663616 \sqrt{3}} \]

[Out]

(249299*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/35831808 - (249299*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/4478976 + (2492
99*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/466560 + (3298*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2))/4455 + (41*(3 + 2*x)
^3*(2 + 5*x + 3*x^2)^(7/2))/110 - ((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(7/2))/33 + ((7405817 + 3365726*x)*(2 + 5*x +
3*x^2)^(7/2))/1496880 - (249299*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(71663616*Sqrt[3])

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Rubi [A]  time = 0.118378, antiderivative size = 206, 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} \[ -\frac{1}{33} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^4+\frac{41}{110} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^3+\frac{3298 \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^2}{4455}+\frac{(3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}}{1496880}+\frac{249299 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac{249299 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac{249299 (6 x+5) \sqrt{3 x^2+5 x+2}}{35831808}-\frac{249299 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{71663616 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)*(3 + 2*x)^4*(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(249299*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/35831808 - (249299*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/4478976 + (2492
99*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/466560 + (3298*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2))/4455 + (41*(3 + 2*x)
^3*(2 + 5*x + 3*x^2)^(7/2))/110 - ((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(7/2))/33 + ((7405817 + 3365726*x)*(2 + 5*x +
3*x^2)^(7/2))/1496880 - (249299*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(71663616*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)^4 \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac{1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac{1}{33} \int (3+2 x)^3 \left (\frac{1127}{2}+369 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac{41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac{1}{990} \int (3+2 x)^2 \left (\frac{53829}{2}+19788 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac{3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac{41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac{\int (3+2 x) \left (\frac{1965801}{2}+721227 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx}{26730}\\ &=\frac{3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac{41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}+\frac{249299 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{12960}\\ &=\frac{249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac{3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac{41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}-\frac{249299 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{186624}\\ &=-\frac{249299 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac{249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac{3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac{41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}+\frac{249299 \int \sqrt{2+5 x+3 x^2} \, dx}{2985984}\\ &=\frac{249299 (5+6 x) \sqrt{2+5 x+3 x^2}}{35831808}-\frac{249299 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac{249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac{3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac{41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}-\frac{249299 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{71663616}\\ &=\frac{249299 (5+6 x) \sqrt{2+5 x+3 x^2}}{35831808}-\frac{249299 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac{249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac{3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac{41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}-\frac{249299 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{35831808}\\ &=\frac{249299 (5+6 x) \sqrt{2+5 x+3 x^2}}{35831808}-\frac{249299 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac{249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac{3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac{41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac{(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}-\frac{249299 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{71663616 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0755306, size = 102, normalized size = 0.5 \[ \frac{-6 \sqrt{3 x^2+5 x+2} \left (180592312320 x^{10}[phone] x^9-1932170526720 x^8-25759323039744 x^7-90095929758720 x^6-172473366866688 x^5-204855126595200 x^4-155155370878800 x^3-73069860056520 x^2-19521700361210 x-2261297826735\right )-95980115 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{82771476480} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)*(3 + 2*x)^4*(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-2261297826735 - 19521700361210*x - 73069860056520*x^2 - 155155370878800*x^3 - 2048
55126595200*x^4 - 172473366866688*x^5 - 90095929758720*x^6 - 25759323039744*x^7 - 1932170526720*x^8 + 87587271
4752*x^9 + 180592312320*x^10) - 95980115*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/82771476480

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Maple [A]  time = 0.014, size = 168, normalized size = 0.8 \begin{align*} -{\frac{16\,{x}^{4}}{33} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{4\,{x}^{3}}{55} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{8762\,{x}^{2}}{891} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{2642401\,x}{106920} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{1246495+1495794\,x}{466560} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{1246495+1495794\,x}{4478976} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{249299\,\sqrt{3}}{214990848}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{1246495+1495794\,x}{35831808}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{5753773}{299376} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3+2*x)^4*(3*x^2+5*x+2)^(5/2),x)

[Out]

-16/33*x^4*(3*x^2+5*x+2)^(7/2)+4/55*x^3*(3*x^2+5*x+2)^(7/2)+8762/891*x^2*(3*x^2+5*x+2)^(7/2)+2642401/106920*x*
(3*x^2+5*x+2)^(7/2)+249299/466560*(5+6*x)*(3*x^2+5*x+2)^(5/2)-249299/4478976*(5+6*x)*(3*x^2+5*x+2)^(3/2)-24929
9/214990848*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+249299/35831808*(5+6*x)*(3*x^2+5*x+2)^(1/2)+
5753773/299376*(3*x^2+5*x+2)^(7/2)

