\(\int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 (3-x+2 x^2)^{3/2}} \, dx\) [193]

Optimal result

Integrand size = 40, antiderivative size = 137 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {369609-175877 x}{154524672 \sqrt {3-x+2 x^2}}-\frac {3667 \sqrt {3-x+2 x^2}}{31104 (5+2 x)^3}+\frac {152885 \sqrt {3-x+2 x^2}}{4478976 (5+2 x)^2}+\frac {430799 \sqrt {3-x+2 x^2}}{107495424 (5+2 x)}-\frac {3505819 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{1289945088 \sqrt {2}} \] Output:

1/154524672*(369609-175877*x)/(2*x^2-x+3)^(1/2)-3667/31104*(2*x^2-x+3)^(1/ 
2)/(5+2*x)^3+152885/4478976*(2*x^2-x+3)^(1/2)/(5+2*x)^2+430799*(2*x^2-x+3) 
^(1/2)/(537477120+214990848*x)-3505819/2579890176*arctanh(1/24*(17-22*x)*2 
^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)
 

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.59 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {\frac {12 \left (1873786587+1257975811 x+441046842 x^2+572739684 x^3+56754760 x^4\right )}{(5+2 x)^3 \sqrt {3-x+2 x^2}}[phone] \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )}{29668737024} \] Input:

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

Output:

((12*(1873786587 + 1257975811*x + 441046842*x^2 + 572739684*x^3 + 56754760 
*x^4))/((5 + 2*x)^3*Sqrt[3 - x + 2*x^2]) + 80633837*Sqrt[2]*ArcTanh[(5 + 2 
*x - Sqrt[6 - 2*x + 4*x^2])/6])/29668737024
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {2177, 27, 2181, 27, 2181, 27, 1228, 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.

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

\(\Big \downarrow \) 2177

\(\displaystyle \frac {2}{23} \int \frac {23 \left (453064 x^3+38587980 x^2+31270710 x+15168577\right )}{26873856 (2 x+5)^4 \sqrt {2 x^2-x+3}}dx+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {453064 x^3+38587980 x^2+31270710 x+15168577}{(2 x+5)^4 \sqrt {2 x^2-x+3}}dx}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {-\frac {1}{216} \int \frac {216 \left (-226532 x^2-12391084 x+3461275\right )}{(2 x+5)^3 \sqrt {2 x^2-x+3}}dx-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\int \frac {-226532 x^2-12391084 x+3461275}{(2 x+5)^3 \sqrt {2 x^2-x+3}}dx-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {\frac {1}{144} \int \frac {72 (2061152 x+1275689)}{(2 x+5)^2 \sqrt {2 x^2-x+3}}dx+\frac {458655 \sqrt {2 x^2-x+3}}{(2 x+5)^2}-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{2} \int \frac {2061152 x+1275689}{(2 x+5)^2 \sqrt {2 x^2-x+3}}dx+\frac {458655 \sqrt {2 x^2-x+3}}{(2 x+5)^2}-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {1}{2} \left (\frac {3505819}{8} \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx+\frac {430799 \sqrt {2 x^2-x+3}}{4 (2 x+5)}\right )+\frac {458655 \sqrt {2 x^2-x+3}}{(2 x+5)^2}-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {1}{2} \left (\frac {430799 \sqrt {2 x^2-x+3}}{4 (2 x+5)}-\frac {3505819}{4} \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}\right )+\frac {458655 \sqrt {2 x^2-x+3}}{(2 x+5)^2}-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{2} \left (\frac {430799 \sqrt {2 x^2-x+3}}{4 (2 x+5)}-\frac {3505819 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{48 \sqrt {2}}\right )+\frac {458655 \sqrt {2 x^2-x+3}}{(2 x+5)^2}-\frac {1584144 \sqrt {2 x^2-x+3}}{(2 x+5)^3}}{13436928}+\frac {369609-175877 x}{154524672 \sqrt {2 x^2-x+3}}\)

Input:

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

Output:

(369609 - 175877*x)/(154524672*Sqrt[3 - x + 2*x^2]) + ((-1584144*Sqrt[3 - 
x + 2*x^2])/(5 + 2*x)^3 + (458655*Sqrt[3 - x + 2*x^2])/(5 + 2*x)^2 + ((430 
799*Sqrt[3 - x + 2*x^2])/(4*(5 + 2*x)) - (3505819*ArcTanh[(17 - 22*x)/(12* 
Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(48*Sqrt[2]))/2)/13436928
 

