3.23.21 \(\int \frac {(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^{11}} \, dx\)

Optimal. Leaf size=209 \[ -\frac {29 \left (3 x^2+5 x+2\right )^{9/2}}{125 (2 x+3)^9}-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}+\frac {1893 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{40000 (2 x+3)^8}-\frac {4417 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{1600000 (2 x+3)^6}+\frac {4417 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{25600000 (2 x+3)^4}-\frac {13251 (8 x+7) \sqrt {3 x^2+5 x+2}}{1024000000 (2 x+3)^2}+\frac {13251 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2048000000 \sqrt {5}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 209, 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 {29 \left (3 x^2+5 x+2\right )^{9/2}}{125 (2 x+3)^9}-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{50 (2 x+3)^{10}}+\frac {1893 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{40000 (2 x+3)^8}-\frac {4417 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{1600000 (2 x+3)^6}+\frac {4417 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{25600000 (2 x+3)^4}-\frac {13251 (8 x+7) \sqrt {3 x^2+5 x+2}}{1024000000 (2 x+3)^2}+\frac {13251 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2048000000 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^11,x]

[Out]

(-13251*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(1024000000*(3 + 2*x)^2) + (4417*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(
25600000*(3 + 2*x)^4) - (4417*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(1600000*(3 + 2*x)^6) + (1893*(7 + 8*x)*(2 +
5*x + 3*x^2)^(7/2))/(40000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(50*(3 + 2*x)^10) - (29*(2 + 5*x + 3*x^
2)^(9/2))/(125*(3 + 2*x)^9) + (13251*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(2048000000*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)^{11}} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {1}{50} \int \frac {\left (-\frac {405}{2}+39 x\right ) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}+\frac {1893}{500} \int \frac {\left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx\\ &=\frac {1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}-\frac {13251 \int \frac {\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{80000}\\ &=-\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac {1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}+\frac {4417 \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{640000}\\ &=\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600000 (3+2 x)^4}-\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac {1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}-\frac {13251 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{51200000}\\ &=-\frac {13251 (7+8 x) \sqrt {2+5 x+3 x^2}}{1024000000 (3+2 x)^2}+\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600000 (3+2 x)^4}-\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac {1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}+\frac {13251 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{2048000000}\\ &=-\frac {13251 (7+8 x) \sqrt {2+5 x+3 x^2}}{1024000000 (3+2 x)^2}+\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600000 (3+2 x)^4}-\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac {1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}-\frac {13251 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{1024000000}\\ &=-\frac {13251 (7+8 x) \sqrt {2+5 x+3 x^2}}{1024000000 (3+2 x)^2}+\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600000 (3+2 x)^4}-\frac {4417 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{1600000 (3+2 x)^6}+\frac {1893 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{40000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{50 (3+2 x)^{10}}-\frac {29 \left (2+5 x+3 x^2\right )^{9/2}}{125 (3+2 x)^9}+\frac {13251 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{2048000000 \sqrt {5}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.16, size = 212, normalized size = 1.01 \begin {gather*} \frac {1}{50} \left (-\frac {58 \left (3 x^2+5 x+2\right )^{9/2}}{5 (2 x+3)^9}-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{(2 x+3)^{10}}+\frac {1893 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac {4417 \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 )}{1024000}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^11,x]

[Out]

((1893*(7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2))/(800*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(3 + 2*x)^10 - (58*
(2 + 5*x + 3*x^2)^(9/2))/(5*(3 + 2*x)^9) - (4417*((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])))/1024000)/50

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.94, size = 106, normalized size = 0.51 \begin {gather*} \frac {13251 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{1024000000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (371791872 x^9+5268182272 x^8+40186580992 x^7+148740043392 x^6+304078211712 x^5+372602220928 x^4+281702072128 x^3+128970753208 x^2+32786922608 x+3544392763\right )}{1024000000 (2 x+3)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^11,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(3544392763 + 32786922608*x + 128970753208*x^2 + 281702072128*x^3 + 372602220928*x^4 +
304078211712*x^5 + 148740043392*x^6 + 40186580992*x^7 + 5268182272*x^8 + 371791872*x^9))/(1024000000*(3 + 2*x)
^10) + (13251*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(1024000000*Sqrt[5])

