3.23.5 \(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{10}} \, dx\)

Optimal. Leaf size=204 \[ -\frac {1321 \left (3 x^2+5 x+2\right )^{7/2}}{5250 (2 x+3)^7}-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{1800 (2 x+3)^8}-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}+\frac {6167 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{120000 (2 x+3)^6}-\frac {6167 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1920000 (2 x+3)^4}+\frac {6167 (8 x+7) \sqrt {3 x^2+5 x+2}}{25600000 (2 x+3)^2}-\frac {6167 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{51200000 \sqrt {5}} \]

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Rubi [A]  time = 0.13, antiderivative size = 204, normalized size of antiderivative = 1.00, 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 = {834, 806, 720, 724, 206} \begin {gather*} -\frac {1321 \left (3 x^2+5 x+2\right )^{7/2}}{5250 (2 x+3)^7}-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{1800 (2 x+3)^8}-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}+\frac {6167 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{120000 (2 x+3)^6}-\frac {6167 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1920000 (2 x+3)^4}+\frac {6167 (8 x+7) \sqrt {3 x^2+5 x+2}}{25600000 (2 x+3)^2}-\frac {6167 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{51200000 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^10,x]

[Out]

(6167*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(25600000*(3 + 2*x)^2) - (6167*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(1920
000*(3 + 2*x)^4) + (6167*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(120000*(3 + 2*x)^6) - (13*(2 + 5*x + 3*x^2)^(7/2)
)/(45*(3 + 2*x)^9) - (527*(2 + 5*x + 3*x^2)^(7/2))/(1800*(3 + 2*x)^8) - (1321*(2 + 5*x + 3*x^2)^(7/2))/(5250*(
3 + 2*x)^7) - (6167*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(51200000*Sqrt[5])

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 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 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 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])

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {1}{45} \int \frac {\left (-\frac {293}{2}+78 x\right ) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}+\frac {\int \frac {\left (\frac {11109}{2}-1581 x\right ) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx}{1800}\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}-\frac {1321 \left (2+5 x+3 x^2\right )^{7/2}}{5250 (3+2 x)^7}+\frac {6167 \int \frac {\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{2000}\\ &=\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{120000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}-\frac {1321 \left (2+5 x+3 x^2\right )^{7/2}}{5250 (3+2 x)^7}-\frac {6167 \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{48000}\\ &=-\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920000 (3+2 x)^4}+\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{120000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}-\frac {1321 \left (2+5 x+3 x^2\right )^{7/2}}{5250 (3+2 x)^7}+\frac {6167 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{1280000}\\ &=\frac {6167 (7+8 x) \sqrt {2+5 x+3 x^2}}{25600000 (3+2 x)^2}-\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920000 (3+2 x)^4}+\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{120000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}-\frac {1321 \left (2+5 x+3 x^2\right )^{7/2}}{5250 (3+2 x)^7}-\frac {6167 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{51200000}\\ &=\frac {6167 (7+8 x) \sqrt {2+5 x+3 x^2}}{25600000 (3+2 x)^2}-\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920000 (3+2 x)^4}+\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{120000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}-\frac {1321 \left (2+5 x+3 x^2\right )^{7/2}}{5250 (3+2 x)^7}+\frac {6167 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{25600000}\\ &=\frac {6167 (7+8 x) \sqrt {2+5 x+3 x^2}}{25600000 (3+2 x)^2}-\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920000 (3+2 x)^4}+\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{120000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}-\frac {1321 \left (2+5 x+3 x^2\right )^{7/2}}{5250 (3+2 x)^7}-\frac {6167 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{51200000 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 207, normalized size = 1.01 \begin {gather*} \frac {1}{45} \left (-\frac {3963 \left (3 x^2+5 x+2\right )^{7/2}}{350 (2 x+3)^7}-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^9}+\frac {18501 \left (\frac {32 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^6}-\frac {2 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}+\frac {3 (8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}+\frac {3 \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )}{256000}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^10,x]

[Out]

