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]

17899/128000000*(7+8*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4-17899/8000000*(7+8*x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^6+7671
/200000*(7+8*x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^8-13/55*(3*x^2+5*x+2)^(9/2)/(3+2*x)^11-621/2750*(3*x^2+5*x+2)^(9/2
)/(3+2*x)^10-3904/20625*(3*x^2+5*x+2)^(9/2)/(3+2*x)^9+53697/51200000000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*
x+2)^(1/2))*5^(1/2)-53697/5120000000*(7+8*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2

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Rubi [A]  time = 0.15, antiderivative size = 234, normalized size of antiderivative = 1.00, 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 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 )^{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*}

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Mathematica [A]  time = 0.31, 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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fricas [A]  time = 0.78, size = 230, normalized size = 0.98 \[ \frac {1772001 \, \sqrt {5} {\left (2048 \, x^{11} + 33792 \, x^{10} + 253440 \, x^{9} + 1140480 \, x^{8} + 3421440 \, x^{7} + 7185024 \, x^{6} + 10777536 \, x^{5} + 11547360 \, x^{4} + 8660520 \, x^{3} + 4330260 \, x^{2} + 1299078 \, x + 177147\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (30557343744 \, x^{10} + 479034140160 \, x^{9} + 3387337708800 \, x^{8} + 14992486229760 \, x^{7} + 41485308553600 \, x^{6} + 72251114756992 \, x^{5} + 80329740407040 \, x^{4} + 56898923222800 \, x^{3} + 24817198954840 \, x^{2} + 6058472990850 \, x + 629890144539\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{3379200000000 \, {\left (2048 \, x^{11} + 33792 \, x^{10} + 253440 \, x^{9} + 1140480 \, x^{8} + 3421440 \, x^{7} + 7185024 \, x^{6} + 10777536 \, x^{5} + 11547360 \, x^{4} + 8660520 \, x^{3} + 4330260 \, x^{2} + 1299078 \, x + 177147\right )}} \]

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)

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giac [B]  time = 0.38, size = 665, normalized size = 2.84 \[ \frac {53697}{51200000000} \, \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 {1814529024 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{21} + 57157664256 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{20} + 57290941171200 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{19} + 557490020440320 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{18} + 3116590396465920 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} - 40571342658595584 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} - 1098653419392131328 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 4929229513296950400 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} - 44860439685628251520 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} - 101067124429527527040 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} - 530008429621517017088 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} - 735944911884403670592 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 2465807894359584887200 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 2226326899649908579920 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 4870616002552398497520 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 2849658548882889760632 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 3959763769847021107884 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 1420163541040959876150 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 1141537424727199856070 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 215130617786249721765 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 76323347715579462729 \, \sqrt {3} x - 4261520459402725896 \, \sqrt {3} + 76323347715579462729 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{168960000000 \, {\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 )}^{11}} \]

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

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maple [B]  time = 0.19, size = 411, normalized size = 1.76 \[ -\frac {53697 \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 )}{51200000000}-\frac {621 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{2816000 \left (x +\frac {3}{2}\right )^{10}}-\frac {7671 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{12800000 \left (x +\frac {3}{2}\right )^{8}}-\frac {7671 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{8000000 \left (x +\frac {3}{2}\right )^{7}}-\frac {14735991 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{4000000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {427819341 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{50000000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {14112083 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{2500000000 \left (x +\frac {3}{2}\right )^{3}}+\frac {80215647 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{12500000000}-\frac {31197957 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{50000000000}-\frac {80215647 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{6250000000 \left (x +\frac {3}{2}\right )}+\frac {519071 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{8000000000}-\frac {53697 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{6400000000}+\frac {53697 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{51200000000}+\frac {17899 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{32000000000}+\frac {53697 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{200000000000}+\frac {7671 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{50000000000}-\frac {237801 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{100000000 \left (x +\frac {3}{2}\right )^{5}}-\frac {48583 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{32000000 \left (x +\frac {3}{2}\right )^{6}}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{112640 \left (x +\frac {3}{2}\right )^{11}}-\frac {61 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{165000 \left (x +\frac {3}{2}\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(7/2)/(2*x+3)^12,x)

[Out]

