Integrand size = 25, antiderivative size = 266 \[ \int \frac {\left (2+3 x+5 x^2\right )^{3/2}}{\left (4+x-2 x^2\right )^4} \, dx=\frac {(983+1150 x) \sqrt {2+3 x+5 x^2}}{13068 \left (4+x-2 x^2\right )^2}-\frac {23 (40605+1414 x) \sqrt {2+3 x+5 x^2}}{133973136 \left (4+x-2 x^2\right )}-\frac {(1-4 x) \left (2+3 x+5 x^2\right )^{3/2}}{99 \left (4+x-2 x^2\right )^3}-\frac {\left (687839+4799663 \sqrt {33}\right ) \text {arctanh}\left (\frac {19-3 \sqrt {33}+2 \left (11-5 \sqrt {33}\right ) x}{2 \sqrt {2 \left (107-11 \sqrt {33}\right )} \sqrt {2+3 x+5 x^2}}\right )}{133973136 \sqrt {66 \left (107-11 \sqrt {33}\right )}}+\frac {\left (687839-4799663 \sqrt {33}\right ) \text {arctanh}\left (\frac {19+3 \sqrt {33}+2 \left (11+5 \sqrt {33}\right ) x}{2 \sqrt {2 \left (107+11 \sqrt {33}\right )} \sqrt {2+3 x+5 x^2}}\right )}{133973136 \sqrt {66 \left (107+11 \sqrt {33}\right )}} \] Output:
1/13068*(983+1150*x)*(5*x^2+3*x+2)^(1/2)/(-2*x^2+x+4)^2-23*(40605+1414*x)* (5*x^2+3*x+2)^(1/2)/(-267946272*x^2+133973136*x+535892544)-1/99*(1-4*x)*(5 *x^2+3*x+2)^(3/2)/(-2*x^2+x+4)^3-1/133973136*(687839+4799663*33^(1/2))*arc tanh(1/2*(19-3*33^(1/2)+2*(11-5*33^(1/2))*x)/(214-22*33^(1/2))^(1/2)/(5*x^ 2+3*x+2)^(1/2))/(7062-726*33^(1/2))^(1/2)+1/133973136*(687839-4799663*33^( 1/2))*arctanh(1/2*(19+3*33^(1/2)+2*(11+5*33^(1/2))*x)/(214+22*33^(1/2))^(1 /2)/(5*x^2+3*x+2)^(1/2))/(7062+726*33^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.79 (sec) , antiderivative size = 582, normalized size of antiderivative = 2.19 \[ \int \frac {\left (2+3 x+5 x^2\right )^{3/2}}{\left (4+x-2 x^2\right )^4} \, dx=\frac {-\frac {1721443144 \sqrt {2+3 x+5 x^2} \left (-22661696-56011564 x-14855765 x^2-7709170 x^3+3605572 x^4+130088 x^5\right )}{\left (4+x-2 x^2\right )^3}-23609510951410660 \text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right )}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]-97536502800 \text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {7727219261 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+5054548389 \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]-99 \text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {4521207380523978672487 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2982791612362287724911 \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]+\text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {448353221338171976800157 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+295789380859309883615861 \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]}{230627136447379584} \] Input:
Integrate[(2 + 3*x + 5*x^2)^(3/2)/(4 + x - 2*x^2)^4,x]
Output:
((-1721443144*Sqrt[2 + 3*x + 5*x^2]*(-22661696 - 56011564*x - 14855765*x^2 - 7709170*x^3 + 3605572*x^4 + 130088*x^5))/(4 + x - 2*x^2)^3 - 2360951095 1410660*RootSum[-22 + 44*Sqrt[5]*#1 - 91*#1^2 + 2*Sqrt[5]*#1^3 + 2*#1^4 & , Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]/(22*Sqrt[5] - 91*#1 + 3*S qrt[5]*#1^2 + 4*#1^3) & ] - 97536502800*RootSum[-22 + 44*Sqrt[5]*#1 - 91*# 1^2 + 2*Sqrt[5]*#1^3 + 2*#1^4 & , (7727219261*Sqrt[5]*Log[-(Sqrt[5]*x) + S qrt[2 + 3*x + 5*x^2] - #1]*#1 + 5054548389*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1^2)/(22*Sqrt[5] - 91*#1 + 3*Sqrt[5]*#1^2 + 4*#1^3) & ] - 99*RootSum[-22 + 44*Sqrt[5]*#1 - 91*#1^2 + 2*Sqrt[5]*#1^3 + 2*#1^4 & , (4 521207380523978672487*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - # 1]*#1 + 2982791612362287724911*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1^2)/(22*Sqrt[5] - 91*#1 + 3*Sqrt[5]*#1^2 + 4*#1^3) & ] + RootSum[-22 + 44*Sqrt[5]*#1 - 91*#1^2 + 2*Sqrt[5]*#1^3 + 2*#1^4 & , (4483532213381719 76800157*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1 + 29578 9380859309883615861*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1^2)/( 22*Sqrt[5] - 91*#1 + 3*Sqrt[5]*#1^2 + 4*#1^3) & ])/230627136447379584
Time = 0.64 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1302, 27, 2132, 25, 2135, 27, 1365, 1154, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^{3/2}}{\left (-2 x^2+x+4\right )^4} \, dx\) |
| \(\Big \downarrow \) 1302 |
| \(\displaystyle \frac {1}{99} \int \frac {\sqrt {5 x^2+3 x+2} \left (80 x^2+114 x+89\right )}{2 \left (-2 x^2+x+4\right )^3}dx-\frac {(1-4 x) \left (5 x^2+3 x+2\right )^{3/2}}{99 \left (-2 x^2+x+4\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{198} \int \frac {\sqrt {5 x^2+3 x+2} \left (80 x^2+114 x+89\right )}{\left (-2 x^2+x+4\right )^3}dx-\frac {(1-4 x) \left (5 x^2+3 x+2\right )^{3/2}}{99 \left (-2 x^2+x+4\right )^3}\) |
| \(\Big \downarrow \) 2132 |
| \(\displaystyle \frac {1}{198} \left (\frac {(1150 x+983) \sqrt {5 x^2+3 x+2}}{66 \left (-2 x^2+x+4\right )^2}-\frac {1}{132} \int -\frac {-3400 x^2-8420 x+291}{\left (-2 x^2+x+4\right )^2 \sqrt {5 x^2+3 x+2}}dx\right )-\frac {(1-4 x) \left (5 x^2+3 x+2\right )^{3/2}}{99 \left (-2 x^2+x+4\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{198} \left (\frac {1}{132} \int \frac {-3400 x^2-8420 x+291}{\left (-2 x^2+x+4\right )^2 \sqrt {5 x^2+3 x+2}}dx+\frac {\sqrt {5 x^2+3 x+2} (1150 x+983)}{66 \left (-2 x^2+x+4\right )^2}\right )-\frac {(1-4 x) \left (5 x^2+3 x+2\right )^{3/2}}{99 \left (-2 x^2+x+4\right )^3}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{198} \left (\frac {1}{132} \left (-\frac {\int -\frac {3 (2743751-9599326 x)}{2 \left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}dx}{15378}-\frac {23 \sqrt {5 x^2+3 x+2} (1414 x+40605)}{5126 \left (-2 x^2+x+4\right )}\right )+\frac {\sqrt {5 x^2+3 x+2} (1150 x+983)}{66 \left (-2 x^2+x+4\right )^2}\right )-\frac {(1-4 x) \left (5 x^2+3 x+2\right )^{3/2}}{99 \left (-2 x^2+x+4\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{198} \left (\frac {1}{132} \left (\frac {\int \frac {2743751-9599326 x}{\left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}dx}{10252}-\frac {23 (1414 x+40605) \sqrt {5 x^2+3 x+2}}{5126 \left (-2 x^2+x+4\right )}\right )+\frac {\sqrt {5 x^2+3 x+2} (1150 x+983)}{66 \left (-2 x^2+x+4\right )^2}\right )-\frac {(1-4 x) \left (5 x^2+3 x+2\right )^{3/2}}{99 \left (-2 x^2+x+4\right )^3}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {1}{198} \left (\frac {1}{132} \left (\frac {-\frac {2}{33} \left (158388879+687839 \sqrt {33}\right ) \int \frac {1}{\left (-4 x-\sqrt {33}+1\right ) \sqrt {5 x^2+3 x+2}}dx-\frac {2}{33} \left (158388879-687839 \sqrt {33}\right ) \int \frac {1}{\left (-4 x+\sqrt {33}+1\right ) \sqrt {5 x^2+3 x+2}}dx}{10252}-\frac {23 (1414 x+40605) \sqrt {5 x^2+3 x+2}}{5126 \left (-2 x^2+x+4\right )}\right )+\frac {\sqrt {5 x^2+3 x+2} (1150 x+983)}{66 \left (-2 x^2+x+4\right )^2}\right )-\frac {(1-4 x) \left (5 x^2+3 