3.23.1 \(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=167 \[ \frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}+\frac {(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^3}-\frac {(26934 x+57845) \sqrt {3 x^2+5 x+2}}{12800 (2 x+3)}+\frac {177}{128} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {137111 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{25600 \sqrt {5}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {810, 812, 843, 621, 206, 724} \begin {gather*} \frac {(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}+\frac {(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^3}-\frac {(26934 x+57845) \sqrt {3 x^2+5 x+2}}{12800 (2 x+3)}+\frac {177}{128} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {137111 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{25600 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^6,x]

[Out]

-((57845 + 26934*x)*Sqrt[2 + 5*x + 3*x^2])/(12800*(3 + 2*x)) + ((17051 + 13074*x)*(2 + 5*x + 3*x^2)^(3/2))/(96
00*(3 + 2*x)^3) + ((119 + 114*x)*(2 + 5*x + 3*x^2)^(5/2))/(80*(3 + 2*x)^5) + (177*Sqrt[3]*ArcTanh[(5 + 6*x)/(2
*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/128 - (137111*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(25600*S
qrt[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 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 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 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 812

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

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx &=\frac {(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}-\frac {1}{160} \int \frac {(437+462 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx\\ &=\frac {(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac {(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}+\frac {\int \frac {(-45914-53868 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{12800}\\ &=-\frac {(57845+26934 x) \sqrt {2+5 x+3 x^2}}{12800 (3+2 x)}+\frac {(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac {(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}-\frac {\int \frac {-725956-849600 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{102400}\\ &=-\frac {(57845+26934 x) \sqrt {2+5 x+3 x^2}}{12800 (3+2 x)}+\frac {(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac {(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}+\frac {531}{128} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx-\frac {137111 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{25600}\\ &=-\frac {(57845+26934 x) \sqrt {2+5 x+3 x^2}}{12800 (3+2 x)}+\frac {(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac {(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}+\frac {531}{64} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )+\frac {137111 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{12800}\\ &=-\frac {(57845+26934 x) \sqrt {2+5 x+3 x^2}}{12800 (3+2 x)}+\frac {(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac {(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}+\frac {177}{128} \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )-\frac {137111 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{25600 \sqrt {5}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 120, normalized size = 0.72 \begin {gather*} \frac {411333 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )+531000 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-\frac {10 \sqrt {3 x^2+5 x+2} \left (172800 x^5+4630848 x^4+21586808 x^3+41641148 x^2+37019838 x+12600183\right )}{(2 x+3)^5}}{384000} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^6,x]

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(12600183 + 37019838*x + 41641148*x^2 + 21586808*x^3 + 4630848*x^4 + 172800*x^5))/
(3 + 2*x)^5 + 411333*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] + 531000*Sqrt[3]*ArcTanh[(5
+ 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/384000

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.79, size = 121, normalized size = 0.72 \begin {gather*} \frac {177}{64} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )-\frac {137111 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{12800 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (-172800 x^5-4630848 x^4-21586808 x^3-41641148 x^2-37019838 x-12600183\right )}{38400 (2 x+3)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^6,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(-12600183 - 37019838*x - 41641148*x^2 - 21586808*x^3 - 4630848*x^4 - 172800*x^5))/(384
00*(3 + 2*x)^5) + (177*Sqrt[3]*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/64 - (137111*ArcTanh[Sqrt[2 +
5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(12800*Sqrt[5])

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 209, normalized size = 1.25 \begin {gather*} \frac {531000 \, \sqrt {3} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 411333 \, \sqrt {5} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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 (172800 \, x^{5} + 4630848 \, x^{4} + 21586808 \, x^{3} + 41641148 \, x^{2} + 37019838 \, x + 12600183\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{768000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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)^6,x, algorithm="fricas")

[Out]

1/768000*(531000*sqrt(3)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x
+ 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 411333*sqrt(5)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*l
og(-(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*(172800*x^5 +
4630848*x^4 + 21586808*x^3 + 41641148*x^2 + 37019838*x + 12600183)*sqrt(3*x^2 + 5*x + 2))/(32*x^5 + 240*x^4 +
720*x^3 + 1080*x^2 + 810*x + 243)

