3.22.85 \(\int \frac {(5-x) (2+5 x+3 x^2)^{3/2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=137 \[ \frac {(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}-\frac {(402 x+845) \sqrt {3 x^2+5 x+2}}{160 (2 x+3)}+\frac {51}{32} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1973 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{320 \sqrt {5}} \]

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Rubi [A]  time = 0.08, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {(342 x+383) \left (3 x^2+5 x+2\right )^{3/2}}{120 (2 x+3)^3}-\frac {(402 x+845) \sqrt {3 x^2+5 x+2}}{160 (2 x+3)}+\frac {51}{32} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1973 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{320 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^4,x]

[Out]

-((845 + 402*x)*Sqrt[2 + 5*x + 3*x^2])/(160*(3 + 2*x)) + ((383 + 342*x)*(2 + 5*x + 3*x^2)^(3/2))/(120*(3 + 2*x
)^3) + (51*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/32 - (1973*ArcTanh[(7 + 8*x)/(2*Sqrt[
5]*Sqrt[2 + 5*x + 3*x^2])])/(320*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 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 )^{3/2}}{(3+2 x)^4} \, dx &=\frac {(383+342 x) \left (2+5 x+3 x^2\right )^{3/2}}{120 (3+2 x)^3}-\frac {1}{80} \int \frac {(361+402 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx\\ &=-\frac {(845+402 x) \sqrt {2+5 x+3 x^2}}{160 (3+2 x)}+\frac {(383+342 x) \left (2+5 x+3 x^2\right )^{3/2}}{120 (3+2 x)^3}+\frac {1}{640} \int \frac {5234+6120 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {(845+402 x) \sqrt {2+5 x+3 x^2}}{160 (3+2 x)}+\frac {(383+342 x) \left (2+5 x+3 x^2\right )^{3/2}}{120 (3+2 x)^3}+\frac {153}{32} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx-\frac {1973}{320} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {(845+402 x) \sqrt {2+5 x+3 x^2}}{160 (3+2 x)}+\frac {(383+342 x) \left (2+5 x+3 x^2\right )^{3/2}}{120 (3+2 x)^3}+\frac {153}{16} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )+\frac {1973}{160} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {(845+402 x) \sqrt {2+5 x+3 x^2}}{160 (3+2 x)}+\frac {(383+342 x) \left (2+5 x+3 x^2\right )^{3/2}}{120 (3+2 x)^3}+\frac {51}{32} \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )-\frac {1973 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{320 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 110, normalized size = 0.80 \begin {gather*} \frac {5919 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )+7650 \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 (720 x^3+13176 x^2+30878 x+19751\right )}{(2 x+3)^3}}{4800} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^4,x]

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(19751 + 30878*x + 13176*x^2 + 720*x^3))/(3 + 2*x)^3 + 5919*Sqrt[5]*ArcTanh[(-7 -
8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] + 7650*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/4800

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IntegrateAlgebraic [A]  time = 0.63, size = 111, normalized size = 0.81 \begin {gather*} \frac {51}{16} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )-\frac {1973 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{160 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (-720 x^3-13176 x^2-30878 x-19751\right )}{480 (2 x+3)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^4,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(-19751 - 30878*x - 13176*x^2 - 720*x^3))/(480*(3 + 2*x)^3) + (51*Sqrt[3]*ArcTanh[Sqrt[
2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/16 - (1973*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(160*Sqrt[5]
)

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fricas [A]  time = 0.42, size = 169, normalized size = 1.23 \begin {gather*} \frac {7650 \, \sqrt {3} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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 ) + 5919 \, \sqrt {5} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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 (720 \, x^{3} + 13176 \, x^{2} + 30878 \, x + 19751\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{9600 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4,x, algorithm="fricas")

[Out]

1/9600*(7650*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120
*x + 49) + 5919*sqrt(5)*(8*x^3 + 36*x^2 + 54*x + 27)*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*(720*x^3 + 13176*x^2 + 30878*x + 19751)*sqrt(3*x^2 + 5*x + 2))/(8*x^3
+ 36*x^2 + 54*x + 27)

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giac [B]  time = 0.34, size = 305, normalized size = 2.23 \begin {gather*} -\frac {1973}{1600} \, \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 {51}{32} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {3}{16} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {62484 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 390510 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 2835190 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 3307455 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 5598195 \, \sqrt {3} x + 1227924 \, \sqrt {3} - 5598195 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{480 \, {\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 )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4,x, algorithm="giac")

[Out]

-1973/1600*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))) - 51/32*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2)) - 5)) - 3/16*sqrt(3*x^2 + 5*x + 2) - 1/480*(62484*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 390510*sq
rt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 2835190*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 3307455*sqrt(3)*
(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 5598195*sqrt(3)*x + 1227924*sqrt(3) - 5598195*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)^3

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maple [A]  time = 0.06, size = 200, normalized size = 1.46 \begin {gather*} \frac {1973 \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 )}{1600}+\frac {51 \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 )}{32}-\frac {37 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{600 \left (x +\frac {3}{2}\right )^{2}}-\frac {158 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{375 \left (x +\frac {3}{2}\right )}-\frac {1973 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{3000}+\frac {121 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{400}-\frac {1973 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{1600}+\frac {79 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{375}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{120 \left (x +\frac {3}{2}\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(3/2)/(2*x+3)^4,x)

[Out]

-37/600/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-158/375/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-1973/3000*(-4*x+
3*(x+3/2)^2-19/4)^(3/2)+121/400*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)+51/32*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(
-4*x+3*(x+3/2)^2-19/4)^(1/2))-1973/1600*(-16*x+12*(x+3/2)^2-19)^(1/2)+1973/1600*5^(1/2)*arctanh(2/5*(-4*x-7/2)
*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))+79/375*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-13/120/(x+3/2)^3*(-4*x+3*
(x+3/2)^2-19/4)^(5/2)

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maxima [A]  time = 1.36, size = 191, normalized size = 1.39 \begin {gather*} \frac {37}{200} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{15 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {37 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{150 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {363}{200} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {51}{32} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {1973}{1600} \, \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 {763}{800} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {79 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{75 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4,x, algorithm="maxima")

[Out]

37/200*(3*x^2 + 5*x + 2)^(3/2) - 13/15*(3*x^2 + 5*x + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 37/150*(3*x^2 +
5*x + 2)^(5/2)/(4*x^2 + 12*x + 9) + 363/200*sqrt(3*x^2 + 5*x + 2)*x + 51/32*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5
*x + 2) + 3*x + 5/2) + 1973/1600*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2
) - 763/800*sqrt(3*x^2 + 5*x + 2) - 79/75*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)

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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 )}^{3/2}}{{\left (2\,x+3\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(3/2))/(2*x + 3)^4,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(3/2))/(2*x + 3)^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**(3/2)/(3+2*x)**4,x)

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-23*x*sqrt(3*x
**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(16*
x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*
x**2 + 216*x + 81), x)

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