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 219

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])
 

rule 1154

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]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(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] - Simp[(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 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 2177

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x + c* 
x^2, x], R = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], 
 x, 0], S = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], 
x, 1]}, Simp[(b*R - 2*a*S + (2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p 
 + 1)*(b^2 - 4*a*c))), x] + Simp[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)*Qx)/(d + e*x 
)^m - ((2*p + 3)*(2*c*R - b*S))/(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 2181

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi 
alRemainder[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] + Simp[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)*Qx + 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]
 

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.53

method	result	size
risch	\(\frac {56754760 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}{2472394752 \left (5+2 x \right )^{3} \sqrt {2 x^{2}-x +3}}-\frac {3505819 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{2579890176}\)	\(73\)
trager	\(\frac {56754760 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}{2472394752 \left (5+2 x \right )^{3} \sqrt {2 x^{2}-x +3}}-\frac {3505819 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {22 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x -17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-24 \sqrt {2 x^{2}-x +3}}{5+2 x}\right )}{2579890176}\)	\(93\)
default	\(\frac {\frac {5 x}{46}-\frac {5}{184}}{\sqrt {2 x^{2}-x +3}}-\frac {3667}{13824 \left (x +\frac {5}{2}\right )^{3} \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}+\frac {314233}{995328 \left (x +\frac {5}{2}\right )^{2} \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}-\frac {3127169}{35831808 \left (x +\frac {5}{2}\right ) \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}+\frac {3505819}{429981696 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}-\frac {261644215 \left (4 x -1\right )}{9889579008 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}-\frac {3505819 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{2579890176}\)	\(151\)

Input:

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

Output:

1/2472394752*(56754760*x^4+572739684*x^3+441046842*x^2+1257975811*x+187378 
6587)/(5+2*x)^3/(2*x^2-x+3)^(1/2)-3505819/2579890176*2^(1/2)*arctanh(1/12* 
(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.03 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {80633837 \, \sqrt {2} {\left (16 \, x^{5} + 112 \, x^{4} + 264 \, x^{3} + 280 \, x^{2} + 325 \, x + 375\right )} \log \left (-\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (56754760 \, x^{4} + 572739684 \, x^{3} + 441046842 \, x^{2} + 1257975811 \, x + 1873786587\right )} \sqrt {2 \, x^{2} - x + 3}}{118674948096 \, {\left (16 \, x^{5} + 112 \, x^{4} + 264 \, x^{3} + 280 \, x^{2} + 325 \, x + 375\right )}} \] Input:

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

Output:

1/118674948096*(80633837*sqrt(2)*(16*x^5 + 112*x^4 + 264*x^3 + 280*x^2 + 3 
25*x + 375)*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*(56754760*x^4 + 572739684*x^3 + 4 
41046842*x^2 + 1257975811*x + 1873786587)*sqrt(2*x^2 - x + 3))/(16*x^5 + 1 
12*x^4 + 264*x^3 + 280*x^2 + 325*x + 375)
 

Sympy [F]

\[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=\int \frac {5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x + 5\right )^{4} \left (2 x^{2} - x + 3\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.58 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {3505819}{2579890176} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {7094345 \, x}{2472394752 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {6128291}{824131584 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {3667}{1728 \, {\left (8 \, \sqrt {2 \, x^{2} - x + 3} x^{3} + 60 \, \sqrt {2 \, x^{2} - x + 3} x^{2} + 150 \, \sqrt {2 \, x^{2} - x + 3} x + 125 \, \sqrt {2 \, x^{2} - x + 3}\right )}} + \frac {314233}{248832 \, {\left (4 \, \sqrt {2 \, x^{2} - x + 3} x^{2} + 20 \, \sqrt {2 \, x^{2} - x + 3} x + 25 \, \sqrt {2 \, x^{2} - x + 3}\right )}} - \frac {3127169}{17915904 \, {\left (2 \, \sqrt {2 \, x^{2} - x + 3} x + 5 \, \sqrt {2 \, x^{2} - x + 3}\right )}} \] Input:

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

Output:

3505819/2579890176*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*s 
qrt(23)/abs(2*x + 5)) + 7094345/2472394752*x/sqrt(2*x^2 - x + 3) + 6128291 
/824131584/sqrt(2*x^2 - x + 3) - 3667/1728/(8*sqrt(2*x^2 - x + 3)*x^3 + 60 
*sqrt(2*x^2 - x + 3)*x^2 + 150*sqrt(2*x^2 - x + 3)*x + 125*sqrt(2*x^2 - x 
+ 3)) + 314233/248832/(4*sqrt(2*x^2 - x + 3)*x^2 + 20*sqrt(2*x^2 - x + 3)* 
x + 25*sqrt(2*x^2 - x + 3)) - 3127169/17915904/(2*sqrt(2*x^2 - x + 3)*x + 
5*sqrt(2*x^2 - x + 3))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (111) = 222\).

