3.2458 \(\int \frac{(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^7} \, dx\)

Optimal. Leaf size=197 \[ -\frac{(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}-\frac{7 (1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{1920 (2 x+3)^5}-\frac{7 (31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{25600 (2 x+3)^3}+\frac{63 (20678 x+44365) \sqrt{3 x^2+5 x+2}}{102400 (2 x+3)}-\frac{8547 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024}+\frac{6620481 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{204800 \sqrt{5}} \]

[Out]

(63*(44365 + 20678*x)*Sqrt[2 + 5*x + 3*x^2])/(102400*(3 + 2*x)) - (7*(40201 + 31174*x)*(2 + 5*x + 3*x^2)^(3/2)
)/(25600*(3 + 2*x)^3) - (7*(1301 + 1046*x)*(2 + 5*x + 3*x^2)^(5/2))/(1920*(3 + 2*x)^5) - ((11 + 3*x)*(2 + 5*x
+ 3*x^2)^(7/2))/(12*(3 + 2*x)^6) - (8547*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/1024 +
(6620481*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(204800*Sqrt[5])

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Rubi [A]  time = 0.131379, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {812, 810, 843, 621, 206, 724} \[ -\frac{(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}-\frac{7 (1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{1920 (2 x+3)^5}-\frac{7 (31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{25600 (2 x+3)^3}+\frac{63 (20678 x+44365) \sqrt{3 x^2+5 x+2}}{102400 (2 x+3)}-\frac{8547 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024}+\frac{6620481 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{204800 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(63*(44365 + 20678*x)*Sqrt[2 + 5*x + 3*x^2])/(102400*(3 + 2*x)) - (7*(40201 + 31174*x)*(2 + 5*x + 3*x^2)^(3/2)
)/(25600*(3 + 2*x)^3) - (7*(1301 + 1046*x)*(2 + 5*x + 3*x^2)^(5/2))/(1920*(3 + 2*x)^5) - ((11 + 3*x)*(2 + 5*x
+ 3*x^2)^(7/2))/(12*(3 + 2*x)^6) - (8547*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/1024 +
(6620481*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(204800*Sqrt[5])

Rule 812

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

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 )^{7/2}}{(3+2 x)^7} \, dx &=-\frac{(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}-\frac{7}{96} \int \frac{(-172-204 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx\\ &=-\frac{7 (1301+1046 x) \left (2+5 x+3 x^2\right )^{5/2}}{1920 (3+2 x)^5}-\frac{(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}+\frac{7 \int \frac{(31932+37032 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx}{15360}\\ &=-\frac{7 (40201+31174 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}-\frac{7 (1301+1046 x) \left (2+5 x+3 x^2\right )^{5/2}}{1920 (3+2 x)^5}-\frac{(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}-\frac{7 \int \frac{(-3816504-4466448 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{1228800}\\ &=\frac{63 (44365+20678 x) \sqrt{2+5 x+3 x^2}}{102400 (3+2 x)}-\frac{7 (40201+31174 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}-\frac{7 (1301+1046 x) \left (2+5 x+3 x^2\right )^{5/2}}{1920 (3+2 x)^5}-\frac{(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}+\frac{7 \int \frac{-60096816-70329600 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{9830400}\\ &=\frac{63 (44365+20678 x) \sqrt{2+5 x+3 x^2}}{102400 (3+2 x)}-\frac{7 (40201+31174 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}-\frac{7 (1301+1046 x) \left (2+5 x+3 x^2\right )^{5/2}}{1920 (3+2 x)^5}-\frac{(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}-\frac{25641 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{1024}+\frac{6620481 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{204800}\\ &=\frac{63 (44365+20678 x) \sqrt{2+5 x+3 x^2}}{102400 (3+2 x)}-\frac{7 (40201+31174 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}-\frac{7 (1301+1046 x) \left (2+5 x+3 x^2\right )^{5/2}}{1920 (3+2 x)^5}-\frac{(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}-\frac{25641}{512} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )-\frac{6620481 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{102400}\\ &=\frac{63 (44365+20678 x) \sqrt{2+5 x+3 x^2}}{102400 (3+2 x)}-\frac{7 (40201+31174 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}-\frac{7 (1301+1046 x) \left (2+5 x+3 x^2\right )^{5/2}}{1920 (3+2 x)^5}-\frac{(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}-\frac{8547 \sqrt{3} \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{1024}+\frac{6620481 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{204800 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.18909, size = 130, normalized size = 0.66 \[ \frac{-\frac{10 \sqrt{3 x^2+5 x+2} \left (2073600 x^7-23155200 x^6-550079616 x^5-2968126160 x^4-7425343520 x^3-9799959120 x^2-6648875480 x-1835461379\right )}{(2 x+3)^6}-19861443 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )-25641000 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{3072000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(-1835461379 - 6648875480*x - 9799959120*x^2 - 7425343520*x^3 - 2968126160*x^4 - 5
50079616*x^5 - 23155200*x^6 + 2073600*x^7))/(3 + 2*x)^6 - 19861443*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[
2 + 5*x + 3*x^2])] - 25641000*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/3072000

