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

Optimal. Leaf size=112 \[ \frac{65991-8779 x}{4292352 \sqrt{2 x^2-x+3}}+\frac{115369 \sqrt{2 x^2-x+3}}{1492992 (2 x+5)}-\frac{3667 \sqrt{2 x^2-x+3}}{20736 (2 x+5)^2}-\frac{52631 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{5971968 \sqrt{2}} \]

[Out]

(65991 - 8779*x)/(4292352*Sqrt[3 - x + 2*x^2]) - (3667*Sqrt[3 - x + 2*x^2])/(20736*(5 + 2*x)^2) + (115369*Sqrt
[3 - x + 2*x^2])/(1492992*(5 + 2*x)) - (52631*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(5971968*
Sqrt[2])

________________________________________________________________________________________

Rubi [A]  time = 0.145885, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1646, 1650, 806, 724, 206} \[ \frac{65991-8779 x}{4292352 \sqrt{2 x^2-x+3}}+\frac{115369 \sqrt{2 x^2-x+3}}{1492992 (2 x+5)}-\frac{3667 \sqrt{2 x^2-x+3}}{20736 (2 x+5)^2}-\frac{52631 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{5971968 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(65991 - 8779*x)/(4292352*Sqrt[3 - x + 2*x^2]) - (3667*Sqrt[3 - x + 2*x^2])/(20736*(5 + 2*x)^2) + (115369*Sqrt
[3 - x + 2*x^2])/(1492992*(5 + 2*x)) - (52631*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(5971968*
Sqrt[2])

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(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*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

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 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 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])

Rubi steps

\begin{align*} \int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^3 \left (3-x+2 x^2\right )^{3/2}} \, dx &=\frac{65991-8779 x}{4292352 \sqrt{3-x+2 x^2}}+\frac{2}{23} \int \frac{\frac{5168261}{746496}+\frac{3637795 x}{186624}+\frac{5620625 x^2}{186624}}{(5+2 x)^3 \sqrt{3-x+2 x^2}} \, dx\\ &=\frac{65991-8779 x}{4292352 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{20736 (5+2 x)^2}-\frac{\int \frac{\frac{842237}{1296}-\frac{4102487 x}{2592}}{(5+2 x)^2 \sqrt{3-x+2 x^2}} \, dx}{1656}\\ &=\frac{65991-8779 x}{4292352 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{20736 (5+2 x)^2}+\frac{115369 \sqrt{3-x+2 x^2}}{1492992 (5+2 x)}+\frac{52631 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{995328}\\ &=\frac{65991-8779 x}{4292352 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{20736 (5+2 x)^2}+\frac{115369 \sqrt{3-x+2 x^2}}{1492992 (5+2 x)}-\frac{52631 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{497664}\\ &=\frac{65991-8779 x}{4292352 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{20736 (5+2 x)^2}+\frac{115369 \sqrt{3-x+2 x^2}}{1492992 (5+2 x)}-\frac{52631 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{5971968 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.296961, size = 84, normalized size = 0.75 \[ \frac{\frac{12 \left (3444340 x^3+3263288 x^2+5842933 x+11594283\right )}{23 (2 x+5)^2 \sqrt{x^2-\frac{x}{2}+\frac{3}{2}}}-52631 \log \left (12 \sqrt{4 x^2-2 x+6}-22 x+17\right )+52631 \log (2 x+5)}{5971968 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((12*(11594283 + 5842933*x + 3263288*x^2 + 3444340*x^3))/(23*(5 + 2*x)^2*Sqrt[3/2 - x/2 + x^2]) + 52631*Log[5
+ 2*x] - 52631*Log[17 - 22*x + 12*Sqrt[6 - 2*x + 4*x^2]])/(5971968*Sqrt[2])

