3.1396 \(\int \frac{(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^{11}} \, dx\)

Optimal. Leaf size=202 \[ -\frac{739619 \left (3 x^2+2\right )^{7/2}}{1260525000 (2 x+3)^7}-\frac{4393 \left (3 x^2+2\right )^{7/2}}{1715000 (2 x+3)^8}-\frac{1171 \left (3 x^2+2\right )^{7/2}}{110250 (2 x+3)^9}-\frac{13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}-\frac{73233 (4-9 x) \left (3 x^2+2\right )^{5/2}}{1050437500 (2 x+3)^6}-\frac{219699 (4-9 x) \left (3 x^2+2\right )^{3/2}}{14706125000 (2 x+3)^4}-\frac{1977291 (4-9 x) \sqrt{3 x^2+2}}{514714375000 (2 x+3)^2}-\frac{5931873 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{257357187500 \sqrt{35}} \]

[Out]

(-1977291*(4 - 9*x)*Sqrt[2 + 3*x^2])/(514714375000*(3 + 2*x)^2) - (219699*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(147061
25000*(3 + 2*x)^4) - (73233*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(1050437500*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(35
0*(3 + 2*x)^10) - (1171*(2 + 3*x^2)^(7/2))/(110250*(3 + 2*x)^9) - (4393*(2 + 3*x^2)^(7/2))/(1715000*(3 + 2*x)^
8) - (739619*(2 + 3*x^2)^(7/2))/(1260525000*(3 + 2*x)^7) - (5931873*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2
])])/(257357187500*Sqrt[35])

________________________________________________________________________________________

Rubi [A]  time = 0.134203, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{739619 \left (3 x^2+2\right )^{7/2}}{1260525000 (2 x+3)^7}-\frac{4393 \left (3 x^2+2\right )^{7/2}}{1715000 (2 x+3)^8}-\frac{1171 \left (3 x^2+2\right )^{7/2}}{110250 (2 x+3)^9}-\frac{13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}-\frac{73233 (4-9 x) \left (3 x^2+2\right )^{5/2}}{1050437500 (2 x+3)^6}-\frac{219699 (4-9 x) \left (3 x^2+2\right )^{3/2}}{14706125000 (2 x+3)^4}-\frac{1977291 (4-9 x) \sqrt{3 x^2+2}}{514714375000 (2 x+3)^2}-\frac{5931873 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{257357187500 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^11,x]

[Out]

(-1977291*(4 - 9*x)*Sqrt[2 + 3*x^2])/(514714375000*(3 + 2*x)^2) - (219699*(4 - 9*x)*(2 + 3*x^2)^(3/2))/(147061
25000*(3 + 2*x)^4) - (73233*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(1050437500*(3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(35
0*(3 + 2*x)^10) - (1171*(2 + 3*x^2)^(7/2))/(110250*(3 + 2*x)^9) - (4393*(2 + 3*x^2)^(7/2))/(1715000*(3 + 2*x)^
8) - (739619*(2 + 3*x^2)^(7/2))/(1260525000*(3 + 2*x)^7) - (5931873*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2
])])/(257357187500*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 )^{5/2}}{(3+2 x)^{11}} \, dx &=-\frac{13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac{1}{350} \int \frac{(-410+117 x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx\\ &=-\frac{13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac{1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}+\frac{\int \frac{(28998-7026 x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx}{110250}\\ &=-\frac{13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac{1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac{4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac{\int \frac{(-1863024+237222 x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx}{30870000}\\ &=-\frac{13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac{1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac{4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac{739619 \left (2+3 x^2\right )^{7/2}}{1260525000 (3+2 x)^7}+\frac{219699 \int \frac{\left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{15006250}\\ &=-\frac{73233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{1050437500 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac{1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac{4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac{739619 \left (2+3 x^2\right )^{7/2}}{1260525000 (3+2 x)^7}+\frac{219699 \int \frac{\left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{105043750}\\ &=-\frac{219699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{14706125000 (3+2 x)^4}-\frac{73233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{1050437500 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac{1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac{4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac{739619 \left (2+3 x^2\right )^{7/2}}{1260525000 (3+2 x)^7}+\frac{1977291 \int \frac{\sqrt{2+3 x^2}}{(3+2 x)^3} \, dx}{7353062500}\\ &=-\frac{1977291 (4-9 x) \sqrt{2+3 x^2}}{514714375000 (3+2 x)^2}-\frac{219699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{14706125000 (3+2 x)^4}-\frac{73233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{1050437500 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac{1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac{4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac{739619 \left (2+3 x^2\right )^{7/2}}{1260525000 (3+2 x)^7}+\frac{5931873 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{257357187500}\\ &=-\frac{1977291 (4-9 x) \sqrt{2+3 x^2}}{514714375000 (3+2 x)^2}-\frac{219699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{14706125000 (3+2 x)^4}-\frac{73233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{1050437500 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac{1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac{4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac{739619 \left (2+3 x^2\right )^{7/2}}{1260525000 (3+2 x)^7}-\frac{5931873 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{257357187500}\\ &=-\frac{1977291 (4-9 x) \sqrt{2+3 x^2}}{514714375000 (3+2 x)^2}-\frac{219699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{14706125000 (3+2 x)^4}-\frac{73233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{1050437500 (3+2 x)^6}-\frac{13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac{1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac{4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac{739619 \left (2+3 x^2\right )^{7/2}}{1260525000 (3+2 x)^7}-\frac{5931873 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{257357187500 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.278751, size = 207, normalized size = 1.02 \[ \frac{1}{350} \left (-\frac{4393 \left (3 x^2+2\right )^{7/2}}{4900 (2 x+3)^8}-\frac{1171 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}-\frac{13 \left (3 x^2+2\right )^{7/2}}{(2 x+3)^{10}}-\frac{31711164625 \left (3 x^2+2\right )^{7/2}+219699 (2 x+3) \left (-945 (9 x-4) \sqrt{3 x^2+2} (2 x+3)^4-3675 (9 x-4) \left (3 x^2+2\right )^{3/2} (2 x+3)^2-17150 (9 x-4) \left (3 x^2+2\right )^{5/2}+162 \sqrt{35} (2 x+3)^6 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )\right )}{154414312500 (2 x+3)^7}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^11,x]