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Maxima [A]  time = 1.53994, size = 265, normalized size = 1.29 \begin{align*} -\frac{16}{33} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x^{4} + \frac{4}{55} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x^{3} + \frac{8762}{891} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x^{2} + \frac{2642401}{106920} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{5753773}{299376} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} + \frac{249299}{77760} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{249299}{93312} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{249299}{746496} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{1246495}{4478976} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{249299}{5971968} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{249299}{214990848} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{1246495}{35831808} \, \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)^4*(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")

[Out]

-16/33*(3*x^2 + 5*x + 2)^(7/2)*x^4 + 4/55*(3*x^2 + 5*x + 2)^(7/2)*x^3 + 8762/891*(3*x^2 + 5*x + 2)^(7/2)*x^2 +
2642401/106920*(3*x^2 + 5*x + 2)^(7/2)*x + 5753773/299376*(3*x^2 + 5*x + 2)^(7/2) + 249299/77760*(3*x^2 + 5*x
+ 2)^(5/2)*x + 249299/93312*(3*x^2 + 5*x + 2)^(5/2) - 249299/746496*(3*x^2 + 5*x + 2)^(3/2)*x - 1246495/44789
76*(3*x^2 + 5*x + 2)^(3/2) + 249299/5971968*sqrt(3*x^2 + 5*x + 2)*x - 249299/214990848*sqrt(3)*log(2*sqrt(3)*s
qrt(3*x^2 + 5*x + 2) + 6*x + 5) + 1246495/35831808*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.60502, size = 483, normalized size = 2.34 \begin{align*} -\frac{1}{13795246080} \,{\left (180592312320 \, x^{10} + 875872714752 \, x^{9} - 1932170526720 \, x^{8} - 25759323039744 \, x^{7} - 90095929758720 \, x^{6} - 172473366866688 \, x^{5} - 204855126595200 \, x^{4} - 155155370878800 \, x^{3} - 73069860056520 \, x^{2} - 19521700361210 \, x - 2261297826735\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{249299}{429981696} \, \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)^4*(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")

[Out]

-1/13795246080*(180592312320*x^10 + 875872714752*x^9 - 1932170526720*x^8 - 25759323039744*x^7 - 90095929758720
*x^6 - 172473366866688*x^5 - 204855126595200*x^4 - 155155370878800*x^3 - 73069860056520*x^2 - 19521700361210*x
- 2261297826735)*sqrt(3*x^2 + 5*x + 2) + 249299/429981696*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 - 12096 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 38421 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 67449 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 70799 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 44295 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 14784 x^{6} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 1304 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 624 x^{8} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 144 x^{9} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 1620 \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)**4*(3*x**2+5*x+2)**(5/2),x)

[Out]

-Integral(-12096*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-38421*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-67
449*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-70799*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-44295*x**5*s
qrt(3*x**2 + 5*x + 2), x) - Integral(-14784*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1304*x**7*sqrt(3*x**2
+ 5*x + 2), x) - Integral(624*x**8*sqrt(3*x**2 + 5*x + 2), x) - Integral(144*x**9*sqrt(3*x**2 + 5*x + 2), x) -
Integral(-1620*sqrt(3*x**2 + 5*x + 2), x)

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Giac [A]  time = 1.15357, size = 134, normalized size = 0.65 \begin{align*} -\frac{1}{13795246080} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (14 \,{\left (48 \,{\left (54 \,{\left (20 \, x + 97\right )} x - 11555\right )} x - 7394353\right )} x - 362075335\right )} x - 24952744049\right )} x - 177825630725\right )} x - 1077467853325\right )} x - 3044577502355\right )} x - 9760850180605\right )} x - 2261297826735\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{249299}{214990848} \, \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)^4*(3*x^2+5*x+2)^(5/2),x, algorithm="giac")

[Out]

-1/13795246080*(2*(12*(6*(8*(6*(36*(14*(48*(54*(20*x + 97)*x - 11555)*x - 7394353)*x - 362075335)*x - 24952744
049)*x - 177825630725)*x - 1077467853325)*x - 3044577502355)*x - 9760850180605)*x - 2261297826735)*sqrt(3*x^2
+ 5*x + 2) + 249299/214990848*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))