3.2463 \(\int \frac{(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^{12}} \, dx\)
Optimal. Leaf size=234 \[ -\frac{3904 \left (3 x^2+5 x+2\right )^{9/2}}{20625 (2 x+3)^9}-\frac{621 \left (3 x^2+5 x+2\right )^{9/2}}{2750 (2 x+3)^{10}}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}+\frac{7671 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{200000 (2 x+3)^8}-\frac{17899 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{8000000 (2 x+3)^6}+\frac{17899 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{128000000 (2 x+3)^4}-\frac{53697 (8 x+7) \sqrt{3 x^2+5 x+2}}{5120000000 (2 x+3)^2}+\frac{53697 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{10240000000 \sqrt{5}} \]
[Out]
(-53697*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(5120000000*(3 + 2*x)^2) + (17899*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/
(128000000*(3 + 2*x)^4) - (17899*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(8000000*(3 + 2*x)^6) + (7671*(7 + 8*x)*(2
+ 5*x + 3*x^2)^(7/2))/(200000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(55*(3 + 2*x)^11) - (621*(2 + 5*x +
3*x^2)^(9/2))/(2750*(3 + 2*x)^10) - (3904*(2 + 5*x + 3*x^2)^(9/2))/(20625*(3 + 2*x)^9) + (53697*ArcTanh[(7 +
8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(10240000000*Sqrt[5])
________________________________________________________________________________________
Rubi [A] time = 0.152495, antiderivative size = 234, normalized size of antiderivative = 1.,
number of steps used = 9, 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} \[ -\frac{3904 \left (3 x^2+5 x+2\right )^{9/2}}{20625 (2 x+3)^9}-\frac{621 \left (3 x^2+5 x+2\right )^{9/2}}{2750 (2 x+3)^{10}}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}+\frac{7671 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{200000 (2 x+3)^8}-\frac{17899 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{8000000 (2 x+3)^6}+\frac{17899 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{128000000 (2 x+3)^4}-\frac{53697 (8 x+7) \sqrt{3 x^2+5 x+2}}{5120000000 (2 x+3)^2}+\frac{53697 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{10240000000 \sqrt{5}} \]
Antiderivative was successfully verified.
[In]
Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^12,x]
[Out]
(-53697*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(5120000000*(3 + 2*x)^2) + (17899*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/
(128000000*(3 + 2*x)^4) - (17899*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(8000000*(3 + 2*x)^6) + (7671*(7 + 8*x)*(2
+ 5*x + 3*x^2)^(7/2))/(200000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(55*(3 + 2*x)^11) - (621*(2 + 5*x +
3*x^2)^(9/2))/(2750*(3 + 2*x)^10) - (3904*(2 + 5*x + 3*x^2)^(9/2))/(20625*(3 + 2*x)^9) + (53697*ArcTanh[(7 +
8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(10240000000*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 )^{7/2}}{(3+2 x)^{12}} \, dx &=-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{1}{55} \int \frac{\left (-\frac{387}{2}+78 x\right ) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{11}} \, dx\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{621 \left (2+5 x+3 x^2\right )^{9/2}}{2750 (3+2 x)^{10}}+\frac{\int \frac{\left (\frac{17835}{2}-1863 x\right ) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx}{2750}\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{621 \left (2+5 x+3 x^2\right )^{9/2}}{2750 (3+2 x)^{10}}-\frac{3904 \left (2+5 x+3 x^2\right )^{9/2}}{20625 (3+2 x)^9}+\frac{7671 \int \frac{\left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx}{2500}\\ &=\frac{7671 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{200000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{621 \left (2+5 x+3 x^2\right )^{9/2}}{2750 (3+2 x)^{10}}-\frac{3904 \left (2+5 x+3 x^2\right )^{9/2}}{20625 (3+2 x)^9}-\frac{53697 \int \frac{\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{400000}\\ &=-\frac{17899 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{8000000 (3+2 x)^6}+\frac{7671 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{200000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{621 \left (2+5 x+3 x^2\right )^{9/2}}{2750 (3+2 x)^{10}}-\frac{3904 \left (2+5 x+3 x^2\right )^{9/2}}{20625 (3+2 x)^9}+\frac{17899 \int \frac{\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{3200000}\\ &=\frac{17899 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{128000000 (3+2 x)^4}-\frac{17899 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{8000000 (3+2 x)^6}+\frac{7671 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{200000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{621 \left (2+5 x+3 x^2\right )^{9/2}}{2750 (3+2 x)^{10}}-\frac{3904 \left (2+5 x+3 x^2\right )^{9/2}}{20625 (3+2 x)^9}-\frac{53697 \int \frac{\sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{256000000}\\ &=-\frac{53697 (7+8 x) \sqrt{2+5 x+3 x^2}}{5120000000 (3+2 x)^2}+\frac{17899 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{128000000 (3+2 x)^4}-\frac{17899 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{8000000 (3+2 x)^6}+\frac{7671 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{200000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{621 \left (2+5 x+3 x^2\right )^{9/2}}{2750 (3+2 x)^{10}}-\frac{3904 \left (2+5 x+3 x^2\right )^{9/2}}{20625 (3+2 x)^9}+\frac{53697 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{10240000000}\\ &=-\frac{53697 (7+8 x) \sqrt{2+5 x+3 x^2}}{5120000000 (3+2 x)^2}+\frac{17899 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{128000000 (3+2 x)^4}-\frac{17899 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{8000000 (3+2 x)^6}+\frac{7671 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{200000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{621 \left (2+5 x+3 x^2\right )^{9/2}}{2750 (3+2 x)^{10}}-\frac{3904 \left (2+5 x+3 x^2\right )^{9/2}}{20625 (3+2 x)^9}-\frac{53697 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{5120000000}\\ &=-\frac{53697 (7+8 x) \sqrt{2+5 x+3 x^2}}{5120000000 (3+2 x)^2}+\frac{17899 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{128000000 (3+2 x)^4}-\frac{17899 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{8000000 (3+2 x)^6}+\frac{7671 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{200000 (3+2 x)^8}-\frac{13 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac{621 \left (2+5 x+3 x^2\right )^{9/2}}{2750 (3+2 x)^{10}}-\frac{3904 \left (2+5 x+3 x^2\right )^{9/2}}{20625 (3+2 x)^9}+\frac{53697 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{10240000000 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.337052, size = 239, normalized size = 1.02 \[ \frac{1}{55} \left (-\frac{3904 \left (3 x^2+5 x+2\right )^{9/2}}{375 (2 x+3)^9}-\frac{621 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}-\frac{13 \left (3 x^2+5 x+2\right )^{9/2}}{(2 x+3)^{11}}+\frac{84381 \left (\frac{2 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^8}-\frac{7 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}+\frac{7 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{960 (2 x+3)^4}-\frac{7 \left (\frac{10 \sqrt{3 x^2+5 x+2} (8 x+7)}{(2 x+3)^2}+\sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )\right )}{128000}\right )}{80000}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^12,x]
[Out]
((-13*(2 + 5*x + 3*x^2)^(9/2))/(3 + 2*x)^11 - (621*(2 + 5*x + 3*x^2)^(9/2))/(50*(3 + 2*x)^10) - (3904*(2 + 5*x
+ 3*x^2)^(9/2))/(375*(3 + 2*x)^9) + (84381*((7*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(960*(3 + 2*x)^4) - (7*(7 +
8*x)*(2 + 5*x + 3*x^2)^(5/2))/(60*(3 + 2*x)^6) + (2*(7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^8 - (7*((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])]))
/128000))/80000)/55
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Maple [B] time = 0.082, size = 411, normalized size = 1.8 \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)^12,x)
[Out]
-237801/100000000/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(9/2)-14735991/4000000000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^
(9/2)-14112083/2500000000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)-427819341/50000000000/(x+3/2)^2*(3*(x+3/2)^2-
4*x-19/4)^(9/2)+80215647/12500000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-31197957/50000000000*(5+6*x)*(3*(x+3
/2)^2-4*x-19/4)^(5/2)-80215647/6250000000/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)+519071/8000000000*(5+6*x)*(3*(x
+3/2)^2-4*x-19/4)^(3/2)-53697/6400000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-53697/51200000000*5^(1/2)*arctan
h(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+7671/50000000000*(3*(x+3/2)^2-4*x-19/4)^(7/2)+53697/200
000000000*(3*(x+3/2)^2-4*x-19/4)^(5/2)+17899/32000000000*(3*(x+3/2)^2-4*x-19/4)^(3/2)+53697/51200000000*(12*(x
+3/2)^2-16*x-19)^(1/2)-13/112640/(x+3/2)^11*(3*(x+3/2)^2-4*x-19/4)^(9/2)-621/2816000/(x+3/2)^10*(3*(x+3/2)^2-4
*x-19/4)^(9/2)-61/165000/(x+3/2)^9*(3*(x+3/2)^2-4*x-19/4)^(9/2)-7671/12800000/(x+3/2)^8*(3*(x+3/2)^2-4*x-19/4)
^(9/2)-7671/8000000/(x+3/2)^7*(3*(x+3/2)^2-4*x-19/4)^(9/2)-48583/32000000/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(9/
2)
________________________________________________________________________________________
Maxima [B] time = 2.30452, size = 878, normalized size = 3.75 \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+5*x+2)^(7/2)/(3+2*x)^12,x, algorithm="maxima")
[Out]
1283458023/50000000000*(3*x^2 + 5*x + 2)^(7/2) - 13/55*(3*x^2 + 5*x + 2)^(9/2)/(2048*x^11 + 33792*x^10 + 25344
0*x^9 + 1140480*x^8 + 3421440*x^7 + 7185024*x^6 + 10777536*x^5 + 11547360*x^4 + 8660520*x^3 + 4330260*x^2 + 12
99078*x + 177147) - 621/2750*(3*x^2 + 5*x + 2)^(9/2)/(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 108864
0*x^6 + 1959552*x^5 + 2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049) - 3904/20625*(3*x^2 + 5*x +
2)^(9/2)/(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 11