((-13*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^9 - (527*(2 + 5*x + 3*x^2)^(7/2))/(40*(3 + 2*x)^8) - (3963*(2 + 5*x +
3*x^2)^(7/2))/(350*(3 + 2*x)^7) + (18501*((3*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^2) - (2*(7 + 8*x)
*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^4 + (32*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^6 + (3*ArcTanh[(-7 -
8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(40*Sqrt[5])))/256000)/45

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IntegrateAlgebraic [A]  time = 0.86, size = 101, normalized size = 0.50 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (333241344 x^8+4204480128 x^7+23288995392 x^6+76435267296 x^5+149661252080 x^4+173974546136 x^3+117870367452 x^2+43246799138 x+6706847909\right )}{1612800000 (2 x+3)^9}-\frac {6167 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{25600000 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^10,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(6706847909 + 43246799138*x + 117870367452*x^2 + 173974546136*x^3 + 149661252080*x^4 +
76435267296*x^5 + 23288995392*x^6 + 4204480128*x^7 + 333241344*x^8))/(1612800000*(3 + 2*x)^9) - (6167*ArcTanh[
Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(25600000*Sqrt[5])

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fricas [A]  time = 0.43, size = 201, normalized size = 0.99 \begin {gather*} \frac {388521 \, \sqrt {5} {\left (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\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 (333241344 \, x^{8} + 4204480128 \, x^{7} + 23288995392 \, x^{6} + 76435267296 \, x^{5} + 149661252080 \, x^{4} + 173974546136 \, x^{3} + 117870367452 \, x^{2} + 43246799138 \, x + 6706847909\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{32256000000 \, {\left (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\right )}} \end {gather*}

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)^10,x, algorithm="fricas")

[Out]

1/32256000000*(388521*sqrt(5)*(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)*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*(333241344*x^8 + 4204480128*x^7 + 23288995392*x^6 + 76435267296*x^5 + 149661252080*x^4
+ 173974546136*x^3 + 117870367452*x^2 + 43246799138*x + 6706847909)*sqrt(3*x^2 + 5*x + 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)

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giac [B]  time = 0.35, size = 563, normalized size = 2.76 \begin {gather*} -\frac {6167}{256000000} \, \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 {99461376 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} + 2536265088 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} - 83954355072 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 341000936640 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 17778066768000 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 177356386111968 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 2399974462831392 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 6844601123556624 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 41172892580130560 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 60936872688585000 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 204498063708405624 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 174436297943297292 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 339439601929212792 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 164994557892929730 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 174936772514694750 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 42504221165006223 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 19065836258759367 \, \sqrt {3} x + 1323473153587704 \, \sqrt {3} - 19065836258759367 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{1612800000 \, {\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 )}^{9}} \end {gather*}

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)^10,x, algorithm="giac")

[Out]

-6167/256000000*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))) + 1/1612800000*(99461376*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^17 + 2536265088*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^16 - 83954355072*(sqrt(3)*x - sqrt(3*x^2 + 5
*x + 2))^15 - 341000936640*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 + 17778066768000*(sqrt(3)*x - sqrt(3
*x^2 + 5*x + 2))^13 + 177356386111968*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 2399974462831392*(sqrt(
3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 6844601123556624*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 411728925
80130560*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 60936872688585000*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))
^8 + 204498063708405624*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 174436297943297292*sqrt(3)*(sqrt(3)*x - sqrt(3
*x^2 + 5*x + 2))^6 + 339439601929212792*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 164994557892929730*sqrt(3)*(sq
rt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 174936772514694750*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 42504221165006
223*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 19065836258759367*sqrt(3)*x + 1323473153587704*sqrt(3) - 1
9065836258759367*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)^9