[Out]

((-13*(2 + 3*x^2)^(7/2))/(3 + 2*x)^10 - (1171*(2 + 3*x^2)^(7/2))/(315*(3 + 2*x)^9) - (4393*(2 + 3*x^2)^(7/2))/
(4900*(3 + 2*x)^8) - (31711164625*(2 + 3*x^2)^(7/2) + 219699*(3 + 2*x)*(-945*(3 + 2*x)^4*(-4 + 9*x)*Sqrt[2 + 3
*x^2] - 3675*(3 + 2*x)^2*(-4 + 9*x)*(2 + 3*x^2)^(3/2) - 17150*(-4 + 9*x)*(2 + 3*x^2)^(5/2) + 162*Sqrt[35]*(3 +
2*x)^6*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])]))/(154414312500*(3 + 2*x)^7))/350

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Maple [B]  time = 0.053, size = 341, normalized size = 1.7 \begin{align*} \text{result too large to display} \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)^11,x)

[Out]

-739619/161347200000/(x+3/2)^7*(3*(x+3/2)^2-9*x-19/4)^(7/2)-73233/33614000000/(x+3/2)^6*(3*(x+3/2)^2-9*x-19/4)
^(7/2)-659097/588245000000/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(7/2)-6371271/10294287500000/(x+3/2)^4*(3*(x+3/2)^
2-9*x-19/4)^(7/2)-65250603/180150031250000/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(7/2)-709847469/3152625546875000/(
x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(7/2)[phone]/55170947070312500*x*(3*(x+3/2)^2-9*x-19/4)^(5/2)-8239371597/
55170947070312500/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(7/2)[phone]/630525109375000*x*(3*(x+3/2)^2-9*x-19/4)^(3/2
)[phone]/18015003125000*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)-5931873/9007501562500*35^(1/2)*arctanh(2/35*(4-9*x)*3
5^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))[phone]/13792736767578125*(3*(x+3/2)^2-9*x-19/4)^(5/2)+5931873/90075015
62500*(12*(x+3/2)^2-36*x-19)^(1/2)+3954582/78815638671875*(3*(x+3/2)^2-9*x-19/4)^(3/2)-13/358400/(x+3/2)^10*(3
*(x+3/2)^2-9*x-19/4)^(7/2)-1171/56448000/(x+3/2)^9*(3*(x+3/2)^2-9*x-19/4)^(7/2)-4393/439040000/(x+3/2)^8*(3*(x
+3/2)^2-9*x-19/4)^(7/2)

