3.1380 \(\int \frac{(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^8} \, dx\)

Optimal. Leaf size=153 \[ -\frac{822 \left (3 x^2+2\right )^{5/2}}{214375 (2 x+3)^5}-\frac{404 \left (3 x^2+2\right )^{5/2}}{25725 (2 x+3)^6}-\frac{13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}-\frac{2689 (4-9 x) \left (3 x^2+2\right )^{3/2}}{6002500 (2 x+3)^4}-\frac{24201 (4-9 x) \sqrt{3 x^2+2}}{210087500 (2 x+3)^2}-\frac{72603 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{105043750 \sqrt{35}} \]

[Out]

(-24201*(4 - 9*x)*Sqrt[2 + 3*x^2])/(210087500*(3 + 2*x)^2) - (2689*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(6002500*(3 +
2*x)^4) - (13*(2 + 3*x^2)^(5/2))/(245*(3 + 2*x)^7) - (404*(2 + 3*x^2)^(5/2))/(25725*(3 + 2*x)^6) - (822*(2 + 3
*x^2)^(5/2))/(214375*(3 + 2*x)^5) - (72603*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(105043750*Sqrt[35])

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Rubi [A]  time = 0.100191, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {835, 807, 721, 725, 206} \[ -\frac{822 \left (3 x^2+2\right )^{5/2}}{214375 (2 x+3)^5}-\frac{404 \left (3 x^2+2\right )^{5/2}}{25725 (2 x+3)^6}-\frac{13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}-\frac{2689 (4-9 x) \left (3 x^2+2\right )^{3/2}}{6002500 (2 x+3)^4}-\frac{24201 (4-9 x) \sqrt{3 x^2+2}}{210087500 (2 x+3)^2}-\frac{72603 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{105043750 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^8,x]

[Out]

(-24201*(4 - 9*x)*Sqrt[2 + 3*x^2])/(210087500*(3 + 2*x)^2) - (2689*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(6002500*(3 +
2*x)^4) - (13*(2 + 3*x^2)^(5/2))/(245*(3 + 2*x)^7) - (404*(2 + 3*x^2)^(5/2))/(25725*(3 + 2*x)^6) - (822*(2 + 3
*x^2)^(5/2))/(214375*(3 + 2*x)^5) - (72603*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(105043750*Sqrt[35])

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 721

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*
d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*
x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2,
0] && GtQ[p, 0]

Rule 725

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

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 \frac{(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx &=-\frac{13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac{1}{245} \int \frac{(-287+78 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx\\ &=-\frac{13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac{404 \left (2+3 x^2\right )^{5/2}}{25725 (3+2 x)^6}+\frac{\int \frac{(13626-2424 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx}{51450}\\ &=-\frac{13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac{404 \left (2+3 x^2\right )^{5/2}}{25725 (3+2 x)^6}-\frac{822 \left (2+3 x^2\right )^{5/2}}{214375 (3+2 x)^5}+\frac{2689 \int \frac{\left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{42875}\\ &=-\frac{2689 (4-9 x) \left (2+3 x^2\right )^{3/2}}{6002500 (3+2 x)^4}-\frac{13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac{404 \left (2+3 x^2\right )^{5/2}}{25725 (3+2 x)^6}-\frac{822 \left (2+3 x^2\right )^{5/2}}{214375 (3+2 x)^5}+\frac{24201 \int \frac{\sqrt{2+3 x^2}}{(3+2 x)^3} \, dx}{3001250}\\ &=-\frac{24201 (4-9 x) \sqrt{2+3 x^2}}{210087500 (3+2 x)^2}-\frac{2689 (4-9 x) \left (2+3 x^2\right )^{3/2}}{6002500 (3+2 x)^4}-\frac{13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac{404 \left (2+3 x^2\right )^{5/2}}{25725 (3+2 x)^6}-\frac{822 \left (2+3 x^2\right )^{5/2}}{214375 (3+2 x)^5}+\frac{72603 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{105043750}\\ &=-\frac{24201 (4-9 x) \sqrt{2+3 x^2}}{210087500 (3+2 x)^2}-\frac{2689 (4-9 x) \left (2+3 x^2\right )^{3/2}}{6002500 (3+2 x)^4}-\frac{13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac{404 \left (2+3 x^2\right )^{5/2}}{25725 (3+2 x)^6}-\frac{822 \left (2+3 x^2\right )^{5/2}}{214375 (3+2 x)^5}-\frac{72603 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{105043750}\\ &=-\frac{24201 (4-9 x) \sqrt{2+3 x^2}}{210087500 (3+2 x)^2}-\frac{2689 (4-9 x) \left (2+3 x^2\right )^{3/2}}{6002500 (3+2 x)^4}-\frac{13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac{404 \left (2+3 x^2\right )^{5/2}}{25725 (3+2 x)^6}-\frac{822 \left (2+3 x^2\right )^{5/2}}{214375 (3+2 x)^5}-\frac{72603 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{105043750 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.149277, size = 161, normalized size = 1.05 \[ \frac{1}{245} \left (-\frac{822 \left (3 x^2+2\right )^{5/2}}{875 (2 x+3)^5}-\frac{404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}-\frac{13 \left (3 x^2+2\right )^{5/2}}{(2 x+3)^7}-\frac{2689 \left (-315 (9 x-4) \sqrt{3 x^2+2} (2 x+3)^2-1225 (9 x-4) \left (3 x^2+2\right )^{3/2}+54 \sqrt{35} (2 x+3)^4 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )\right )}{30012500 (2 x+3)^4}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^8,x]