8098*x + 19683) - 7671/50000*(3*x^2 + 5*x + 2)^(9/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 +
108864*x^3 + 81648*x^2 + 34992*x + 6561) - 7671/62500*(3*x^2 + 5*x + 2)^(9/2)/(128*x^7 + 1344*x^6 + 6048*x^5
+ 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 48583/500000*(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) - 237801/3125000*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 +
720*x^3 + 1080*x^2 + 810*x + 243) - 14735991/250000000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 2
16*x + 81) - 14112083/312500000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 427819341/12500000000*(
3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 93593871/25000000000*(3*x^2 + 5*x + 2)^(5/2)*x - 623905443/2000000
00000*(3*x^2 + 5*x + 2)^(5/2) - 80215647/2500000000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 1557213/4000000000*(3*
x^2 + 5*x + 2)^(3/2)*x + 10399319/32000000000*(3*x^2 + 5*x + 2)^(3/2) - 161091/3200000000*sqrt(3*x^2 + 5*x + 2
)*x - 53697/51200000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 10202
43/25600000000*sqrt(3*x^2 + 5*x + 2)
________________________________________________________________________________________
Fricas [A] time = 1.70533, size = 929, normalized size = 3.97 \begin{align*} \frac{1772001 \, \sqrt{5}{\left (2048 \, x^{11} + 33792 \, x^{10} + 253440 \, x^{9} + 1140480 \, x^{8} + 3421440 \, x^{7} + 7185024 \, x^{6} + 10777536 \, x^{5} + 11547360 \, x^{4} + 8660520 \, x^{3} + 4330260 \, x^{2} + 1299078 \, x + 177147\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 (30557343744 \, x^{10} + 479034140160 \, x^{9} + 3387337708800 \, x^{8} + 14992486229760 \, x^{7} + 41485308553600 \, x^{6} + 72251114756992 \, x^{5} + 80329740407040 \, x^{4} + 56898923222800 \, x^{3} + 24817198954840 \, x^{2} + 6058472990850 \, x + 629890144539\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{3379200000000 \,{\left (2048 \, x^{11} + 33792 \, x^{10} + 253440 \, x^{9} + 1140480 \, x^{8} + 3421440 \, x^{7} + 7185024 \, x^{6} + 10777536 \, x^{5} + 11547360 \, x^{4} + 8660520 \, x^{3} + 4330260 \, x^{2} + 1299078 \, x + 177147\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)^12,x, algorithm="fricas")
[Out]
1/3379200000000*(1772001*sqrt(5)*(2048*x^11 + 33792*x^10 + 253440*x^9 + 1140480*x^8 + 3421440*x^7 + 7185024*x^
6 + 10777536*x^5 + 11547360*x^4 + 8660520*x^3 + 4330260*x^2 + 1299078*x + 177147)*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*(30557343744*x^10 + 479034140160*x^9 + 338
7337708800*x^8 + 14992486229760*x^7 + 41485308553600*x^6 + 72251114756992*x^5 + 80329740407040*x^4 + 568989232
22800*x^3 + 24817198954840*x^2 + 6058472990850*x + 629890144539)*sqrt(3*x^2 + 5*x + 2))/(2048*x^11 + 33792*x^1
0 + 253440*x^9 + 1140480*x^8 + 3421440*x^7 + 7185024*x^6 + 10777536*x^5 + 11547360*x^4 + 8660520*x^3 + 4330260
*x^2 + 1299078*x + 177147)
________________________________________________________________________________________
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)**12,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.30352, size = 898, normalized size = 3.84 \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+5*x+2)^(7/2)/(3+2*x)^12,x, algorithm="giac")
[Out]
53697/51200000000*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))) - 1/168960000000*(1814529024*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^21 + 57157664256*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^20 + 57290941171200*(sqrt(3)*x - sqrt
(3*x^2 + 5*x + 2))^19 + 557490020440320*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^18 + 3116590396465920*(sqr
t(3)*x - sqrt(3*x^2 + 5*x + 2))^17 - 40571342658595584*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^16 - 109865
3419392131328*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^15 - 4929229513296950400*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5
*x + 2))^14 - 44860439685628251520*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 - 101067124429527527040*sqrt(3)*(sqr
t(3)*x - sqrt(3*x^2 + 5*x + 2))^12 - 530008429621517017088*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 - 7359449118
84403670592*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 2465807894359584887200*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2))^9 - 2226326899649908579920*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 4870616002552398497520*(s
qrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 2849658548882889760632*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 3
959763769847021107884*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 1420163541040959876150*sqrt(3)*(sqrt(3)*x - sqrt
(3*x^2 + 5*x + 2))^4 - 1141537424727199856070*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 215130617786249721765*sq
rt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 76323347715579462729*sqrt(3)*x - 4261520459402725896*sqrt(3) + 7
6323347715579462729*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 - s
qrt(3*x^2 + 5*x + 2)) + 11)^11