Time = 0.22 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.98 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=-\frac {3505819}{2579890176} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {3505819}{2579890176} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {175877 \, x - 369609}{154524672 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {\sqrt {2} {\left (10398764 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} - 303070900 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} - 529738052 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} + 3644644652 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} - 2612608649 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1052284471\right )}}{214990848 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{3}} \] Input:

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

Output:

-3505819/2579890176*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 
- x + 3))) + 3505819/2579890176*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) 
+ 2*sqrt(2*x^2 - x + 3))) - 1/154524672*(175877*x - 369609)/sqrt(2*x^2 - x 
 + 3) - 1/214990848*sqrt(2)*(10398764*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x 
+ 3))^5 - 303070900*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^4 - 529738052*sqrt(2 
)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 + 3644644652*(sqrt(2)*x - sqrt(2*x^2 
 - x + 3))^2 - 2612608649*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1052 
284471)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - s 
qrt(2*x^2 - x + 3)) - 11)^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.48 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {1362114240 \sqrt {2 x^{2}-x +3}\, x^{4}[phone] \sqrt {2 x^{2}-x +3}\, x^{3}[phone] \sqrt {2 x^{2}-x +3}\, x^{2}[phone] \sqrt {2 x^{2}-x +3}\, x [phone] \sqrt {2 x^{2}-x +3}[phone] \sqrt {2}\, \mathrm {log}\left (12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{5}[phone] \sqrt {2}\, \mathrm {log}\left (12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{4}[phone] \sqrt {2}\, \mathrm {log}\left (12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{3}[phone] \sqrt {2}\, \mathrm {log}\left (12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{2}[phone] \sqrt {2}\, \mathrm {log}\left (12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x [phone] \sqrt {2}\, \mathrm {log}\left (12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right )-1290141392 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{5}-9030989744 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{4}-21287332968 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{3}-22577474360 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{2}-26205997025 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x -30237688875 \sqrt {2}\, \mathrm {log}\left (2 x +5\right )}{949399584768 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

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

Output:

(1362114240*sqrt(2*x**2 - x + 3)*x**4 + 13745752416*sqrt(2*x**2 - x + 3)*x 
**3 + 10585124208*sqrt(2*x**2 - x + 3)*x**2 + 30191419464*sqrt(2*x**2 - x 
+ 3)*x + 44970878088*sqrt(2*x**2 - x + 3) + 1290141392*sqrt(2)*log(12*sqrt 
(2*x**2 - x + 3)*sqrt(2) + 22*x - 17)*x**5 + 9030989744*sqrt(2)*log(12*sqr 
t(2*x**2 - x + 3)*sqrt(2) + 22*x - 17)*x**4 + 21287332968*sqrt(2)*log(12*s 
qrt(2*x**2 - x + 3)*sqrt(2) + 22*x - 17)*x**3 + 22577474360*sqrt(2)*log(12 
*sqrt(2*x**2 - x + 3)*sqrt(2) + 22*x - 17)*x**2 + 26205997025*sqrt(2)*log( 
12*sqrt(2*x**2 - x + 3)*sqrt(2) + 22*x - 17)*x + 30237688875*sqrt(2)*log(1 
2*sqrt(2*x**2 - x + 3)*sqrt(2) + 22*x - 17) - 1290141392*sqrt(2)*log(2*x + 
 5)*x**5 - 9030989744*sqrt(2)*log(2*x + 5)*x**4 - 21287332968*sqrt(2)*log( 
2*x + 5)*x**3 - 22577474360*sqrt(2)*log(2*x + 5)*x**2 - 26205997025*sqrt(2 
)*log(2*x + 5)*x - 30237688875*sqrt(2)*log(2*x + 5))/(59337474048*(16*x**5 
 + 112*x**4 + 264*x**3 + 280*x**2 + 325*x + 375))