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 215, normalized size = 1.03 \begin {gather*} \frac {13251 \, \sqrt {5} {\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 {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 (371791872 \, x^{9} + 5268182272 \, x^{8} + 40186580992 \, x^{7} + 148740043392 \, x^{6} + 304078211712 \, x^{5} + 372602220928 \, x^{4} + 281702072128 \, x^{3} + 128970753208 \, x^{2} + 32786922608 \, x + 3544392763\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{20480000000 \, {\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 {gather*}

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)^11,x, algorithm="fricas")

[Out]

1/20480000000*(13251*sqrt(5)*(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6 + 1959552*x^5 + 24
49440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049)*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*(371791872*x^9 + 5268182272*x^8 + 40186580992*x^7 + 148740043392*x
^6 + 304078211712*x^5 + 372602220928*x^4 + 281702072128*x^3 + 128970753208*x^2 + 32786922608*x + 3544392763)*s
qrt(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)

________________________________________________________________________________________

giac [B]  time = 0.41, size = 614, normalized size = 2.94 \begin {gather*} \frac {13251}{10240000000} \, \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 {6784512 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{19} + 83137358592 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{18} + 2689605043456 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} + 9174489217536 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} - 53080570863872 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 898783135722624 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} - 13174687008250752 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} - 40507172795248512 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} - 270169596727110016 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} - 458790099197766656 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 1833183533173743552 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 1939024456450048032 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 4903074367120921776 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 3280073192617110456 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 5164856211259534888 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 2082844158764403144 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 1869656136275991262 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 391066159205340747 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 153124376229353121 \, \sqrt {3} x - 9387541838830536 \, \sqrt {3} + 153124376229353121 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{1024000000 \, {\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 )}^{10}} \end {gather*}

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)^11,x, algorithm="giac")

[Out]

13251/10240000000*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))) - 1/1024000000*(6784512*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^19 + 83137358592*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^18 + 2689605043456*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^17 + 9174489217536*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^16 - 53080570863872*(sqrt(3)*x - s
qrt(3*x^2 + 5*x + 2))^15 - 898783135722624*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 - 13174687008250752*
(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 - 40507172795248512*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 - 27
0169596727110016*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 - 458790099197766656*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2))^10 - 1833183533173743552*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 1939024456450048032*sqrt(3)*(sqrt(
3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 4903074367120921776*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 3280073192617110
456*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 5164856211259534888*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5
- 2082844158764403144*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 1869656136275991262*(sqrt(3)*x - sqrt(3*
x^2 + 5*x + 2))^3 - 391066159205340747*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 153124376229353121*sqrt
(3)*x - 9387541838830536*sqrt(3) + 153124376229353121*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)^10

________________________________________________________________________________________

maple [B]  time = 0.15, size = 390, normalized size = 1.87 \begin {gather*} -\frac {13251 \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 )}{10240000000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{51200 \left (x +\frac {3}{2}\right )^{10}}-\frac {1893 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{2560000 \left (x +\frac {3}{2}\right )^{8}}-\frac {1893 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{1600000 \left (x +\frac {3}{2}\right )^{7}}-\frac {3636453 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{800000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {105574503 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{10000000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {3482489 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{500000000 \left (x +\frac {3}{2}\right )^{3}}+\frac {19795101 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{2500000000}-\frac {7698831 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{10000000000}-\frac {19795101 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{1250000000 \left (x +\frac {3}{2}\right )}+\frac {128093 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{1600000000}-\frac {13251 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{1280000000}+\frac {13251 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{10240000000}+\frac {4417 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{6400000000}+\frac {13251 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{40000000000}+\frac {1893 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{10000000000}-\frac {58683 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{20000000 \left (x +\frac {3}{2}\right )^{5}}-\frac {11989 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{6400000 \left (x +\frac {3}{2}\right )^{6}}-\frac {29 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{64000 \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)^(7/2)/(2*x+3)^11,x)

[Out]