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Maxima [B]  time = 1.62037, size = 671, normalized size = 3.32 \begin{align*} \text{result too large to display} \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)^11,x, algorithm="maxima")

[Out]

2129542407/3152625546875000*(3*x^2 + 2)^(5/2) - 13/350*(3*x^2 + 2)^(7/2)/(1024*x^10 + 15360*x^9 + 103680*x^8 +
414720*x^7 + 1088640*x^6 + 1959552*x^5 + 2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049) - 1171/1
10250*(3*x^2 + 2)^(7/2)/(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 +
314928*x^2 + 118098*x + 19683) - 4393/1715000*(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) - 739619/1260525000*(3*x^2 + 2)^(7/2)/(128*x^7 + 1344*x^6
+ 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 73233/525218750*(3*x^2 + 2)^(7/2)/(64*x^6
+ 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 659097/18382656250*(3*x^2 + 2)^(7/2)/(32*x^5 + 24
0*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 6371271/643392968750*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2
+ 216*x + 81) - 65250603/22518753906250*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 709847469/78815638671
8750*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) + 694029141/630525109375000*(3*x^2 + 2)^(3/2)*x + 3954582/7881563867
1875*(3*x^2 + 2)^(3/2) - 8239371597/3152625546875000*(3*x^2 + 2)^(5/2)/(2*x + 3) + 53386857/18015003125000*sqr
t(3*x^2 + 2)*x + 5931873/9007501562500*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3))
+ 5931873/4503750781250*sqrt(3*x^2 + 2)

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Fricas [A]  time = 2.08214, size = 826, normalized size = 4.09 \begin{align*} \frac{53386857 \, \sqrt{35}{\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\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 (7968937464 \, x^{9} + 101311348104 \, x^{8} + 544524933294 \, x^{7} + 1541962687104 \, x^{6} - 3078520541586 \, x^{5} + 11369945485836 \, x^{4} + 4704132871221 \, x^{3} + 18888919063956 \, x^{2} + 5421307926571 \, x + 5288003538036\right )} \sqrt{3 \, x^{2} + 2}}{162135028125000 \,{\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\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)^11,x, algorithm="fricas")

[Out]

1/162135028125000*(53386857*sqrt(35)*(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6 + 1959552*
x^5 + 2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 9
3*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(7968937464*x^9 + 101311348104*x^8 + 544524933294*x^7 + 1541962687
104*x^6 - 3078520541586*x^5 + 11369945485836*x^4 + 4704132871221*x^3 + 18888919063956*x^2 + 5421307926571*x +
5288003538036)*sqrt(3*x^2 + 2))/(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6 + 1959552*x^5 +
2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049)

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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)**(5/2)/(3+2*x)**11,x)

[Out]

Timed out

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Giac [B]  time = 1.32908, size = 733, normalized size = 3.63 \begin{align*} \frac{5931873}{9007501562500} \, \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 (168728832 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{19} + 4808771712 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{18} + 180483607296 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{17} + 2449600006086 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{16} + 1950011203428 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{15} + 11324343251586 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{14} - 129748494414672 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{13} - 114750161469717 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{12} - 790683925144266 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{11} - 64560900263031 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{10} - 520582739768172 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 409007369125548 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} - 2437545878994816 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} + 775661489485344 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 927787935017088 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 53888888658816 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} - 63600137874432 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 6293205518848 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 1046970832896 \, \sqrt{3} x + 25185777664 \, \sqrt{3} + 1046970832896 \, \sqrt{3 \, x^{2} + 2}\right )}}{65883440000000 \,{\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 )}^{10}} \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)^11,x, algorithm="giac")

[Out]

5931873/9007501562500*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x
- sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 9/65883440000000*(168728832*(sqrt(3)*x - sqrt(3*x^2 + 2))^19 +
4808771712*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^18 + 180483607296*(sqrt(3)*x - sqrt(3*x^2 + 2))^17 + 24496000
06086*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^16 + 1950011203428*(sqrt(3)*x - sqrt(3*x^2 + 2))^15 + 113243432515
86*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^14 - 129748494414672*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 - 1147501614697
17*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^12 - 790683925144266*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 - 6456090026303
1*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 - 520582739768172*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 409007369125548
*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 - 2437545878994816*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 775661489485344*
sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 927787935017088*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 53888888658816*sqr
t(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 63600137874432*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 6293205518848*sqrt(3)*
(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 1046970832896*sqrt(3)*x + 25185777664*sqrt(3) + 1046970832896*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)^10