\(\int \frac {(5-x) (3+2 x)^4}{(2+5 x+3 x^2)^{5/2}} \, dx\) [2516]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [F]
   Maxima [B] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [F(-1)]

Optimal result

Integrand size = 27, antiderivative size = 115 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {6848}{9} \sqrt {2+5 x+3 x^2}+\frac {152 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{27 \sqrt {3}} \]

[Out]

-2/9*(3+2*x)^3*(121+139*x)/(3*x^2+5*x+2)^(3/2)+152/81*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)
+4/27*(3+2*x)*(6809+7976*x)/(3*x^2+5*x+2)^(1/2)-6848/9*(3*x^2+5*x+2)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {832, 654, 635, 212} \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {152 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{27 \sqrt {3}}-\frac {2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 (7976 x+6809) (2 x+3)}{27 \sqrt {3 x^2+5 x+2}}-\frac {6848}{9} \sqrt {3 x^2+5 x+2} \]

[In]

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

[Out]

(-2*(3 + 2*x)^3*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (4*(3 + 2*x)*(6809 + 7976*x))/(27*Sqrt[2 + 5*x +
3*x^2]) - (6848*Sqrt[2 + 5*x + 3*x^2])/9 + (152*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(27*Sqrt
[3])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
 /; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

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 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
 1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 832

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {2}{9} \int \frac {(3+2 x)^2 (-117+272 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx \\ & = -\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {4}{27} \int \frac {-12802-15408 x}{\sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {6848}{9} \sqrt {2+5 x+3 x^2}+\frac {152}{27} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {6848}{9} \sqrt {2+5 x+3 x^2}+\frac {304}{27} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right ) \\ & = -\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {6848}{9} \sqrt {2+5 x+3 x^2}+\frac {152 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{27 \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.72 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2}{81} \left (\frac {3 \sqrt {2+5 x+3 x^2} \left (-30819-118153 x-146180 x^2-58720 x^3+72 x^4\right )}{(1+x)^2 (2+3 x)^2}-152 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )\right ) \]

[In]

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

[Out]

(-2*((3*Sqrt[2 + 5*x + 3*x^2]*(-30819 - 118153*x - 146180*x^2 - 58720*x^3 + 72*x^4))/((1 + x)^2*(2 + 3*x)^2) -
 152*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)]))/81

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.57

method result size
risch \(-\frac {2 \left (72 x^{4}-58720 x^{3}-146180 x^{2}-118153 x -30819\right )}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {152 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{81}\) \(65\)
trager \(-\frac {2 \left (72 x^{4}-58720 x^{3}-146180 x^{2}-118153 x -30819\right )}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {152 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{81}\) \(77\)
default \(-\frac {16181 \left (5+6 x \right )}{1458 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {\frac {295120}{243}+\frac {118048 x}{81}}{\sqrt {3 x^{2}+5 x +2}}-\frac {145763}{1458 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {16 x^{4}}{3 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {152 x^{3}}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {2380 x^{2}}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {14639 x}{81 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {152 x}{27 \sqrt {3 x^{2}+5 x +2}}+\frac {380}{81 \sqrt {3 x^{2}+5 x +2}}+\frac {152 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{81}\) \(178\)

[In]

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

[Out]

-2/27*(72*x^4-58720*x^3-146180*x^2-118153*x-30819)/(3*x^2+5*x+2)^(3/2)+152/81*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+
5*x+2)^(1/2))*3^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.02 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (38 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) - 3 \, {\left (72 \, x^{4} - 58720 \, x^{3} - 146180 \, x^{2} - 118153 \, x - 30819\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{81 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

[In]

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

[Out]

2/81*(38*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 +
 120*x + 49) - 3*(72*x^4 - 58720*x^3 - 146180*x^2 - 118153*x - 30819)*sqrt(3*x^2 + 5*x + 2))/(9*x^4 + 30*x^3 +
 37*x^2 + 20*x + 4)

Sympy [F]

\[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \left (- \frac {999 x}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {864 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {264 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {16 x^{4}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \frac {16 x^{5}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {405}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]

[In]

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

[Out]

-Integral(-999*x/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x +
 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-864*x**2/(9*x**4*sqrt(3*x**2 + 5
*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sq
rt(3*x**2 + 5*x + 2)), x) - Integral(-264*x**3/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2)
 + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(16*
x**4/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*s
qrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(16*x**5/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x
**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*
x + 2)), x) - Integral(-405/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x
**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (93) = 186\).

Time = 0.26 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.86 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {16 \, x^{4}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {152}{81} \, x {\left (\frac {1410 \, x}{\sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {9 \, x^{2}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} + \frac {1175}{\sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {55 \, x}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {46}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}\right )} + \frac {152}{81} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {71440}{81} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {60704 \, x}{81 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {920 \, x^{2}}{9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {15680}{27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {13066 \, x}{81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {6766}{81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

-16/3*x^4/(3*x^2 + 5*x + 2)^(3/2) - 152/81*x*(1410*x/sqrt(3*x^2 + 5*x + 2) + 9*x^2/(3*x^2 + 5*x + 2)^(3/2) + 1
175/sqrt(3*x^2 + 5*x + 2) - 55*x/(3*x^2 + 5*x + 2)^(3/2) - 46/(3*x^2 + 5*x + 2)^(3/2)) + 152/81*sqrt(3)*log(2*
sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 71440/81*sqrt(3*x^2 + 5*x + 2) - 60704/81*x/sqrt(3*x^2 + 5*x + 2) -
 920/9*x^2/(3*x^2 + 5*x + 2)^(3/2) - 15680/27/sqrt(3*x^2 + 5*x + 2) - 13066/81*x/(3*x^2 + 5*x + 2)^(3/2) - 676
6/81/(3*x^2 + 5*x + 2)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.59 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {152}{81} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {2 \, {\left ({\left (4 \, {\left (2 \, {\left (9 \, x - 7340\right )} x - 36545\right )} x - 118153\right )} x - 30819\right )}}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

-152/81*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 2/27*((4*(2*(9*x - 7340)*x - 36
545)*x - 118153)*x - 30819)/(3*x^2 + 5*x + 2)^(3/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {{\left (2\,x+3\right )}^4\,\left (x-5\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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