\(\int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx\) [1526]

Optimal result

Integrand size = 28, antiderivative size = 129 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx=\frac {31 \sqrt {1-2 x} \sqrt {3+5 x}}{147 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {1159}{63} \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )-\frac {37}{63} \sqrt {\frac {5}{7}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:

31/147*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)+11/7*(3+5*x)^(3/2)/(1-2*x 
)^(1/2)/(2+3*x)^(1/2)+1159/441*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35* 
1155^(1/2))*35^(1/2)-37/441*EllipticF(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*115 
5^(1/2))*35^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.42 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.91 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx=\frac {3 \sqrt {2+3 x} \sqrt {3+5 x} (724+1093 x)-1159 i \sqrt {33-66 x} (2+3 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+1190 i \sqrt {33-66 x} (2+3 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{441 \sqrt {1-2 x} (2+3 x)} \] Input:

Integrate[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)),x]
 

Output:

(3*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(724 + 1093*x) - (1159*I)*Sqrt[33 - 66*x]*( 
2 + 3*x)*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (1190*I)*Sqrt[33 - 
66*x]*(2 + 3*x)*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(441*Sqrt[1 - 
 2*x]*(2 + 3*x))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {109, 27, 167, 27, 176, 123, 129}

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.

\(\displaystyle \int \frac {(5 x+3)^{5/2}}{(1-2 x)^{3/2} (3 x+2)^{3/2}} \, dx\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}-\frac {1}{7} \int \frac {\sqrt {5 x+3} (340 x+237)}{2 \sqrt {1-2 x} (3 x+2)^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}-\frac {1}{14} \int \frac {\sqrt {5 x+3} (340 x+237)}{\sqrt {1-2 x} (3 x+2)^{3/2}}dx\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {1}{14} \left (\frac {62 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}-\frac {2}{21} \int \frac {5 (2318 x+1459)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {62 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}-\frac {5}{21} \int \frac {2318 x+1459}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{14} \left (\frac {62 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}-\frac {5}{21} \left (\frac {341}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2318}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{14} \left (\frac {62 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}-\frac {5}{21} \left (\frac {341}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2318}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{14} \left (\frac {62 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}-\frac {5}{21} \left (-\frac {62}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {2318}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}\)

Input:

Int[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)),x]
 

Output:

(11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) + ((62*Sqrt[1 - 2*x]* 
Sqrt[3 + 5*x])/(21*Sqrt[2 + 3*x]) - (5*((-2318*Sqrt[11/3]*EllipticE[ArcSin 
[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (62*Sqrt[11/3]*EllipticF[ArcSin[Sqr 
t[3/7]*Sqrt[1 - 2*x]], 35/33])/5))/21)/14
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || 
IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 123

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] 
/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, 
b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !L 
tQ[-(b*c - a*d)/d, 0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d 
), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])
 

rule 129

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ 
Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - 
a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ 
[(b*e - a*f)/b, 0] && PosQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d 
*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(( 
-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ 
[((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f 
/b]))
 

rule 167

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
 

rule 176

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* 
Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f   Int[Sqrt[e + f*x]/(Sqrt[a + b*x 
]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f   Int[1/(Sqrt[a + b*x]*Sqrt[c 
 + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim 
plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
 

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.03

method	result	size
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (1023 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2318 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-32790 x^{2}-41394 x -13032\right )}{26460 x^{3}+20286 x^{2}-6174 x -5292}\)	\(133\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-18-30 x \right ) \left (\frac {181}{441}+\frac {1093 x}{1764}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}-\frac {7295 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{6174 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {5795 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{3087 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(201\)

Input:

int((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/882*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(1023*2^(1/2)*(2+3*x)^(1/2 
)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2)) 
-2318*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28 
+42*x)^(1/2),1/2*70^(1/2))-32790*x^2-41394*x-13032)/(30*x^3+23*x^2-7*x-6)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.64 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx=-\frac {135 \, {\left (1093 \, x + 724\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 19499 \, \sqrt {-30} {\left (6 \, x^{2} + x - 2\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 52155 \, \sqrt {-30} {\left (6 \, x^{2} + x - 2\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{19845 \, {\left (6 \, x^{2} + x - 2\right )}} \] Input:

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(3/2),x, algorithm="fricas")
 

Output:

-1/19845*(135*(1093*x + 724)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 
19499*sqrt(-30)*(6*x^2 + x - 2)*weierstrassPInverse(1159/675, 38998/91125, 
 x + 23/90) + 52155*sqrt(-30)*(6*x^2 + x - 2)*weierstrassZeta(1159/675, 38 
998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(6*x^2 
+ x - 2)
 

Sympy [F]

\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((3+5*x)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**(3/2),x)
 

Output:

Integral((5*x + 3)**(5/2)/((1 - 2*x)**(3/2)*(3*x + 2)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(3/2),x, algorithm="maxima")
 

Output:

integrate((5*x + 3)^(5/2)/((3*x + 2)^(3/2)*(-2*x + 1)^(3/2)), x)
 

Giac [F]

\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(3/2),x, algorithm="giac")
 

Output:

integrate((5*x + 3)^(5/2)/((3*x + 2)^(3/2)*(-2*x + 1)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}} \,d x \] Input:

int((5*x + 3)^(5/2)/((1 - 2*x)^(3/2)*(3*x + 2)^(3/2)),x)
 

Output:

int((5*x + 3)^(5/2)/((1 - 2*x)^(3/2)*(3*x + 2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx=\frac {200 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x -330 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}-3300 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{180 x^{5}+168 x^{4}-79 x^{3}-89 x^{2}+8 x +12}d x \right ) x^{2}-550 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{180 x^{5}+168 x^{4}-79 x^{3}-89 x^{2}+8 x +12}d x \right ) x +1100 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{180 x^{5}+168 x^{4}-79 x^{3}-89 x^{2}+8 x +12}d x \right )-1782 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{180 x^{5}+168 x^{4}-79 x^{3}-89 x^{2}+8 x +12}d x \right ) x^{2}-297 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{180 x^{5}+168 x^{4}-79 x^{3}-89 x^{2}+8 x +12}d x \right ) x +594 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{180 x^{5}+168 x^{4}-79 x^{3}-89 x^{2}+8 x +12}d x \right )}{144 x^{2}+24 x -48} \] Input:

int((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(3/2),x)
 

Output:

(200*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 330*sqrt(3*x + 2)*sq 
rt(5*x + 3)*sqrt( - 2*x + 1) - 3300*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( 
 - 2*x + 1)*x**2)/(180*x**5 + 168*x**4 - 79*x**3 - 89*x**2 + 8*x + 12),x)* 
x**2 - 550*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(180*x* 
*5 + 168*x**4 - 79*x**3 - 89*x**2 + 8*x + 12),x)*x + 1100*int((sqrt(3*x + 
2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(180*x**5 + 168*x**4 - 79*x**3 - 8 
9*x**2 + 8*x + 12),x) - 1782*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x 
+ 1))/(180*x**5 + 168*x**4 - 79*x**3 - 89*x**2 + 8*x + 12),x)*x**2 - 297*i 
nt((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(180*x**5 + 168*x**4 - 7 
9*x**3 - 89*x**2 + 8*x + 12),x)*x + 594*int((sqrt(3*x + 2)*sqrt(5*x + 3)*s 
qrt( - 2*x + 1))/(180*x**5 + 168*x**4 - 79*x**3 - 89*x**2 + 8*x + 12),x))/ 
(24*(6*x**2 + x - 2))