x+2\right )^{3/2}}{99 \left (-2 x^2+x+4\right )^3}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{198} \left (\frac {1}{132} \left (\frac {\frac {4}{33} \left (158388879+687839 \sqrt {33}\right ) \int \frac {1}{8 \left (107-11 \sqrt {33}\right )-\frac {\left (2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19\right )^2}{5 x^2+3 x+2}}d\left (-\frac {2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19}{\sqrt {5 x^2+3 x+2}}\right )+\frac {4}{33} \left (158388879-687839 \sqrt {33}\right ) \int \frac {1}{8 \left (107+11 \sqrt {33}\right )-\frac {\left (2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19\right )^2}{5 x^2+3 x+2}}d\left (-\frac {2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19}{\sqrt {5 x^2+3 x+2}}\right )}{10252}-\frac {23 (1414 x+40605) \sqrt {5 x^2+3 x+2}}{5126 \left (-2 x^2+x+4\right )}\right )+\frac {\sqrt {5 x^2+3 x+2} (1150 x+983)}{66 \left (-2 x^2+x+4\right )^2}\right )-\frac {(1-4 x) \left (5 x^2+3 x+2\right )^{3/2}}{99 \left (-2 x^2+x+4\right )^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{198} \left (\frac {1}{132} \left (\frac {-\frac {1}{33} \sqrt {\frac {2}{107-11 \sqrt {33}}} \left (158388879+687839 \sqrt {33}\right ) \text {arctanh}\left (\frac {2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19}{2 \sqrt {2 \left (107-11 \sqrt {33}\right )} \sqrt {5 x^2+3 x+2}}\right )-\frac {1}{33} \left (158388879-687839 \sqrt {33}\right ) \sqrt {\frac {2}{107+11 \sqrt {33}}} \text {arctanh}\left (\frac {2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19}{2 \sqrt {2 \left (107+11 \sqrt {33}\right )} \sqrt {5 x^2+3 x+2}}\right )}{10252}-\frac {23 (1414 x+40605) \sqrt {5 x^2+3 x+2}}{5126 \left (-2 x^2+x+4\right )}\right )+\frac {\sqrt {5 x^2+3 x+2} (1150 x+983)}{66 \left (-2 x^2+x+4\right )^2}\right )-\frac {(1-4 x) \left (5 x^2+3 x+2\right )^{3/2}}{99 \left (-2 x^2+x+4\right )^3}\) |
Input:
Int[(2 + 3*x + 5*x^2)^(3/2)/(4 + x - 2*x^2)^4,x]
Output:
-1/99*((1 - 4*x)*(2 + 3*x + 5*x^2)^(3/2))/(4 + x - 2*x^2)^3 + (((983 + 115 0*x)*Sqrt[2 + 3*x + 5*x^2])/(66*(4 + x - 2*x^2)^2) + ((-23*(40605 + 1414*x )*Sqrt[2 + 3*x + 5*x^2])/(5126*(4 + x - 2*x^2)) + (-1/33*(Sqrt[2/(107 - 11 *Sqrt[33])]*(158388879 + 687839*Sqrt[33])*ArcTanh[(19 - 3*Sqrt[33] + 2*(11 - 5*Sqrt[33])*x)/(2*Sqrt[2*(107 - 11*Sqrt[33])]*Sqrt[2 + 3*x + 5*x^2])]) - ((158388879 - 687839*Sqrt[33])*Sqrt[2/(107 + 11*Sqrt[33])]*ArcTanh[(19 + 3*Sqrt[33] + 2*(11 + 5*Sqrt[33])*x)/(2*Sqrt[2*(107 + 11*Sqrt[33])]*Sqrt[2 + 3*x + 5*x^2])])/33)/10252)/132)/198
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-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]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x _)^2)^(q_), x_Symbol] :> Simp[(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*((d + e *x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p + 3) + b*e*q + (2*b*f*q + 2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim p[(2*c*g - h*(b - q))/q Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] , x] - Simp[(2*c*g - h*(b + q))/q Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f *x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0 ] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. )*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] , C = Coeff[Px, x, 2]}, Simp[(A*b*c - 2*a*B*c + a*b*C - (c*(b*B - 2*A*c) - C*(b^2 - 2*a*c))*x)*(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^q/(c*(b^2 - 4*a*c)*(p + 1))), x] - Simp[1/(c*(b^2 - 4*a*c)*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[e*q*(A*b*c - 2*a*B*c + a*b*C) - d*(c*(b*B - 2*A*c)*(2*p + 3) + C*(2*a*c - b^2*(p + 2))) + (2*f*q*(A*b*c - 2*a*B*c + a*b*C) - e*(c*(b*B - 2*A*c)*(2*p + q + 3) + C*(2*a*c*(q + 1) - b^2*(p + q + 2))))*x - f*(c*(b*B - 2*A*c)*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. )*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] , C = Coeff[Px, x, 2]}, Simp[(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^( q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))*( (A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b *e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c* e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x), x] + Simp[1/((b^2 - 4 *a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)) Int[(a + b*x + c *x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B )*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a *c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*( c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4))) *x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; F reeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
Time = 4.44 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(\frac {\left (130088 x^{5}+3605572 x^{4}-7709170 x^{3}-14855765 x^{2}-56011564 x -22661696\right ) \sqrt {5 x^{2}+3 x +2}}{133973136 \left (2 x^{2}-x -4\right )^{3}}-\frac {\left (-687839+4799663 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {214+22 \sqrt {33}+8 \left (\frac {11}{2}+\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )}{\sqrt {214+22 \sqrt {33}}\, \sqrt {80 \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )^{2}+16 \left (\frac {11}{2}+\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )+214+22 \sqrt {33}}}\right )}{4421113488 \sqrt {214+22 \sqrt {33}}}-\frac {\left (687839+4799663 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {214-22 \sqrt {33}+8 \left (\frac {11}{2}-\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )}{\sqrt {214-22 \sqrt {33}}\, \sqrt {80 \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )^{2}+16 \left (\frac {11}{2}-\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )+214-22 \sqrt {33}}}\right )}{4421113488 \sqrt {214-22 \sqrt {33}}}\) | \(241\) |
| trager | \(\text {Expression too large to display}\) | \(493\) |
| default | \(\text {Expression too large to display}\) | \(7090\) |
Input:
int((5*x^2+3*x+2)^(3/2)/(-2*x^2+x+4)^4,x,method=_RETURNVERBOSE)
Output:
1/133973136*(130088*x^5+3605572*x^4-7709170*x^3-14855765*x^2-56011564*x-22 661696)/(2*x^2-x-4)^3*(5*x^2+3*x+2)^(1/2)-1/4421113488*(-687839+4799663*33 ^(1/2))*33^(1/2)/(214+22*33^(1/2))^(1/2)*arctanh(8*(107/4+11/4*33^(1/2)+(1 1/2+5/2*33^(1/2))*(x-1/4*33^(1/2)-1/4))/(214+22*33^(1/2))^(1/2)/(80*(x-1/4 *33^(1/2)-1/4)^2+16*(11/2+5/2*33^(1/2))*(x-1/4*33^(1/2)-1/4)+214+22*33^(1/ 2))^(1/2))-1/4421113488*(687839+4799663*33^(1/2))*33^(1/2)/(214-22*33^(1/2 ))^(1/2)*arctanh(8*(107/4-11/4*33^(1/2)+(11/2-5/2*33^(1/2))*(x-1/4+1/4*33^ (1/2)))/(214-22*33^(1/2))^(1/2)/(80*(x-1/4+1/4*33^(1/2))^2+16*(11/2-5/2*33 ^(1/2))*(x-1/4+1/4*33^(1/2))+214-22*33^(1/2))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (212) = 424\).