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maple [A]  time = 0.10, size = 332, normalized size = 1.63 \begin {gather*} \frac {6167 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{256000000}-\frac {6167 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{1920000 \left (x +\frac {3}{2}\right )^{6}}-\frac {1321 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{672000 \left (x +\frac {3}{2}\right )^{7}}-\frac {376187 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{30000000 \left (x +\frac {3}{2}\right )^{3}}-\frac {6167 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{1200000 \left (x +\frac {3}{2}\right )^{5}}-\frac {11464453 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{600000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {3583027 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{250000000}-\frac {178843 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{120000000}-\frac {3583027 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{125000000 \left (x +\frac {3}{2}\right )}+\frac {6167 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{32000000}-\frac {6167 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{256000000}-\frac {129507 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{16000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {6167 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{480000000}-\frac {6167 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1000000000}-\frac {527 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{460800 \left (x +\frac {3}{2}\right )^{8}}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{23040 \left (x +\frac {3}{2}\right )^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(5/2)/(2*x+3)^10,x)

[Out]

-6167/1920000/(x+3/2)^6*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-1321/672000/(x+3/2)^7*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-3761
87/30000000/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-6167/1200000/(x+3/2)^5*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-11464
453/600000000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(7/2)+3583027/250000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-
178843/120000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-3583027/125000000/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)
+6167/32000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)+6167/256000000*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-1
6*x+12*(x+3/2)^2-19)^(1/2))-6167/256000000*(-16*x+12*(x+3/2)^2-19)^(1/2)-129507/16000000/(x+3/2)^4*(-4*x+3*(x+
3/2)^2-19/4)^(7/2)-6167/480000000*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-6167/1000000000*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-
527/460800/(x+3/2)^8*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-13/23040/(x+3/2)^9*(-4*x+3*(x+3/2)^2-19/4)^(7/2)

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maxima [B]  time = 1.40, size = 484, normalized size = 2.37 \begin {gather*} \frac {11464453}{200000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{45 \, {\left (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\right )}} - \frac {527 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{1800 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac {1321 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{5250 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {6167 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{30000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {6167 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{37500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {129507 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{1000000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {376187 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{3750000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {11464453 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{150000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {178843}{20000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {3583027}{480000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {3583027 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{50000000 \, {\left (2 \, x + 3\right )}} + \frac {18501}{16000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {6167}{256000000} \, \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 {117173}{128000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

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)^10,x, algorithm="maxima")

[Out]

11464453/200000000*(3*x^2 + 5*x + 2)^(5/2) - 13/45*(3*x^2 + 5*x + 2)^(7/2)/(512*x^9 + 6912*x^8 + 41472*x^7 + 1
45152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x + 19683) - 527/1800*(3*x^2 + 5*x + 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) - 13
21/5250*(3*x^2 + 5*x + 2)^(7/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x +
2187) - 6167/30000*(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)
- 6167/37500*(3*x^2 + 5*x + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 129507/1000000*(
3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 376187/3750000*(3*x^2 + 5*x + 2)^(7/2)/(8*x^
3 + 36*x^2 + 54*x + 27) - 11464453/150000000*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) - 178843/20000000*(3*x
^2 + 5*x + 2)^(3/2)*x - 3583027/480000000*(3*x^2 + 5*x + 2)^(3/2) - 3583027/50000000*(3*x^2 + 5*x + 2)^(5/2)/(
2*x + 3) + 18501/16000000*sqrt(3*x^2 + 5*x + 2)*x + 6167/256000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/a
bs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 117173/128000000*sqrt(3*x^2 + 5*x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{10}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^10,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^10, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 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}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 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}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 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}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 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}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 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}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 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}\, dx \end {gather*}

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)**10,x)

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**8 + 414720*x**7 + 1088640*x**6 + 195
9552*x**5 + 2449440*x**4 + 2099520*x**3 + 1180980*x**2 + 393660*x + 59049), x) - Integral(-96*x*sqrt(3*x**2 +
5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**8 + 414720*x**7 + 1088640*x**6 + 1959552*x**5 + 2449440*x**4 + 2
099520*x**3 + 1180980*x**2 + 393660*x + 59049), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 1
5360*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), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 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),
x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 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), x) - Integral(9*x**5*sqrt
(3*x**2 + 5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**8 + 414720*x**7 + 1088640*x**6 + 1959552*x**5 + 244944
0*x**4 + 2099520*x**3 + 1180980*x**2 + 393660*x + 59049), x)

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