3.2454 \(\int \frac{(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^3} \, dx\)

Optimal. Leaf size=181 \[ -\frac{(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}+\frac{7 (121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{240 (2 x+3)}+\frac{7 (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}}{4608}+\frac{7 (167495-349806 x) \sqrt{3 x^2+5 x+2}}{36864}-\frac{12443893 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{73728 \sqrt{3}}+\frac{44625 \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1024} \]

[Out]

(7*(167495 - 349806*x)*Sqrt[2 + 5*x + 3*x^2])/36864 + (7*(805 - 17394*x)*(2 + 5*x + 3*x^2)^(3/2))/4608 + (7*(5
84 + 121*x)*(2 + 5*x + 3*x^2)^(5/2))/(240*(3 + 2*x)) - ((21 + x)*(2 + 5*x + 3*x^2)^(7/2))/(12*(3 + 2*x)^2) - (
12443893*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(73728*Sqrt[3]) + (44625*Sqrt[5]*ArcTanh[(7 + 8
*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/1024

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Rubi [A]  time = 0.128258, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {812, 814, 843, 621, 206, 724} \[ -\frac{(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}+\frac{7 (121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{240 (2 x+3)}+\frac{7 (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}}{4608}+\frac{7 (167495-349806 x) \sqrt{3 x^2+5 x+2}}{36864}-\frac{12443893 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{73728 \sqrt{3}}+\frac{44625 \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1024} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(167495 - 349806*x)*Sqrt[2 + 5*x + 3*x^2])/36864 + (7*(805 - 17394*x)*(2 + 5*x + 3*x^2)^(3/2))/4608 + (7*(5
84 + 121*x)*(2 + 5*x + 3*x^2)^(5/2))/(240*(3 + 2*x)) - ((21 + x)*(2 + 5*x + 3*x^2)^(7/2))/(12*(3 + 2*x)^2) - (
12443893*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(73728*Sqrt[3]) + (44625*Sqrt[5]*ArcTanh[(7 + 8
*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/1024

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 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*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] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 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 )^{7/2}}{(3+2 x)^3} \, dx &=-\frac{(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac{7}{96} \int \frac{(-404-484 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx\\ &=\frac{7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}+\frac{7}{768} \int \frac{(-19488-23192 x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx\\ &=\frac{7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac{7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac{7 \int \frac{(2361672+2798448 x) \sqrt{2+5 x+3 x^2}}{3+2 x} \, dx}{73728}\\ &=\frac{7 (167495-349806 x) \sqrt{2+5 x+3 x^2}}{36864}+\frac{7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac{7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}+\frac{7 \int \frac{-145828656-170659104 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{3538944}\\ &=\frac{7 (167495-349806 x) \sqrt{2+5 x+3 x^2}}{36864}+\frac{7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac{7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac{12443893 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{73728}+\frac{223125 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{1024}\\ &=\frac{7 (167495-349806 x) \sqrt{2+5 x+3 x^2}}{36864}+\frac{7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac{7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac{12443893 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{36864}-\frac{223125}{512} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{7 (167495-349806 x) \sqrt{2+5 x+3 x^2}}{36864}+\frac{7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac{7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac{12443893 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{73728 \sqrt{3}}+\frac{44625 \sqrt{5} \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{1024}\\ \end{align*}

Mathematica [A]  time = 0.114115, size = 130, normalized size = 0.72 \[ \frac{-\frac{6 \sqrt{3 x^2+5 x+2} \left (414720 x^7-926208 x^6-6830784 x^5-15112992 x^4-12848072 x^3-19284852 x^2-89867034 x-91912653\right )}{(2 x+3)^2}-48195000 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )-62219465 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{1105920} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*Sqrt[2 + 5*x + 3*x^2]*(-91912653 - 89867034*x - 19284852*x^2 - 12848072*x^3 - 15112992*x^4 - 6830784*x^5
- 926208*x^6 + 414720*x^7))/(3 + 2*x)^2 - 48195000*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]
)] - 62219465*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/1105920