[Out]

((-13*(2 + 3*x^2)^(5/2))/(3 + 2*x)^7 - (404*(2 + 3*x^2)^(5/2))/(105*(3 + 2*x)^6) - (822*(2 + 3*x^2)^(5/2))/(87
5*(3 + 2*x)^5) - (2689*(-315*(3 + 2*x)^2*(-4 + 9*x)*Sqrt[2 + 3*x^2] - 1225*(-4 + 9*x)*(2 + 3*x^2)^(3/2) + 54*S
qrt[35]*(3 + 2*x)^4*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])]))/(30012500*(3 + 2*x)^4))/245

________________________________________________________________________________________

Maple [A]  time = 0.017, size = 245, normalized size = 1.6 \begin{align*} -{\frac{101}{411600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{411}{3430000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{2689}{48020000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{24201}{840350000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{250077}{14706125000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{2831517}{257357187500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{96804}{64339296875} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{653427\,x}{7353062500}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{72603}{3676531250}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{72603\,\sqrt{35}}{3676531250}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{8494551\,x}{257357187500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{13}{31360} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^8,x)

[Out]

-101/411600/(x+3/2)^6*(3*(x+3/2)^2-9*x-19/4)^(5/2)-411/3430000/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(5/2)-2689/480
20000/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(5/2)-24201/840350000/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(5/2)-250077/147
06125000/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(5/2)-2831517/257357187500/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(5/2)+9680
4/64339296875*(3*(x+3/2)^2-9*x-19/4)^(3/2)+653427/7353062500*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)+72603/3676531250*(
12*(x+3/2)^2-36*x-19)^(1/2)-72603/3676531250*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/
2))+8494551/257357187500*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)-13/31360/(x+3/2)^7*(3*(x+3/2)^2-9*x-19/4)^(5/2)

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Maxima [B]  time = 1.5685, size = 405, normalized size = 2.65 \begin{align*} \frac{750231}{14706125000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{245 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac{404 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{25725 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{822 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{214375 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{2689 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{3001250 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{24201 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{105043750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{250077 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{3676531250 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{653427}{7353062500} \, \sqrt{3 \, x^{2} + 2} x + \frac{72603}{3676531250} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{72603}{1838265625} \, \sqrt{3 \, x^{2} + 2} - \frac{2831517 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{14706125000 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^8,x, algorithm="maxima")

[Out]

750231/14706125000*(3*x^2 + 2)^(3/2) - 13/245*(3*x^2 + 2)^(5/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 2
2680*x^3 + 20412*x^2 + 10206*x + 2187) - 404/25725*(3*x^2 + 2)^(5/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 +
4860*x^2 + 2916*x + 729) - 822/214375*(3*x^2 + 2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)
- 2689/3001250*(3*x^2 + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 24201/105043750*(3*x^2 + 2)^(5/2)
/(8*x^3 + 36*x^2 + 54*x + 27) - 250077/3676531250*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) + 653427/7353062500*sqr
t(3*x^2 + 2)*x + 72603/3676531250*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 72
603/1838265625*sqrt(3*x^2 + 2) - 2831517/14706125000*(3*x^2 + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 2.27875, size = 552, normalized size = 3.61 \begin{align*} \frac{217809 \, \sqrt{35}{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (-\frac{\sqrt{35} \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \,{\left (5104296 \, x^{6} + 44301924 \, x^{5} + 148868010 \, x^{4} - 98810025 \, x^{3} + 740031210 \, x^{2} + 256388969 \, x + 471103116\right )} \sqrt{3 \, x^{2} + 2}}{22059187500 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^8,x, algorithm="fricas")

[Out]

1/22059187500*(217809*sqrt(35)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x +
2187)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(5104296*x^6 + 4
4301924*x^5 + 148868010*x^4 - 98810025*x^3 + 740031210*x^2 + 256388969*x + 471103116)*sqrt(3*x^2 + 2))/(128*x^
7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)

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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+2)**(3/2)/(3+2*x)**8,x)

[Out]

Timed out

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Giac [B]  time = 1.21478, size = 551, normalized size = 3.6 \begin{align*} \frac{72603}{3676531250} \, \sqrt{35} \log \left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{9 \,{\left (258144 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{13} + 5033808 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{12} + 225898166 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{11} + 26360013 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{10} + 555459995 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} - 2679767547 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} - 4252091247 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} - 6029804778 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} + 11677158028 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} - 7324195080 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 2245361152 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 675266496 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 174039168 \, \sqrt{3} x - 6049536 \, \sqrt{3} - 174039168 \, \sqrt{3 \, x^{2} + 2}\right )}}{3361400000 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^8,x, algorithm="giac")

[Out]

72603/3676531250*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqr
t(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 9/3361400000*(258144*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 + 5033808*sqrt
(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^12 + 225898166*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 + 26360013*sqrt(3)*(sqrt(3)*
x - sqrt(3*x^2 + 2))^10 + 555459995*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 - 2679767547*sqrt(3)*(sqrt(3)*x - sqrt(3*x
^2 + 2))^8 - 4252091247*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 6029804778*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 +
11677158028*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 7324195080*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 2245361152
*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 675266496*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 174039168*sqrt(3)*x - 6
049536*sqrt(3) - 174039168*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x
^2 + 2)) - 2)^7