3.2440 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=165 \[ -\frac{(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}+\frac{5 (43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{48 (2 x+3)^2}-\frac{5 (343 x+736) \sqrt{3 x^2+5 x+2}}{64 (2 x+3)}+\frac{13505 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{256 \sqrt{3}}-\frac{3487}{256} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

(-5*(736 + 343*x)*Sqrt[2 + 5*x + 3*x^2])/(64*(3 + 2*x)) + (5*(93 + 43*x)*(2 + 5*x + 3*x^2)^(3/2))/(48*(3 + 2*x
)^2) - ((8 + x)*(2 + 5*x + 3*x^2)^(5/2))/(6*(3 + 2*x)^3) + (13505*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x +
3*x^2])])/(256*Sqrt[3]) - (3487*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/256

________________________________________________________________________________________

Rubi [A]  time = 0.109866, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {812, 843, 621, 206, 724} \[ -\frac{(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}+\frac{5 (43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{48 (2 x+3)^2}-\frac{5 (343 x+736) \sqrt{3 x^2+5 x+2}}{64 (2 x+3)}+\frac{13505 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{256 \sqrt{3}}-\frac{3487}{256} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(736 + 343*x)*Sqrt[2 + 5*x + 3*x^2])/(64*(3 + 2*x)) + (5*(93 + 43*x)*(2 + 5*x + 3*x^2)^(3/2))/(48*(3 + 2*x
)^2) - ((8 + x)*(2 + 5*x + 3*x^2)^(5/2))/(6*(3 + 2*x)^3) + (13505*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x +
3*x^2])])/(256*Sqrt[3]) - (3487*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/256

Rule 812

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

Rule 843

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

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])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx &=-\frac{(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}-\frac{5}{72} \int \frac{(-216-258 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx\\ &=\frac{5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac{(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac{5}{768} \int \frac{(-7032-8232 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^2} \, dx\\ &=-\frac{5 (736+343 x) \sqrt{2+5 x+3 x^2}}{64 (3+2 x)}+\frac{5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac{(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}-\frac{5 \int \frac{-110784-129648 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{6144}\\ &=-\frac{5 (736+343 x) \sqrt{2+5 x+3 x^2}}{64 (3+2 x)}+\frac{5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac{(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac{13505}{256} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx-\frac{17435}{256} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{5 (736+343 x) \sqrt{2+5 x+3 x^2}}{64 (3+2 x)}+\frac{5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac{(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac{13505}{128} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )+\frac{17435}{128} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=-\frac{5 (736+343 x) \sqrt{2+5 x+3 x^2}}{64 (3+2 x)}+\frac{5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac{(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac{13505 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{256 \sqrt{3}}-\frac{3487}{256} \sqrt{5} \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.110783, size = 120, normalized size = 0.73 \[ \frac{1}{768} \left (-\frac{4 \sqrt{3 x^2+5 x+2} \left (288 x^5-1896 x^4+1944 x^3+64332 x^2+143533 x+89224\right )}{(2 x+3)^3}+10461 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )+13505 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*Sqrt[2 + 5*x + 3*x^2]*(89224 + 143533*x + 64332*x^2 + 1944*x^3 - 1896*x^4 + 288*x^5))/(3 + 2*x)^3 + 10461
*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] + 13505*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 1
5*x + 9*x^2])])/768

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 237, normalized size = 1.4 \begin{align*}{\frac{67}{600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{197}{125} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{3487}{1000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{1645+1974\,x}{240} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{2215+2658\,x}{128}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{13505\,\sqrt{3}}{768}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }-{\frac{3487}{480} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{3487}{256}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}+{\frac{3487\,\sqrt{5}}{256}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }+{\frac{985+1182\,x}{250} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{13}{120} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

67/600/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(7/2)-197/125/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-3487/1000*(3*(x+3/2
)^2-4*x-19/4)^(5/2)+329/240*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)+443/128*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+
13505/768*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)-3487/480*(3*(x+3/2)^2-4*x-19/4)^(3/2)
-3487/256*(12*(x+3/2)^2-16*x-19)^(1/2)+3487/256*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^
(1/2))+197/250*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)-13/120/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(7/2)

________________________________________________________________________________________

Maxima [A]  time = 1.53929, size = 297, normalized size = 1.8 \begin{align*} -\frac{67}{200} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{15 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac{67 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{150 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{329}{40} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{197}{480} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{197 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{50 \,{\left (2 \, x + 3\right )}} + \frac{1329}{64} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{13505}{768} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{3487}{256} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{159}{16} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^4,x, algorithm="maxima")

[Out]

-67/200*(3*x^2 + 5*x + 2)^(5/2) - 13/15*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) + 67/150*(3*x^2 +
5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) + 329/40*(3*x^2 + 5*x + 2)^(3/2)*x - 197/480*(3*x^2 + 5*x + 2)^(3/2) - 197/
50*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) + 1329/64*sqrt(3*x^2 + 5*x + 2)*x + 13505/768*sqrt(3)*log(sqrt(3)*sqrt(3*
x^2 + 5*x + 2) + 3*x + 5/2) + 3487/256*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x +
3) - 2) - 159/16*sqrt(3*x^2 + 5*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.51789, size = 509, normalized size = 3.08 \begin{align*} \frac{13505 \, \sqrt{3}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 10461 \, \sqrt{5}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 8 \,{\left (288 \, x^{5} - 1896 \, x^{4} + 1944 \, x^{3} + 64332 \, x^{2} + 143533 \, x + 89224\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{1536 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^4,x, algorithm="fricas")

[Out]

1/1536*(13505*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 12
0*x + 49) + 10461*sqrt(5)*(8*x^3 + 36*x^2 + 54*x + 27)*log(-(4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) - 124*x
^2 - 212*x - 89)/(4*x^2 + 12*x + 9)) - 8*(288*x^5 - 1896*x^4 + 1944*x^3 + 64332*x^2 + 143533*x + 89224)*sqrt(3
*x^2 + 5*x + 2))/(8*x^3 + 36*x^2 + 54*x + 27)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{20 \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx - \int - \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx - \int - \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx - \int - \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx - \int - \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-96*x*sqrt(3*x
**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(16
*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 +
216*x**2 + 216*x + 81), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x +
81), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x)

________________________________________________________________________________________

Giac [B]  time = 1.29896, size = 425, normalized size = 2.58 \begin{align*} -\frac{1}{128} \,{\left (2 \,{\left (12 \, x - 133\right )} x + 1197\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{3487}{256} \, \sqrt{5} \log \left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac{13505}{768} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac{203604 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 1334970 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 10053790 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 12051375 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 20819415 \, \sqrt{3} x + 4639299 \, \sqrt{3} - 20819415 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{384 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^4,x, algorithm="giac")

[Out]

-1/128*(2*(12*x - 133)*x + 1197)*sqrt(3*x^2 + 5*x + 2) - 3487/256*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6
*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 135
05/768*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 1/384*(203604*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2))^5 + 1334970*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 10053790*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^3 + 12051375*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 20819415*sqrt(3)*x + 4639299*sqrt(3)
- 20819415*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^
2 + 5*x + 2)) + 11)^3