________________________________________________________________________________________

giac [B]  time = 0.40, size = 407, normalized size = 2.44 \begin {gather*} -\frac {137111}{128000} \, \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 {177}{128} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {9}{64} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {27201072 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 316934472 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 4873277176 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 14374341276 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 80473660448 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 98380998102 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 236231795506 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 119385279741 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 103767800973 \, \sqrt {3} x + 13144069068 \, \sqrt {3} - 103767800973 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{38400 \, {\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 )}^{5}} \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)^6,x, algorithm="giac")

[Out]

-137111/128000*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))) - 177/128*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3
*x^2 + 5*x + 2)) - 5)) - 9/64*sqrt(3*x^2 + 5*x + 2) - 1/38400*(27201072*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9
+ 316934472*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 4873277176*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 +
14374341276*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 80473660448*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5
+ 98380998102*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 236231795506*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)
)^3 + 119385279741*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 103767800973*sqrt(3)*x + 13144069068*sqrt(3
) - 103767800973*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)^5

________________________________________________________________________________________

maple [B]  time = 0.06, size = 279, normalized size = 1.67 \begin {gather*} \frac {137111 \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 )}{128000}+\frac {177 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\right )}{128}-\frac {521 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{15000 \left (x +\frac {3}{2}\right )^{3}}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{800 \left (x +\frac {3}{2}\right )^{5}}-\frac {9349 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{300000 \left (x +\frac {3}{2}\right )^{2}}+\frac {11491 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{125000}+\frac {6281 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{60000}-\frac {11491 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{62500 \left (x +\frac {3}{2}\right )}+\frac {4361 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{16000}-\frac {137111 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{128000}-\frac {131 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{8000 \left (x +\frac {3}{2}\right )^{4}}-\frac {137111 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{240000}-\frac {137111 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{500000} \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)^6,x)

[Out]

-521/15000/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-13/800/(x+3/2)^5*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-9349/300000/
(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(7/2)+11491/125000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(5/2)+6281/60000*(6*x+5)*
(-4*x+3*(x+3/2)^2-19/4)^(3/2)-11491/62500/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)+4361/16000*(6*x+5)*(-4*x+3*(x+
3/2)^2-19/4)^(1/2)+137111/128000*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))+177/128
*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(-4*x+3*(x+3/2)^2-19/4)^(1/2))-137111/128000*(-16*x+12*(x+3/2)^2-19)^(1/2)-1
31/8000/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-137111/240000*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-137111/500000*(-4*
x+3*(x+3/2)^2-19/4)^(5/2)

________________________________________________________________________________________

maxima [B]  time = 1.53, size = 297, normalized size = 1.78 \begin {gather*} \frac {9349}{100000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{25 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {131 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {521 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{1875 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {9349 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{75000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {6281}{10000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {11491}{240000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {11491 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{25000 \, {\left (2 \, x + 3\right )}} + \frac {13083}{8000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {177}{128} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {137111}{128000} \, \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 {49891}{64000} \, \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)^6,x, algorithm="maxima")

[Out]

9349/100000*(3*x^2 + 5*x + 2)^(5/2) - 13/25*(3*x^2 + 5*x + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 8
10*x + 243) - 131/500*(3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 521/1875*(3*x^2 + 5*x
+ 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 9349/75000*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) + 6281/10000*
(3*x^2 + 5*x + 2)^(3/2)*x - 11491/240000*(3*x^2 + 5*x + 2)^(3/2) - 11491/25000*(3*x^2 + 5*x + 2)^(5/2)/(2*x +
3) + 13083/8000*sqrt(3*x^2 + 5*x + 2)*x + 177/128*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 137
111/128000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 49891/64000*sqrt(3
*x^2 + 5*x + 2)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^6} \,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)^6,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^6, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, 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)**6,x)

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729),
x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x +
729), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 +
2916*x + 729), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4
860*x**2 + 2916*x + 729), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320
*x**3 + 4860*x**2 + 2916*x + 729), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4
+ 4320*x**3 + 4860*x**2 + 2916*x + 729), x)

________________________________________________________________________________________