3.2429 \(\int \frac{(5-x) (2+5 x+3 x^2)^{3/2}}{(3+2 x)^7} \, dx\)

Optimal. Leaf size=149 \[ -\frac{167 \left (3 x^2+5 x+2\right )^{5/2}}{375 (2 x+3)^5}-\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}+\frac{1141 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{12000 (2 x+3)^4}-\frac{1141 (8 x+7) \sqrt{3 x^2+5 x+2}}{160000 (2 x+3)^2}+\frac{1141 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{320000 \sqrt{5}} \]

[Out]

(-1141*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(160000*(3 + 2*x)^2) + (1141*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(12000
*(3 + 2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(30*(3 + 2*x)^6) - (167*(2 + 5*x + 3*x^2)^(5/2))/(375*(3 + 2*x)^5
) + (1141*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(320000*Sqrt[5])

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Rubi [A]  time = 0.0819372, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {834, 806, 720, 724, 206} \[ -\frac{167 \left (3 x^2+5 x+2\right )^{5/2}}{375 (2 x+3)^5}-\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}+\frac{1141 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{12000 (2 x+3)^4}-\frac{1141 (8 x+7) \sqrt{3 x^2+5 x+2}}{160000 (2 x+3)^2}+\frac{1141 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{320000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1141*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(160000*(3 + 2*x)^2) + (1141*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(12000
*(3 + 2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(30*(3 + 2*x)^6) - (167*(2 + 5*x + 3*x^2)^(5/2))/(375*(3 + 2*x)^5
) + (1141*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(320000*Sqrt[5])

Rule 834

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

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 720

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

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+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx &=-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{30 (3+2 x)^6}-\frac{1}{30} \int \frac{\left (-\frac{217}{2}+39 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{30 (3+2 x)^6}-\frac{167 \left (2+5 x+3 x^2\right )^{5/2}}{375 (3+2 x)^5}+\frac{1141}{300} \int \frac{\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx\\ &=\frac{1141 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{12000 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{30 (3+2 x)^6}-\frac{167 \left (2+5 x+3 x^2\right )^{5/2}}{375 (3+2 x)^5}-\frac{1141 \int \frac{\sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{8000}\\ &=-\frac{1141 (7+8 x) \sqrt{2+5 x+3 x^2}}{160000 (3+2 x)^2}+\frac{1141 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{12000 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{30 (3+2 x)^6}-\frac{167 \left (2+5 x+3 x^2\right )^{5/2}}{375 (3+2 x)^5}+\frac{1141 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{320000}\\ &=-\frac{1141 (7+8 x) \sqrt{2+5 x+3 x^2}}{160000 (3+2 x)^2}+\frac{1141 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{12000 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{30 (3+2 x)^6}-\frac{167 \left (2+5 x+3 x^2\right )^{5/2}}{375 (3+2 x)^5}-\frac{1141 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{160000}\\ &=-\frac{1141 (7+8 x) \sqrt{2+5 x+3 x^2}}{160000 (3+2 x)^2}+\frac{1141 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{12000 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{30 (3+2 x)^6}-\frac{167 \left (2+5 x+3 x^2\right )^{5/2}}{375 (3+2 x)^5}+\frac{1141 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{320000 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0989014, size = 151, normalized size = 1.01 \[ \frac{1}{30} \left (-\frac{334 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}-\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^6}+\frac{1141 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{400 (2 x+3)^4}-\frac{3423 \left (\frac{10 \sqrt{3 x^2+5 x+2} (8 x+7)}{(2 x+3)^2}+\sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )\right )}{160000}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1141*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(400*(3 + 2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^6 - (334*
(2 + 5*x + 3*x^2)^(5/2))/(25*(3 + 2*x)^5) - (3423*((10*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 + Sqrt[5]*
ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])]))/160000)/30

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Maple [A]  time = 0.014, size = 232, normalized size = 1.6 \begin{align*} -{\frac{13}{1920} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{167}{12000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{1141}{48000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{1141}{30000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{35371}{600000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{33089}{375000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{1141}{3000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{5705+6846\,x}{200000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{1141}{1600000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{1141\,\sqrt{5}}{1600000}{\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{165445+198534\,x}{750000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/1920/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(5/2)-167/12000/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1141/48000/(x
+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1141/30000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(5/2)-35371/600000/(x+3/2)^2*
(3*(x+3/2)^2-4*x-19/4)^(5/2)-33089/375000/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(5/2)+1141/3000000*(3*(x+3/2)^2-4*x-1
9/4)^(3/2)-1141/200000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+1141/1600000*(12*(x+3/2)^2-16*x-19)^(1/2)-1141/160
0000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+33089/750000*(5+6*x)*(3*(x+3/2)^2-4*
x-19/4)^(3/2)

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Maxima [B]  time = 1.50986, size = 387, normalized size = 2.6 \begin{align*} \frac{35371}{200000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{30 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{167 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{375 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{1141 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{3000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{1141 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{3750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{35371 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{150000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{3423}{100000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{1141}{1600000} \, \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{21679}{800000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{33089 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{150000 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35371/200000*(3*x^2 + 5*x + 2)^(3/2) - 13/30*(3*x^2 + 5*x + 2)^(5/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 +
4860*x^2 + 2916*x + 729) - 167/375*(3*x^2 + 5*x + 2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 2
43) - 1141/3000*(3*x^2 + 5*x + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 1141/3750*(3*x^2 + 5*x + 2)
^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 35371/150000*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12*x + 9) - 3423/100000*sq
rt(3*x^2 + 5*x + 2)*x - 1141/1600000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3)
- 2) - 21679/800000*sqrt(3*x^2 + 5*x + 2) - 33089/150000*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 1.62987, size = 475, normalized size = 3.19 \begin{align*} \frac{3423 \, \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 (95616 \, x^{5} + 799120 \, x^{4} + 3065440 \, x^{3} + 4479600 \, x^{2} + 2526920 \, x + 412679\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{9600000 \,{\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)^(3/2)/(3+2*x)^7,x, algorithm="fricas")

[Out]

1/9600000*(3423*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*(95616*x^5 + 799120*x^4 + 3065440
*x^3 + 4479600*x^2 + 2526920*x + 412679)*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]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{10 \sqrt{3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\, dx - \int - \frac{23 x \sqrt{3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\, dx - \int - \frac{10 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\, dx - \int \frac{3 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*x**2
+ 10206*x + 2187), x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 +
22680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**
6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x
+ 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187), x)

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Giac [B]  time = 1.24477, size = 554, normalized size = 3.72 \begin{align*} \frac{1141}{1600000} \, \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{109536 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 6127344 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 70129360 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 83080800 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 3334681440 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 9802137888 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 47432214576 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 48106882440 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 94851959950 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 39436262415 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 28403540997 \, \sqrt{3} x - 3009604608 \, \sqrt{3} + 28403540997 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{480000 \,{\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)^(3/2)/(3+2*x)^7,x, algorithm="giac")

[Out]

1141/1600000*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))) - 1/480000*(109536*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11
+ 6127344*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 70129360*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 83
080800*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 3334681440*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 9802
137888*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 47432214576*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 481
06882440*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 94851959950*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 3
9436262415*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 28403540997*sqrt(3)*x - 3009604608*sqrt(3) + 284035
40997*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