-13/51200/(x+3/2)^10*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-1893/2560000/(x+3/2)^8*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-1893/1
600000/(x+3/2)^7*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-3636453/800000000/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-10557
4503/10000000000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-3482489/500000000/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(
9/2)[phone]/2500000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-7698831/10000000000*(6*x+5)*(-4*x+3*(x+3/2)^2-1
9/4)^(5/2)-19795101/1250000000/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(9/2)+128093/1600000000*(6*x+5)*(-4*x+3*(x+3/2)
^2-19/4)^(3/2)-13251/1280000000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-13251/10240000000*5^(1/2)*arctanh(2/5*(-
4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))+13251/10240000000*(-16*x+12*(x+3/2)^2-19)^(1/2)+4417/640000000
0*(-4*x+3*(x+3/2)^2-19/4)^(3/2)+13251/40000000000*(-4*x+3*(x+3/2)^2-19/4)^(5/2)+1893/10000000000*(-4*x+3*(x+3/
2)^2-19/4)^(7/2)-58683/20000000/(x+3/2)^5*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-11989/6400000/(x+3/2)^6*(-4*x+3*(x+3/2
)^2-19/4)^(9/2)-29/64000/(x+3/2)^9*(-4*x+3*(x+3/2)^2-19/4)^(9/2)

________________________________________________________________________________________

maxima [B]  time = 1.50, size = 579, normalized size = 2.77 \begin {gather*} \frac {316723509}{10000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50 \, {\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 {29 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{125 \, {\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 {1893 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{10000 \, {\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 {1893 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{12500 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {11989 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{100000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {58683 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{625000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {3636453 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50000000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {3482489 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{62500000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {105574503 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{2500000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {23096493}{5000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {153963369}{40000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {19795101 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{500000000 \, {\left (2 \, x + 3\right )}} + \frac {384279}{800000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {2566277}{6400000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {39753}{640000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {13251}{10240000000} \, \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 {251769}{5120000000} \, \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)^(7/2)/(3+2*x)^11,x, algorithm="maxima")

[Out]

316723509/10000000000*(3*x^2 + 5*x + 2)^(7/2) - 13/50*(3*x^2 + 5*x + 2)^(9/2)/(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) - 2
9/125*(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 + 118098*x + 19683) - 1893/10000*(3*x^2 + 5*x + 2)^(9/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48
384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561) - 1893/12500*(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) - 11989/100000*(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) - 58683/625000*(3*x^2 + 5*x + 2)^(9/2)/
(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 3636453/50000000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*
x^3 + 216*x^2 + 216*x + 81) - 3482489/62500000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 10557450
3/2500000000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 23096493/5000000000*(3*x^2 + 5*x + 2)^(5/2)*x - 1539
63369/40000000000*(3*x^2 + 5*x + 2)^(5/2) - 19795101/500000000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 384279/8000
00000*(3*x^2 + 5*x + 2)^(3/2)*x + 2566277/6400000000*(3*x^2 + 5*x + 2)^(3/2) - 39753/640000000*sqrt(3*x^2 + 5*
x + 2)*x - 13251/10240000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) -
251769/5120000000*sqrt(3*x^2 + 5*x + 2)

________________________________________________________________________________________

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 )}^{7/2}}{{\left (2\,x+3\right )}^{11}} \,d x \end {gather*}

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)^11,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^11, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{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 )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{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 )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{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 )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{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 )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{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 )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{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 )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{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}\, dx \end {gather*}

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)**11,x)

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(2048*x**11 + 33792*x**10 + 253440*x**9 + 1140480*x**8 + 3421440*x**7 + 7
185024*x**6 + 10777536*x**5 + 11547360*x**4 + 8660520*x**3 + 4330260*x**2 + 1299078*x + 177147), x) - Integral
(-292*x*sqrt(3*x**2 + 5*x + 2)/(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), x) - Integral(-870*
x**2*sqrt(3*x**2 + 5*x + 2)/(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), x) - Integral(-1339*x*
*3*sqrt(3*x**2 + 5*x + 2)/(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), x) - Integral(-1090*x**4
*sqrt(3*x**2 + 5*x + 2)/(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), x) - Integral(-396*x**5*sq
rt(3*x**2 + 5*x + 2)/(2048*x**11 + 33792*x**10 + 253440*x**9 + 1140480*x**8 + 3421440*x**7 + 7185024*x**6 + 10
777536*x**5 + 11547360*x**4 + 8660520*x**3 + 4330260*x**2 + 1299078*x + 177147), x) - Integral(27*x**7*sqrt(3*
x**2 + 5*x + 2)/(2048*x**11 + 33792*x**10 + 253440*x**9 + 1140480*x**8 + 3421440*x**7 + 7185024*x**6 + 1077753
6*x**5 + 11547360*x**4 + 8660520*x**3 + 4330260*x**2 + 1299078*x + 177147), x)

________________________________________________________________________________________