3.2443 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^7} \, dx\)

Optimal. Leaf size=167 \[ \frac{(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}+\frac{(328 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{1920 (2 x+3)^4}+\frac{(10952 x+14083) \sqrt{3 x^2+5 x+2}}{25600 (2 x+3)^2}-\frac{9}{128} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{13931 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{51200 \sqrt{5}} \]

[Out]

((14083 + 10952*x)*Sqrt[2 + 5*x + 3*x^2])/(25600*(3 + 2*x)^2) + ((437 + 328*x)*(2 + 5*x + 3*x^2)^(3/2))/(1920*
(3 + 2*x)^4) + ((109 + 116*x)*(2 + 5*x + 3*x^2)^(5/2))/(120*(3 + 2*x)^6) - (9*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqr
t[3]*Sqrt[2 + 5*x + 3*x^2])])/128 + (13931*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(51200*Sqrt[5
])

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Rubi [A]  time = 0.103488, antiderivative size = 167, 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 = {810, 843, 621, 206, 724} \[ \frac{(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}+\frac{(328 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{1920 (2 x+3)^4}+\frac{(10952 x+14083) \sqrt{3 x^2+5 x+2}}{25600 (2 x+3)^2}-\frac{9}{128} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{13931 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{51200 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((14083 + 10952*x)*Sqrt[2 + 5*x + 3*x^2])/(25600*(3 + 2*x)^2) + ((437 + 328*x)*(2 + 5*x + 3*x^2)^(3/2))/(1920*
(3 + 2*x)^4) + ((109 + 116*x)*(2 + 5*x + 3*x^2)^(5/2))/(120*(3 + 2*x)^6) - (9*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqr
t[3]*Sqrt[2 + 5*x + 3*x^2])])/128 + (13931*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(51200*Sqrt[5
])

Rule 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

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)^7} \, dx &=\frac{(109+116 x) \left (2+5 x+3 x^2\right )^{5/2}}{120 (3+2 x)^6}-\frac{1}{240} \int \frac{(215+180 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx\\ &=\frac{(437+328 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920 (3+2 x)^4}+\frac{(109+116 x) \left (2+5 x+3 x^2\right )^{5/2}}{120 (3+2 x)^6}+\frac{\int \frac{(-18330-21600 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{38400}\\ &=\frac{(14083+10952 x) \sqrt{2+5 x+3 x^2}}{25600 (3+2 x)^2}+\frac{(437+328 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920 (3+2 x)^4}+\frac{(109+116 x) \left (2+5 x+3 x^2\right )^{5/2}}{120 (3+2 x)^6}-\frac{\int \frac{1108140+1296000 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{3072000}\\ &=\frac{(14083+10952 x) \sqrt{2+5 x+3 x^2}}{25600 (3+2 x)^2}+\frac{(437+328 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920 (3+2 x)^4}+\frac{(109+116 x) \left (2+5 x+3 x^2\right )^{5/2}}{120 (3+2 x)^6}-\frac{27}{128} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{13931 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{51200}\\ &=\frac{(14083+10952 x) \sqrt{2+5 x+3 x^2}}{25600 (3+2 x)^2}+\frac{(437+328 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920 (3+2 x)^4}+\frac{(109+116 x) \left (2+5 x+3 x^2\right )^{5/2}}{120 (3+2 x)^6}-\frac{27}{64} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )-\frac{13931 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{25600}\\ &=\frac{(14083+10952 x) \sqrt{2+5 x+3 x^2}}{25600 (3+2 x)^2}+\frac{(437+328 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920 (3+2 x)^4}+\frac{(109+116 x) \left (2+5 x+3 x^2\right )^{5/2}}{120 (3+2 x)^6}-\frac{9}{128} \sqrt{3} \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )+\frac{13931 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{51200 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.155853, size = 120, normalized size = 0.72 \[ \frac{\frac{10 \sqrt{3 x^2+5 x+2} \left (1351296 x^5+7629680 x^4+18217760 x^3+22854480 x^2+14921560 x+4015849\right )}{(2 x+3)^6}-41793 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )-54000 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{768000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((10*Sqrt[2 + 5*x + 3*x^2]*(4015849 + 14921560*x + 22854480*x^2 + 18217760*x^3 + 7629680*x^4 + 1351296*x^5))/(
3 + 2*x)^6 - 41793*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] - 54000*Sqrt[3]*ArcTanh[(5 +
6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/768000