-621/2816000/(x+3/2)^10*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-7671/12800000/(x+3/2)^8*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-76
71/8000000/(x+3/2)^7*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-14735991/4000000000/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(9/2)
-427819341/50000000000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-14112083/2500000000/(x+3/2)^3*(-4*x+3*(x+3/2)^2
-19/4)^(9/2)[phone]/12500000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-31197957/50000000000*(6*x+5)*(-4*x+3*(
x+3/2)^2-19/4)^(5/2)-80215647/6250000000/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(9/2)+519071/8000000000*(6*x+5)*(-4*x
+3*(x+3/2)^2-19/4)^(3/2)-53697/6400000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-53697/51200000000*5^(1/2)*arct
anh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))+53697/51200000000*(-16*x+12*(x+3/2)^2-19)^(1/2)+1789
9/32000000000*(-4*x+3*(x+3/2)^2-19/4)^(3/2)+53697/200000000000*(-4*x+3*(x+3/2)^2-19/4)^(5/2)+7671/50000000000*
(-4*x+3*(x+3/2)^2-19/4)^(7/2)-237801/100000000/(x+3/2)^5*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-48583/32000000/(x+3/2)^
6*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-13/112640/(x+3/2)^11*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-61/165000/(x+3/2)^9*(-4*x+3
*(x+3/2)^2-19/4)^(9/2)

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maxima [B]  time = 1.42, size = 650, normalized size = 2.78 \[ \frac {1283458023}{50000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{55 \, {\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 )}} - \frac {621 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{2750 \, {\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 )}} - \frac {3904 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{20625 \, {\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 {7671 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50000 \, {\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 {7671 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{62500 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {48583 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{500000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {237801 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{3125000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {14735991 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{250000000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {14112083 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{312500000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {427819341 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{12500000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {93593871}{25000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {623905443}{200000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {80215647 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{2500000000 \, {\left (2 \, x + 3\right )}} + \frac {1557213}{4000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {10399319}{32000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {161091}{3200000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {53697}{51200000000} \, \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 {1020243}{25600000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^{12}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^12,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^12, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\, dx \]

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]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(4096*x**12 + 73728*x**11 + 608256*x**10 + 3041280*x**9 + 10264320*x**8 +
24634368*x**7 + 43110144*x**6 + 55427328*x**5 + 51963120*x**4 + 34642080*x**3 + 15588936*x**2 + 4251528*x + 5
31441), x) - Integral(-292*x*sqrt(3*x**2 + 5*x + 2)/(4096*x**12 + 73728*x**11 + 608256*x**10 + 3041280*x**9 +
10264320*x**8 + 24634368*x**7 + 43110144*x**6 + 55427328*x**5 + 51963120*x**4 + 34642080*x**3 + 15588936*x**2
+ 4251528*x + 531441), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(4096*x**12 + 73728*x**11 + 608256*x**10
+ 3041280*x**9 + 10264320*x**8 + 24634368*x**7 + 43110144*x**6 + 55427328*x**5 + 51963120*x**4 + 34642080*x**
3 + 15588936*x**2 + 4251528*x + 531441), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(4096*x**12 + 73728*x
**11 + 608256*x**10 + 3041280*x**9 + 10264320*x**8 + 24634368*x**7 + 43110144*x**6 + 55427328*x**5 + 51963120*
x**4 + 34642080*x**3 + 15588936*x**2 + 4251528*x + 531441), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(4
096*x**12 + 73728*x**11 + 608256*x**10 + 3041280*x**9 + 10264320*x**8 + 24634368*x**7 + 43110144*x**6 + 554273
28*x**5 + 51963120*x**4 + 34642080*x**3 + 15588936*x**2 + 4251528*x + 531441), x) - Integral(-396*x**5*sqrt(3*
x**2 + 5*x + 2)/(4096*x**12 + 73728*x**11 + 608256*x**10 + 3041280*x**9 + 10264320*x**8 + 24634368*x**7 + 4311
0144*x**6 + 55427328*x**5 + 51963120*x**4 + 34642080*x**3 + 15588936*x**2 + 4251528*x + 531441), x) - Integral
(27*x**7*sqrt(3*x**2 + 5*x + 2)/(4096*x**12 + 73728*x**11 + 608256*x**10 + 3041280*x**9 + 10264320*x**8 + 2463
4368*x**7 + 43110144*x**6 + 55427328*x**5 + 51963120*x**4 + 34642080*x**3 + 15588936*x**2 + 4251528*x + 531441
), x)

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