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Maple [A]  time = 0.012, size = 253, normalized size = 1.4 \begin{align*} -{\frac{13}{40} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{27}{10} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{51}{8} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}-{\frac{5635+6762\,x}{480} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{101465+121758\,x}{4608} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{2040535+2448642\,x}{36864}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{12443893\,\sqrt{3}}{221184}\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{357}{32} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{2975}{128} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{44625}{1024}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{44625\,\sqrt{5}}{1024}{\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{135+162\,x}{20} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/40/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(9/2)+27/10/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)+51/8*(3*(x+3/2)^2-4*x
-19/4)^(7/2)-1127/480*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)-20293/4608*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-408
107/36864*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-12443893/221184*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)
^(1/2))*3^(1/2)+357/32*(3*(x+3/2)^2-4*x-19/4)^(5/2)+2975/128*(3*(x+3/2)^2-4*x-19/4)^(3/2)+44625/1024*(12*(x+3/
2)^2-16*x-19)^(1/2)-44625/1024*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))-27/20*(5+6
*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)

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Maxima [A]  time = 2.16644, size = 294, normalized size = 1.62 \begin{align*} \frac{39}{40} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{10 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{1127}{80} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{7}{12} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{4 \,{\left (2 \, x + 3\right )}} - \frac{20293}{768} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{5635}{4608} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{408107}{6144} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{12443893}{221184} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{44625}{1024} \, \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{1172465}{36864} \, \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)^(7/2)/(3+2*x)^3,x, algorithm="maxima")

[Out]

39/40*(3*x^2 + 5*x + 2)^(7/2) - 13/10*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 1127/80*(3*x^2 + 5*x + 2)^(
5/2)*x - 7/12*(3*x^2 + 5*x + 2)^(5/2) + 27/4*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) - 20293/768*(3*x^2 + 5*x + 2)^(
3/2)*x + 5635/4608*(3*x^2 + 5*x + 2)^(3/2) - 408107/6144*sqrt(3*x^2 + 5*x + 2)*x - 12443893/221184*sqrt(3)*log
(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 44625/1024*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3
) + 5/2/abs(2*x + 3) - 2) + 1172465/36864*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.63705, size = 544, normalized size = 3.01 \begin{align*} \frac{62219465 \, \sqrt{3}{\left (4 \, x^{2} + 12 \, x + 9\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 ) + 48195000 \, \sqrt{5}{\left (4 \, x^{2} + 12 \, x + 9\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 ) - 12 \,{\left (414720 \, x^{7} - 926208 \, x^{6} - 6830784 \, x^{5} - 15112992 \, x^{4} - 12848072 \, x^{3} - 19284852 \, x^{2} - 89867034 \, x - 91912653\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{2211840 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^3,x, algorithm="fricas")

[Out]

1/2211840*(62219465*sqrt(3)*(4*x^2 + 12*x + 9)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x
+ 49) + 48195000*sqrt(5)*(4*x^2 + 12*x + 9)*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)) - 12*(414720*x^7 - 926208*x^6 - 6830784*x^5 - 15112992*x^4 - 12848072*x^3 - 19284852
*x^2 - 89867034*x - 91912653)*sqrt(3*x^2 + 5*x + 2))/(4*x^2 + 12*x + 9)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.35276, size = 377, normalized size = 2.08 \begin{align*} -\frac{1}{184320} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \,{\left (30 \, x - 157\right )} x - 725\right )} x - 67409\right )} x + 1173065\right )} x - 8219517\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{44625}{1024} \, \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{12443893}{221184} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac{25 \,{\left (5878 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 22241 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 75807 \, \sqrt{3} x + 27061 \, \sqrt{3} - 75807 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{512 \,{\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 )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^3,x, algorithm="giac")

[Out]

-1/184320*(2*(12*(18*(8*(30*x - 157)*x - 725)*x - 67409)*x + 1173065)*x - 8219517)*sqrt(3*x^2 + 5*x + 2) + 446
25/1024*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*s
qrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) + 12443893/221184*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2)) - 5)) + 25/512*(5878*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 22241*sqrt(3)*(sqrt(3)*x - sqrt
(3*x^2 + 5*x + 2))^2 + 75807*sqrt(3)*x + 27061*sqrt(3) - 75807*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)^2