3.1394 \(\int \frac{(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^9} \, dx\)

Optimal. Leaf size=158 \[ -\frac{773 \left (3 x^2+2\right )^{7/2}}{68600 (2 x+3)^7}-\frac{13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}-\frac{233 (4-9 x) \left (3 x^2+2\right )^{5/2}}{171500 (2 x+3)^6}-\frac{699 (4-9 x) \left (3 x^2+2\right )^{3/2}}{2401000 (2 x+3)^4}-\frac{6291 (4-9 x) \sqrt{3 x^2+2}}{84035000 (2 x+3)^2}-\frac{18873 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42017500 \sqrt{35}} \]

[Out]

(-6291*(4 - 9*x)*Sqrt[2 + 3*x^2])/(84035000*(3 + 2*x)^2) - (699*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(2401000*(3 + 2*x
)^4) - (233*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(171500*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(280*(3 + 2*x)^8) - (77
3*(2 + 3*x^2)^(7/2))/(68600*(3 + 2*x)^7) - (18873*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(42017500*Sqr
t[35])

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Rubi [A]  time = 0.0829256, antiderivative size = 158, 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} \[ -\frac{773 \left (3 x^2+2\right )^{7/2}}{68600 (2 x+3)^7}-\frac{13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}-\frac{233 (4-9 x) \left (3 x^2+2\right )^{5/2}}{171500 (2 x+3)^6}-\frac{699 (4-9 x) \left (3 x^2+2\right )^{3/2}}{2401000 (2 x+3)^4}-\frac{6291 (4-9 x) \sqrt{3 x^2+2}}{84035000 (2 x+3)^2}-\frac{18873 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42017500 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^9,x]

[Out]

(-6291*(4 - 9*x)*Sqrt[2 + 3*x^2])/(84035000*(3 + 2*x)^2) - (699*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(2401000*(3 + 2*x
)^4) - (233*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(171500*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(280*(3 + 2*x)^8) - (77
3*(2 + 3*x^2)^(7/2))/(68600*(3 + 2*x)^7) - (18873*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(42017500*Sqr
t[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 )^{5/2}}{(3+2 x)^9} \, dx &=-\frac{13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac{1}{280} \int \frac{(-328+39 x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx\\ &=-\frac{13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac{773 \left (2+3 x^2\right )^{7/2}}{68600 (3+2 x)^7}+\frac{699 \int \frac{\left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{2450}\\ &=-\frac{233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{171500 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac{773 \left (2+3 x^2\right )^{7/2}}{68600 (3+2 x)^7}+\frac{699 \int \frac{\left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{17150}\\ &=-\frac{699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{2401000 (3+2 x)^4}-\frac{233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{171500 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac{773 \left (2+3 x^2\right )^{7/2}}{68600 (3+2 x)^7}+\frac{6291 \int \frac{\sqrt{2+3 x^2}}{(3+2 x)^3} \, dx}{1200500}\\ &=-\frac{6291 (4-9 x) \sqrt{2+3 x^2}}{84035000 (3+2 x)^2}-\frac{699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{2401000 (3+2 x)^4}-\frac{233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{171500 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac{773 \left (2+3 x^2\right )^{7/2}}{68600 (3+2 x)^7}+\frac{18873 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{42017500}\\ &=-\frac{6291 (4-9 x) \sqrt{2+3 x^2}}{84035000 (3+2 x)^2}-\frac{699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{2401000 (3+2 x)^4}-\frac{233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{171500 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac{773 \left (2+3 x^2\right )^{7/2}}{68600 (3+2 x)^7}-\frac{18873 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{42017500}\\ &=-\frac{6291 (4-9 x) \sqrt{2+3 x^2}}{84035000 (3+2 x)^2}-\frac{699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{2401000 (3+2 x)^4}-\frac{233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{171500 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{280 (3+2 x)^8}-\frac{773 \left (2+3 x^2\right )^{7/2}}{68600 (3+2 x)^7}-\frac{18873 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{42017500 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.209291, size = 144, normalized size = 0.91 \[ \frac{1}{280} \left (-\frac{773 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}-\frac{13 \left (3 x^2+2\right )^{7/2}}{(2 x+3)^8}+\frac{466 (9 x-4) \left (3 x^2+2\right )^{5/2}}{1225 (2 x+3)^6}+\frac{699 \left (\frac{35 \sqrt{3 x^2+2} \left (1269 x^3+408 x^2+927 x-604\right )}{(2 x+3)^4}-54 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )\right )}{10504375}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^9,x]

[Out]

((466*(-4 + 9*x)*(2 + 3*x^2)^(5/2))/(1225*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(3 + 2*x)^8 - (773*(2 + 3*x^2)
^(7/2))/(245*(3 + 2*x)^7) + (699*((35*Sqrt[2 + 3*x^2]*(-604 + 927*x + 408*x^2 + 1269*x^3))/(3 + 2*x)^4 - 54*Sq
rt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])]))/10504375)/280

________________________________________________________________________________________

Maple [B]  time = 0.026, size = 299, normalized size = 1.9 \begin{align*} -{\frac{773}{8780800} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-7}}-{\frac{233}{5488000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{2097}{96040000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{20271}{1680700000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{207603}{29412250000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{2258469}{514714375000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{78643791\,x}{9007501562500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{26214597}{9007501562500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{2208141\,x}{102942875000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{169857\,x}{2941225000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{18873\,\sqrt{35}}{1470612500}{\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{150984}{2251875390625} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{18873}{1470612500}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}+{\frac{12582}{12867859375} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{13}{71680} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^9,x)