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Maple [B]  time = 0.016, size = 337, 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+5*x+2)^(7/2)/(3+2*x)^7,x)

[Out]

-21/4000/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(9/2)-1143/80000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(9/2)-459/50000/(x
+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)-63693/1000000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(9/2)-47169/250000*(5+6*x)
*(3*(x+3/2)^2-4*x-19/4)^(7/2)-349461/1000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)+47169/125000/(x+3/2)*(3*(x+3
/2)^2-4*x-19/4)^(9/2)-104517/160000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-210231/128000*(5+6*x)*(3*(x+3/2)^2-4*
x-19/4)^(1/2)-8547/1024*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)-6620481/1024000*5^(1/2)
*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+945783/1000000*(3*(x+3/2)^2-4*x-19/4)^(7/2)+6620
481/4000000*(3*(x+3/2)^2-4*x-19/4)^(5/2)+2206827/640000*(3*(x+3/2)^2-4*x-19/4)^(3/2)+6620481/1024000*(12*(x+3/
2)^2-16*x-19)^(1/2)-13/1920/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(9/2)

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Maxima [B]  time = 2.05624, size = 502, normalized size = 2.55 \begin{align*} \frac{191079}{1000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{30 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{21 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{125 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{1143 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{5000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{459 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{6250 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{63693 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{250000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{1048383}{500000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{368739}{4000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{47169 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{50000 \,{\left (2 \, x + 3\right )}} - \frac{313551}{80000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{116487}{640000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{630693}{64000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{8547}{1024} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{6620481}{1024000} \, \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{2415861}{512000} \, \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)^(7/2)/(3+2*x)^7,x, algorithm="maxima")

[Out]

191079/1000000*(3*x^2 + 5*x + 2)^(7/2) - 13/30*(3*x^2 + 5*x + 2)^(9/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3
+ 4860*x^2 + 2916*x + 729) - 21/125*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x +
243) - 1143/5000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 459/6250*(3*x^2 + 5*x + 2)
^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 63693/250000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 1048383/500000
*(3*x^2 + 5*x + 2)^(5/2)*x - 368739/4000000*(3*x^2 + 5*x + 2)^(5/2) + 47169/50000*(3*x^2 + 5*x + 2)^(7/2)/(2*x
+ 3) - 313551/80000*(3*x^2 + 5*x + 2)^(3/2)*x + 116487/640000*(3*x^2 + 5*x + 2)^(3/2) - 630693/64000*sqrt(3*x
^2 + 5*x + 2)*x - 8547/1024*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 6620481/1024000*sqrt(5)*l
og(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 2415861/512000*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.51256, size = 759, normalized size = 3.85 \begin{align*} \frac{25641000 \, \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 ) + 19861443 \, \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 (2073600 \, x^{7} - 23155200 \, x^{6} - 550079616 \, x^{5} - 2968126160 \, x^{4} - 7425343520 \, x^{3} - 9799959120 \, x^{2} - 6648875480 \, x - 1835461379\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{6144000 \,{\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)^(7/2)/(3+2*x)^7,x, algorithm="fricas")

[Out]

1/6144000*(25641000*sqrt(3)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)*log(-4*sqrt(3)*
sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 19861443*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*(2073600*x^7 - 23155200*x^6 - 550079616*x^5 - 2968126160*x^4 - 7425343520*x^3 - 9799959120*x^
2 - 6648875480*x - 1835461379)*sqrt(3*x^2 + 5*x + 2))/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 291
6*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)**(7/2)/(3+2*x)**7,x)

[Out]

Timed out

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Giac [B]  time = 1.29607, size = 625, normalized size = 3.17 \begin{align*} -\frac{9}{512} \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x - 121\right )} + \frac{6620481}{1024000} \, \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{8547}{1024} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac{1761054624 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 26119839696 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 522182992240 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 2060002389600 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 16013156565600 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 28585665528288 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 107556795368496 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 94759944627240 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 172447244925750 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 68627763126675 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 48408731804817 \, \sqrt{3} x + 5098539730008 \, \sqrt{3} - 48408731804817 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{307200 \,{\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)^(7/2)/(3+2*x)^7,x, algorithm="giac")

[Out]

-9/512*sqrt(3*x^2 + 5*x + 2)*(6*x - 121) + 6620481/1024000*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))) + 8547/1024*
sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) + 1/307200*(1761054624*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2))^11 + 26119839696*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 522182992240*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2))^9 + 2060002389600*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 16013156565600*(sqrt(
3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 28585665528288*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 1075567953684
96*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 94759944627240*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 1724
47244925750*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 68627763126675*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))
^2 + 48408731804817*sqrt(3)*x + 5098539730008*sqrt(3) - 48408731804817*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