________________________________________________________________________________________

Maple [A]  time = 0.058, size = 144, normalized size = 1.3 \begin{align*} -{\frac{5}{16}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{-149+596\,x}{368}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{3667}{4608} \left ( x+{\frac{5}{2}} \right ) ^{-2}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}+{\frac{196043}{165888} \left ( x+{\frac{5}{2}} \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}+{\frac{52631}{1990656}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}+{\frac{-19399069+77596276\,x}{45785088}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}-{\frac{52631\,\sqrt{2}}{11943936}{\it Artanh} \left ({\frac{\sqrt{2}}{12} \left ({\frac{17}{2}}-11\,x \right ){\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/16/(2*x^2-x+3)^(1/2)-149/368*(-1+4*x)/(2*x^2-x+3)^(1/2)-3667/4608/(x+5/2)^2/(2*(x+5/2)^2-11*x-19/2)^(1/2)+1
96043/165888/(x+5/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2)+52631/1990656/(2*(x+5/2)^2-11*x-19/2)^(1/2)[phone]/457850
88*(-1+4*x)/(2*(x+5/2)^2-11*x-19/2)^(1/2)-52631/11943936*2^(1/2)*arctanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2
-11*x-19/2)^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.51339, size = 201, normalized size = 1.79 \begin{align*} \frac{52631}{11943936} \, \sqrt{2} \operatorname{arsinh}\left (\frac{22 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 5 \right |}} - \frac{17 \, \sqrt{23}}{23 \,{\left | 2 \, x + 5 \right |}}\right ) + \frac{861085 \, x}{11446272 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{1163201}{3815424 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{3667}{1152 \,{\left (4 \, \sqrt{2 \, x^{2} - x + 3} x^{2} + 20 \, \sqrt{2 \, x^{2} - x + 3} x + 25 \, \sqrt{2 \, x^{2} - x + 3}\right )}} + \frac{196043}{82944 \,{\left (2 \, \sqrt{2 \, x^{2} - x + 3} x + 5 \, \sqrt{2 \, x^{2} - x + 3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

52631/11943936*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) + 861085/11446272*
x/sqrt(2*x^2 - x + 3) - 1163201/3815424/sqrt(2*x^2 - x + 3) - 3667/1152/(4*sqrt(2*x^2 - x + 3)*x^2 + 20*sqrt(2
*x^2 - x + 3)*x + 25*sqrt(2*x^2 - x + 3)) + 196043/82944/(2*sqrt(2*x^2 - x + 3)*x + 5*sqrt(2*x^2 - x + 3))

________________________________________________________________________________________

Fricas [A]  time = 1.3407, size = 379, normalized size = 3.38 \begin{align*} \frac{1210513 \, \sqrt{2}{\left (8 \, x^{4} + 36 \, x^{3} + 42 \, x^{2} + 35 \, x + 75\right )} \log \left (-\frac{24 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \,{\left (3444340 \, x^{3} + 3263288 \, x^{2} + 5842933 \, x + 11594283\right )} \sqrt{2 \, x^{2} - x + 3}}{549421056 \,{\left (8 \, x^{4} + 36 \, x^{3} + 42 \, x^{2} + 35 \, x + 75\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/549421056*(1210513*sqrt(2)*(8*x^4 + 36*x^3 + 42*x^2 + 35*x + 75)*log(-(24*sqrt(2)*sqrt(2*x^2 - x + 3)*(22*x
- 17) + 1060*x^2 - 1036*x + 1153)/(4*x^2 + 20*x + 25)) + 48*(3444340*x^3 + 3263288*x^2 + 5842933*x + 11594283)
*sqrt(2*x^2 - x + 3))/(8*x^4 + 36*x^3 + 42*x^2 + 35*x + 75)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x + 5\right )^{3} \left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.15733, size = 297, normalized size = 2.65 \begin{align*} -\frac{52631}{11943936} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{52631}{11943936} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{8779 \, x - 65991}{4292352 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{\sqrt{2}{\left (3594214 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} + 19874490 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} - 30140067 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 19989859\right )}}{2985984 \,{\left (2 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 11\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-52631/11943936*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 52631/11943936*sqrt(2)*log(
abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 1/4292352*(8779*x - 65991)/sqrt(2*x^2 - x + 3) + 1/2
985984*sqrt(2)*(3594214*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 + 19874490*(sqrt(2)*x - sqrt(2*x^2 - x + 3
))^2 - 30140067*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 19989859)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 +
10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^2