Time = 0.12 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.63 \[ \int \frac {\left (2+3 x+5 x^2\right )^{3/2}}{\left (4+x-2 x^2\right )^4} \, dx =\text {Too large to display} \] Input:
integrate((5*x^2+3*x+2)^(3/2)/(-2*x^2+x+4)^4,x, algorithm="fricas")
Output:
-1/1071785088*((8*x^6 - 12*x^5 - 42*x^4 + 47*x^3 + 84*x^2 - 48*x - 64)*sqr t(68742792623043/233*sqrt(33) + 20947563518842067/7689)*log(-(sqrt(5*x^2 + 3*x + 2)*sqrt(68742792623043/233*sqrt(33) + 20947563518842067/7689)*(4806 69349*sqrt(33) - 1335207093) + 94967514957232*sqrt(33)*(3*x + 4) - 7312498 651706864*x - 1899350299144640)/x) - (8*x^6 - 12*x^5 - 42*x^4 + 47*x^3 + 8 4*x^2 - 48*x - 64)*sqrt(68742792623043/233*sqrt(33) + 20947563518842067/76 89)*log((sqrt(5*x^2 + 3*x + 2)*sqrt(68742792623043/233*sqrt(33) + 20947563 518842067/7689)*(480669349*sqrt(33) - 1335207093) - 94967514957232*sqrt(33 )*(3*x + 4) + 7312498651706864*x + 1899350299144640)/x) + (8*x^6 - 12*x^5 - 42*x^4 + 47*x^3 + 84*x^2 - 48*x - 64)*sqrt(-68742792623043/233*sqrt(33) + 20947563518842067/7689)*log((sqrt(5*x^2 + 3*x + 2)*(480669349*sqrt(33) + 1335207093)*sqrt(-68742792623043/233*sqrt(33) + 20947563518842067/7689) + 94967514957232*sqrt(33)*(3*x + 4) + 7312498651706864*x + 1899350299144640 )/x) - (8*x^6 - 12*x^5 - 42*x^4 + 47*x^3 + 84*x^2 - 48*x - 64)*sqrt(-68742 792623043/233*sqrt(33) + 20947563518842067/7689)*log(-(sqrt(5*x^2 + 3*x + 2)*(480669349*sqrt(33) + 1335207093)*sqrt(-68742792623043/233*sqrt(33) + 2 0947563518842067/7689) - 94967514957232*sqrt(33)*(3*x + 4) - 7312498651706 864*x - 1899350299144640)/x) - 8*(130088*x^5 + 3605572*x^4 - 7709170*x^3 - 14855765*x^2 - 56011564*x - 22661696)*sqrt(5*x^2 + 3*x + 2))/(8*x^6 - 12* x^5 - 42*x^4 + 47*x^3 + 84*x^2 - 48*x - 64)
\[ \int \frac {\left (2+3 x+5 x^2\right )^{3/2}}{\left (4+x-2 x^2\right )^4} \, dx=\int \frac {\left (5 x^{2} + 3 x + 2\right )^{\frac {3}{2}}}{\left (2 x^{2} - x - 4\right )^{4}}\, dx \] Input:
integrate((5*x**2+3*x+2)**(3/2)/(-2*x**2+x+4)**4,x)
Output:
Integral((5*x**2 + 3*x + 2)**(3/2)/(2*x**2 - x - 4)**4, x)
\[ \int \frac {\left (2+3 x+5 x^2\right )^{3/2}}{\left (4+x-2 x^2\right )^4} \, dx=\int { \frac {{\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}}}{{\left (2 \, x^{2} - x - 4\right )}^{4}} \,d x } \] Input:
integrate((5*x^2+3*x+2)^(3/2)/(-2*x^2+x+4)^4,x, algorithm="maxima")
Output:
integrate((5*x^2 + 3*x + 2)^(3/2)/(2*x^2 - x - 4)^4, x)
Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (212) = 424\).