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Maple [B]  time = 0.015, size = 300, normalized size = 1.8 \begin{align*} -{\frac{13}{1920} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{23}{2400} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{249}{16000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{709}{30000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{22271}{600000} \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{30445+36534\,x}{250000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{6089}{125000} \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{5005+6006\,x}{120000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{2155+2586\,x}{32000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{9\,\sqrt{3}}{128}\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{13931\,\sqrt{5}}{256000}{\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{13931}{1000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{13931}{480000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{13931}{256000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}} \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)^7,x)

[Out]

-13/1920/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(7/2)-23/2400/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(7/2)-249/16000/(x+3/
2)^4*(3*(x+3/2)^2-4*x-19/4)^(7/2)-709/30000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(7/2)-22271/600000/(x+3/2)^2*(3*(
x+3/2)^2-4*x-19/4)^(7/2)+6089/250000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)-6089/125000/(x+3/2)*(3*(x+3/2)^2-4*x
-19/4)^(7/2)-1001/120000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-431/32000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-9
/128*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)-13931/256000*5^(1/2)*arctanh(2/5*(-7/2-4*x
)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+13931/1000000*(3*(x+3/2)^2-4*x-19/4)^(5/2)+13931/480000*(3*(x+3/2)^2-4
*x-19/4)^(3/2)+13931/256000*(12*(x+3/2)^2-16*x-19)^(1/2)

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Maxima [B]  time = 1.79316, size = 463, normalized size = 2.77 \begin{align*} \frac{22271}{200000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{30 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{23 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{75 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{249 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{1000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{709 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{3750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{22271 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{150000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{1001}{20000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{6089}{480000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{6089 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{50000 \,{\left (2 \, x + 3\right )}} - \frac{1293}{16000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{9}{128} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{13931}{256000} \, \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{5311}{128000} \, \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)^7,x, algorithm="maxima")

[Out]

22271/200000*(3*x^2 + 5*x + 2)^(5/2) - 13/30*(3*x^2 + 5*x + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 +
4860*x^2 + 2916*x + 729) - 23/75*(3*x^2 + 5*x + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243
) - 249/1000*(3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 709/3750*(3*x^2 + 5*x + 2)^(7/
2)/(8*x^3 + 36*x^2 + 54*x + 27) - 22271/150000*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) - 1001/20000*(3*x^2
+ 5*x + 2)^(3/2)*x - 6089/480000*(3*x^2 + 5*x + 2)^(3/2) - 6089/50000*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) - 1293
/16000*sqrt(3*x^2 + 5*x + 2)*x - 9/128*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 13931/256000*s
qrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 5311/128000*sqrt(3*x^2 + 5*x +
2)

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Fricas [A]  time = 1.53943, size = 693, normalized size = 4.15 \begin{align*} \frac{54000 \, \sqrt{3}{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\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 ) + 41793 \, \sqrt{5}{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\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 ) + 20 \,{\left (1351296 \, x^{5} + 7629680 \, x^{4} + 18217760 \, x^{3} + 22854480 \, x^{2} + 14921560 \, x + 4015849\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{1536000 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\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)^7,x, algorithm="fricas")

[Out]

1/1536000*(54000*sqrt(3)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)*log(-4*sqrt(3)*sqr
t(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 41793*sqrt(5)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 +
4860*x^2 + 2916*x + 729)*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)) + 20*(1351296*x^5 + 7629680*x^4 + 18217760*x^3 + 22854480*x^2 + 14921560*x + 4015849)*sqrt(3*x^2 + 5*x +
2))/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)

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

[Out]

Timed out

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Giac [B]  time = 1.29021, size = 599, normalized size = 3.59 \begin{align*} \frac{13931}{256000} \, \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{9}{128} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac{20435424 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 269619696 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 4893810640 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 17834042400 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 129909086880 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 219870810528 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 791797675536 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 672745449240 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 1187868124850 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 460902113505 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 318216938187 \, \sqrt{3} x + 32907940848 \, \sqrt{3} - 318216938187 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{76800 \,{\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 )}^{6}} \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)^7,x, algorithm="giac")

[Out]

13931/256000*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))) + 9/128*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2)) - 5)) + 1/76800*(20435424*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 269619696*sqrt(3)*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2))^10 + 4893810640*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 17834042400*sqrt(3)*(sqrt(3)*x
- sqrt(3*x^2 + 5*x + 2))^8 + 129909086880*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 219870810528*sqrt(3)*(sqrt(
3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 791797675536*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 672745449240*sqrt(3)*(s
qrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 1187868124850*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 460902113505*sqrt(
3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 318216938187*sqrt(3)*x + 32907940848*sqrt(3) - 318216938187*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)
^6