[Out]

-773/8780800/(x+3/2)^7*(3*(x+3/2)^2-9*x-19/4)^(7/2)-233/5488000/(x+3/2)^6*(3*(x+3/2)^2-9*x-19/4)^(7/2)-2097/96
040000/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(7/2)-20271/1680700000/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(7/2)-207603/2
9412250000/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(7/2)-2258469/514714375000/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(7/2)+
78643791/9007501562500*x*(3*(x+3/2)^2-9*x-19/4)^(5/2)-26214597/9007501562500/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(7
/2)+2208141/102942875000*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)+169857/2941225000*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)-18873
/1470612500*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))+150984/2251875390625*(3*(x+3/
2)^2-9*x-19/4)^(5/2)+18873/1470612500*(12*(x+3/2)^2-36*x-19)^(1/2)+12582/12867859375*(3*(x+3/2)^2-9*x-19/4)^(3
/2)-13/71680/(x+3/2)^8*(3*(x+3/2)^2-9*x-19/4)^(7/2)

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Maxima [B]  time = 1.59894, size = 508, normalized size = 3.22 \begin{align*} \frac{6775407}{514714375000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{280 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac{773 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{68600 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac{233 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{85750 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{2097 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{3001250 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{20271 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{105043750 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{207603 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{3676531250 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{2258469 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{128678593750 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{2208141}{102942875000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{12582}{12867859375} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{26214597 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{514714375000 \,{\left (2 \, x + 3\right )}} + \frac{169857}{2941225000} \, \sqrt{3 \, x^{2} + 2} x + \frac{18873}{1470612500} \, \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{18873}{735306250} \, \sqrt{3 \, x^{2} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^9,x, algorithm="maxima")

[Out]

6775407/514714375000*(3*x^2 + 2)^(5/2) - 13/280*(3*x^2 + 2)^(7/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5
+ 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561) - 773/68600*(3*x^2 + 2)^(7/2)/(128*x^7 + 1344*x^6 + 604
8*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 233/85750*(3*x^2 + 2)^(7/2)/(64*x^6 + 576*x^5 +
2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 2097/3001250*(3*x^2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 +
1080*x^2 + 810*x + 243) - 20271/105043750*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 207603/
3676531250*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 2258469/128678593750*(3*x^2 + 2)^(7/2)/(4*x^2 + 12
*x + 9) + 2208141/102942875000*(3*x^2 + 2)^(3/2)*x + 12582/12867859375*(3*x^2 + 2)^(3/2) - 26214597/5147143750
00*(3*x^2 + 2)^(5/2)/(2*x + 3) + 169857/2941225000*sqrt(3*x^2 + 2)*x + 18873/1470612500*sqrt(35)*arcsinh(3/2*s
qrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 18873/735306250*sqrt(3*x^2 + 2)

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Fricas [A]  time = 1.95732, size = 603, normalized size = 3.82 \begin{align*} \frac{18873 \, \sqrt{35}{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\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 (49626 \, x^{7} + 2206008 \, x^{6} + 210306726 \, x^{5} + 33613440 \, x^{4} + 226355535 \, x^{3} - 178164896 \, x^{2} - 38788883 \, x - 104577556\right )} \sqrt{3 \, x^{2} + 2}}{2941225000 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^9,x, algorithm="fricas")

[Out]

1/2941225000*(18873*sqrt(35)*(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2
+ 34992*x + 6561)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 35*(496
26*x^7 + 2206008*x^6 + 210306726*x^5 + 33613440*x^4 + 226355535*x^3 - 178164896*x^2 - 38788883*x - 104577556)*
sqrt(3*x^2 + 2))/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x +
6561)

________________________________________________________________________________________

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)**(5/2)/(3+2*x)**9,x)

[Out]

Timed out

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Giac [B]  time = 1.24645, size = 612, normalized size = 3.87 \begin{align*} \frac{18873}{1470612500} \, \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{27 \,{\left (178944 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{15} + 46043740 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{14} + 30787400 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{13} + 191125270 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{12} - 3328877720 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{11} - 2893694188 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{10} - 13787031160 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 522152825 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} - 28541438480 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} + 10194100560 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 23140527424 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} - 4295198880 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 1726278400 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 3033847040 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 39843840 \, \sqrt{3} x - 470528 \, \sqrt{3} - 39843840 \, \sqrt{3 \, x^{2} + 2}\right )}}{10756480000 \,{\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 )}^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^9,x, algorithm="giac")

[Out]

18873/1470612500*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))) - 27/10756480000*(178944*(sqrt(3)*x - sqrt(3*x^2 + 2))^15 + 46043740*s
qrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^14 + 30787400*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 + 191125270*sqrt(3)*(sqrt(
3)*x - sqrt(3*x^2 + 2))^12 - 3328877720*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 - 2893694188*sqrt(3)*(sqrt(3)*x - sqr
t(3*x^2 + 2))^10 - 13787031160*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 522152825*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2
))^8 - 28541438480*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 10194100560*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 231
40527424*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 4295198880*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 1726278400*(sq
rt(3)*x - sqrt(3*x^2 + 2))^3 - 3033847040*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 39843840*sqrt(3)*x - 47052
8*sqrt(3) - 39843840*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)^8