3.2437 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{3+2 x} \, dx\)

Optimal. Leaf size=146 \[ \frac{1}{360} (209-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac{(25-5586 x) \left (3 x^2+5 x+2\right )^{3/2}}{3456}+\frac{(51455-106734 x) \sqrt{3 x^2+5 x+2}}{27648}-\frac{543811 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{55296 \sqrt{3}}+\frac{325}{128} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

((51455 - 106734*x)*Sqrt[2 + 5*x + 3*x^2])/27648 + ((25 - 5586*x)*(2 + 5*x + 3*x^2)^(3/2))/3456 + ((209 - 30*x
)*(2 + 5*x + 3*x^2)^(5/2))/360 - (543811*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(55296*Sqrt[3])
+ (325*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/128

________________________________________________________________________________________

Rubi [A]  time = 0.0999991, antiderivative size = 146, normalized size of antiderivative = 1., 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 = {814, 843, 621, 206, 724} \[ \frac{1}{360} (209-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac{(25-5586 x) \left (3 x^2+5 x+2\right )^{3/2}}{3456}+\frac{(51455-106734 x) \sqrt{3 x^2+5 x+2}}{27648}-\frac{543811 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{55296 \sqrt{3}}+\frac{325}{128} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((51455 - 106734*x)*Sqrt[2 + 5*x + 3*x^2])/27648 + ((25 - 5586*x)*(2 + 5*x + 3*x^2)^(3/2))/3456 + ((209 - 30*x
)*(2 + 5*x + 3*x^2)^(5/2))/360 - (543811*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(55296*Sqrt[3])
+ (325*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/128

Rule 814

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

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 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 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]

Rubi steps

\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{3+2 x} \, dx &=\frac{1}{360} (209-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{144} \int \frac{(1623+1862 x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx\\ &=\frac{(25-5586 x) \left (2+5 x+3 x^2\right )^{3/2}}{3456}+\frac{1}{360} (209-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac{\int \frac{(-179802-213468 x) \sqrt{2+5 x+3 x^2}}{3+2 x} \, dx}{13824}\\ &=\frac{(51455-106734 x) \sqrt{2+5 x+3 x^2}}{27648}+\frac{(25-5586 x) \left (2+5 x+3 x^2\right )^{3/2}}{3456}+\frac{1}{360} (209-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac{\int \frac{11153196+13051464 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{663552}\\ &=\frac{(51455-106734 x) \sqrt{2+5 x+3 x^2}}{27648}+\frac{(25-5586 x) \left (2+5 x+3 x^2\right )^{3/2}}{3456}+\frac{1}{360} (209-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac{543811 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{55296}+\frac{1625}{128} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{(51455-106734 x) \sqrt{2+5 x+3 x^2}}{27648}+\frac{(25-5586 x) \left (2+5 x+3 x^2\right )^{3/2}}{3456}+\frac{1}{360} (209-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac{543811 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{27648}-\frac{1625}{64} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{(51455-106734 x) \sqrt{2+5 x+3 x^2}}{27648}+\frac{(25-5586 x) \left (2+5 x+3 x^2\right )^{3/2}}{3456}+\frac{1}{360} (209-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac{543811 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{55296 \sqrt{3}}+\frac{325}{128} \sqrt{5} \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0688082, size = 113, normalized size = 0.77 \[ \frac{-6 \sqrt{3 x^2+5 x+2} \left (103680 x^5-376704 x^4-1311120 x^3-1624872 x^2-583490 x-580299\right )-2106000 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )-2719055 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{829440} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-580299 - 583490*x - 1624872*x^2 - 1311120*x^3 - 376704*x^4 + 103680*x^5) - 2106000
*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] - 2719055*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 +
15*x + 9*x^2])])/829440

________________________________________________________________________________________

Maple [B]  time = 0.007, size = 239, normalized size = 1.6 \begin{align*} -{\frac{5+6\,x}{72} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{25+30\,x}{3456} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{25+30\,x}{27648}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{5\,\sqrt{3}}{165888}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{13}{20} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{65+78\,x}{48} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{1235+1482\,x}{384}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{7553\,\sqrt{3}}{2304}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }+{\frac{65}{48} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{325}{128}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{325\,\sqrt{5}}{128}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(5+6*x)*(3*x^2+5*x+2)^(5/2)+5/3456*(5+6*x)*(3*x^2+5*x+2)^(3/2)-5/27648*(5+6*x)*(3*x^2+5*x+2)^(1/2)+5/165
888*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+13/20*(3*(x+3/2)^2-4*x-19/4)^(5/2)-13/48*(5+6*x)*(3*
(x+3/2)^2-4*x-19/4)^(3/2)-247/384*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-7553/2304*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(
x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)+65/48*(3*(x+3/2)^2-4*x-19/4)^(3/2)+325/128*(12*(x+3/2)^2-16*x-19)^(1/2)-325/
128*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.53316, size = 212, normalized size = 1.45 \begin{align*} -\frac{1}{12} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{209}{360} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{931}{576} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{25}{3456} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{17789}{4608} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{543811}{165888} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{325}{128} \, \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{51455}{27648} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

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),x, algorithm="maxima")

[Out]

-1/12*(3*x^2 + 5*x + 2)^(5/2)*x + 209/360*(3*x^2 + 5*x + 2)^(5/2) - 931/576*(3*x^2 + 5*x + 2)^(3/2)*x + 25/345
6*(3*x^2 + 5*x + 2)^(3/2) - 17789/4608*sqrt(3*x^2 + 5*x + 2)*x - 543811/165888*sqrt(3)*log(sqrt(3)*sqrt(3*x^2
+ 5*x + 2) + 3*x + 5/2) - 325/128*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) -
2) + 51455/27648*sqrt(3*x^2 + 5*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.53768, size = 419, normalized size = 2.87 \begin{align*} -\frac{1}{138240} \,{\left (103680 \, x^{5} - 376704 \, x^{4} - 1311120 \, x^{3} - 1624872 \, x^{2} - 583490 \, x - 580299\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{543811}{331776} \, \sqrt{3} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + \frac{325}{256} \, \sqrt{5} \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 ) \end{align*}

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),x, algorithm="fricas")

[Out]

-1/138240*(103680*x^5 - 376704*x^4 - 1311120*x^3 - 1624872*x^2 - 583490*x - 580299)*sqrt(3*x^2 + 5*x + 2) + 54
3811/331776*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 325/256*sqrt(5)*lo
g((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))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{20 \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx - \int - \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx - \int - \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx - \int - \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx - \int - \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \end{align*}

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),x)

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Int
egral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x)
- Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(2*x + 3),
x)

________________________________________________________________________________________

Giac [A]  time = 1.21794, size = 197, normalized size = 1.35 \begin{align*} -\frac{1}{138240} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \,{\left (30 \, x - 109\right )} x - 3035\right )} x - 67703\right )} x - 291745\right )} x - 580299\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{325}{128} \, \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{543811}{165888} \, \sqrt{3} \log \left ({\left | -6 \, \sqrt{3} x - 5 \, \sqrt{3} + 6 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) \end{align*}

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),x, algorithm="giac")

[Out]

-1/138240*(2*(12*(18*(8*(30*x - 109)*x - 3035)*x - 67703)*x - 291745)*x - 580299)*sqrt(3*x^2 + 5*x + 2) + 325/
128*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))) + 543811/165888*sqrt(3)*log(abs(-6*sqrt(3)*x - 5*sqrt(3) + 6*sqrt(3
*x^2 + 5*x + 2)))