Time = 0.18 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.80 \[ \int \frac {\left (2+3 x+5 x^2\right )^{3/2}}{\left (4+x-2 x^2\right )^4} \, dx=-\frac {38397304 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{11} - 106968264 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{10} - 4215429092 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{9} - 27627624304 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{8} - 293612574754 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{7} - 304441105682 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{6} - 962237337353 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{5} - 368536654772 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} - 353763969768 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} - 4216436664 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} + 48130637060 \, \sqrt {5} x + 5546394128 \, \sqrt {5} - 48130637060 \, \sqrt {5 \, x^{2} + 3 \, x + 2}}{133973136 \, {\left (2 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} - 2 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} - 91 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - 44 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} - 22\right )}^{3}} - 0.00189338515268464 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 8.38267526007000\right ) - 0.00392277827816914 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 0.312157316296000\right ) + 0.00189338515268464 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 0.842024981991000\right ) + 0.00392277827816914 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 4.99242498429000\right ) \] Input:
integrate((5*x^2+3*x+2)^(3/2)/(-2*x^2+x+4)^4,x, algorithm="giac")
Output:
-1/133973136*(38397304*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^11 - 106968264* sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^10 - 4215429092*(sqrt(5)*x - s qrt(5*x^2 + 3*x + 2))^9 - 27627624304*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3* x + 2))^8 - 293612574754*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^7 - 304441105 682*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^6 - 962237337353*(sqrt(5)* x - sqrt(5*x^2 + 3*x + 2))^5 - 368536654772*sqrt(5)*(sqrt(5)*x - sqrt(5*x^ 2 + 3*x + 2))^4 - 353763969768*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^3 - 421 6436664*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 + 48130637060*sqrt(5 )*x + 5546394128*sqrt(5) - 48130637060*sqrt(5*x^2 + 3*x + 2))/(2*(sqrt(5)* x - sqrt(5*x^2 + 3*x + 2))^4 - 2*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2 ))^3 - 91*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 - 44*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2)) - 22)^3 - 0.00189338515268464*log(-sqrt(5)*x + sqrt (5*x^2 + 3*x + 2) + 8.38267526007000) - 0.00392277827816914*log(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) - 0.312157316296000) + 0.00189338515268464*log(-s qrt(5)*x + sqrt(5*x^2 + 3*x + 2) - 0.842024981991000) + 0.0039227782781691 4*log(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) - 4.99242498429000)
Timed out. \[ \int \frac {\left (2+3 x+5 x^2\right )^{3/2}}{\left (4+x-2 x^2\right )^4} \, dx=\int \frac {{\left (5\,x^2+3\,x+2\right )}^{3/2}}{{\left (-2\,x^2+x+4\right )}^4} \,d x \] Input:
int((3*x + 5*x^2 + 2)^(3/2)/(x - 2*x^2 + 4)^4,x)
Output:
int((3*x + 5*x^2 + 2)^(3/2)/(x - 2*x^2 + 4)^4, x)
\[ \int \frac {\left (2+3 x+5 x^2\right )^{3/2}}{\left (4+x-2 x^2\right )^4} \, dx=\int \frac {\left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}{\left (-2 x^{2}+x +4\right )^{4}}d x \] Input:
int((5*x^2+3*x+2)^(3/2)/(-2*x^2+x+4)^4,x)
Output:
int((5*x^2+3*x+2)^(3/2)/(-2*x^2+x+4)^4,x)