\(\int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 (3-x+2 x^2)^{5/2}} \, dx\) [200]

Optimal result

Integrand size = 40, antiderivative size = 160 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}-\frac {27754539-31190998 x}{31986607104 \sqrt {3-x+2 x^2}}-\frac {3667 \sqrt {3-x+2 x^2}}{559872 (5+2 x)^3}-\frac {89137 \sqrt {3-x+2 x^2}}{80621568 (5+2 x)^2}+\frac {475357 \sqrt {3-x+2 x^2}}{1934917632 (5+2 x)}+\frac {4778789 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{7739670528 \sqrt {2}} \] Output:

1/463574016*(369609-175877*x)/(2*x^2-x+3)^(3/2)-1/31986607104*(27754539-31 
190998*x)/(2*x^2-x+3)^(1/2)-3667/559872*(2*x^2-x+3)^(1/2)/(5+2*x)^3-89137/ 
80621568*(2*x^2-x+3)^(1/2)/(5+2*x)^2+475357*(2*x^2-x+3)^(1/2)/(9674588160+ 
3869835264*x)+4778789/15479341056*arctanh(1/24*(17-22*x)*2^(1/2)/(2*x^2-x+ 
3)^(1/2))*2^(1/2)
 

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.57 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {\frac {12 \left (-95241881529+73621973154 x-6702882569 x^2+27484986184 x^3+46210466520 x^4+34872810880 x^5+6664404208 x^6\right )}{(5+2 x)^3 \left (3-x+2 x^2\right )^{3/2}}-2527979381 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )}{4094285709312} \] Input:

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

Output:

((12*(-95241881529 + 73621973154*x - 6702882569*x^2 + 27484986184*x^3 + 46 
210466520*x^4 + 34872810880*x^5 + 6664404208*x^6))/((5 + 2*x)^3*(3 - x + 2 
*x^2)^(3/2)) - 2527979381*Sqrt[2]*ArcTanh[(5 + 2*x - Sqrt[6 - 2*x + 4*x^2] 
)/6])/4094285709312
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2177, 27, 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.

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

\(\Big \downarrow \) 2177

\(\displaystyle \frac {2}{69} \int \frac {-11256128 x^4-81299864 x^3+2240465820 x^2+1454170990 x+606939313}{26873856 (2 x+5)^4 \left (2 x^2-x+3\right )^{3/2}}dx+\frac {369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {-11256128 x^4-81299864 x^3+2240465820 x^2+1454170990 x+606939313}{(2 x+5)^4 \left (2 x^2-x+3\right )^{3/2}}dx}{927148032}+\frac {369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 2177

\(\displaystyle \frac {\frac {2}{23} \int -\frac {1058 \left (301804 x^3+3955080 x^2+20194167 x+9095911\right )}{3 (2 x+5)^4 \sqrt {2 x^2-x+3}}dx-\frac {2 (27754539-31190998 x)}{69 \sqrt {2 x^2-x+3}}}{927148032}+\frac {369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {92}{3} \int \frac {301804 x^3+3955080 x^2+20194167 x+9095911}{(2 x+5)^4 \sqrt {2 x^2-x+3}}dx-\frac {2 (27754539-31190998 x)}{69 \sqrt {2 x^2-x+3}}}{927148032}+\frac {369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {-\frac {92}{3} \left (\frac {198018 \sqrt {2 x^2-x+3}}{(2 x+5)^3}-\frac {1}{216} \int -\frac {216 \left (150902 x^2+2392357 x+2631056\right )}{(2 x+5)^3 \sqrt {2 x^2-x+3}}dx\right )-\frac {2 (27754539-31190998 x)}{69 \sqrt {2 x^2-x+3}}}{927148032}+\frac {369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {92}{3} \left (\int \frac {150902 x^2+2392357 x+2631056}{(2 x+5)^3 \sqrt {2 x^2-x+3}}dx+\frac {198018 \sqrt {2 x^2-x+3}}{(2 x+5)^3}\right )-\frac {2 (27754539-31190998 x)}{69 \sqrt {2 x^2-x+3}}}{927148032}+\frac {369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {-\frac {92}{3} \left (-\frac {1}{144} \int -\frac {9 (2276860 x+9970363)}{(2 x+5)^2 \sqrt {2 x^2-x+3}}dx+\frac {267411 \sqrt {2 x^2-x+3}}{8 (2 x+5)^2}+\frac {198018 \sqrt {2 x^2-x+3}}{(2 x+5)^3}\right )-\frac {2 (27754539-31190998 x)}{69 \sqrt {2 x^2-x+3}}}{927148032}+\frac {369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {92}{3} \left (\frac {1}{16} \int \frac {2276860 x+9970363}{(2 x+5)^2 \sqrt {2 x^2-x+3}}dx+\frac {267411 \sqrt {2 x^2-x+3}}{8 (2 x+5)^2}+\frac {198018 \sqrt {2 x^2-x+3}}{(2 x+5)^3}\right )-\frac {2 (27754539-31190998 x)}{69 \sqrt {2 x^2-x+3}}}{927148032}+\frac {369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {-\frac {92}{3} \left (\frac {1}{16} \left (\frac {14336367}{8} \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {475357 \sqrt {2 x^2-x+3}}{4 (2 x+5)}\right )+\frac {267411 \sqrt {2 x^2-x+3}}{8 (2 x+5)^2}+\frac {198018 \sqrt {2 x^2-x+3}}{(2 x+5)^3}\right )-\frac {2 (27754539-31190998 x)}{69 \sqrt {2 x^2-x+3}}}{927148032}+\frac {369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {-\frac {92}{3} \left (\frac {1}{16} \left (-\frac {14336367}{4} \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}-\frac {475357 \sqrt {2 x^2-x+3}}{4 (2 x+5)}\right )+\frac {267411 \sqrt {2 x^2-x+3}}{8 (2 x+5)^2}+\frac {198018 \sqrt {2 x^2-x+3}}{(2 x+5)^3}\right )-\frac {2 (27754539-31190998 x)}{69 \sqrt {2 x^2-x+3}}}{927148032}+\frac {369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {92}{3} \left (\frac {1}{16} \left (-\frac {4778789 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{16 \sqrt {2}}-\frac {475357 \sqrt {2 x^2-x+3}}{4 (2 x+5)}\right )+\frac {267411 \sqrt {2 x^2-x+3}}{8 (2 x+5)^2}+\frac {198018 \sqrt {2 x^2-x+3}}{(2 x+5)^3}\right )-\frac {2 (27754539-31190998 x)}{69 \sqrt {2 x^2-x+3}}}{927148032}+\frac {369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}\)

Input:

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

Output:

(369609 - 175877*x)/(463574016*(3 - x + 2*x^2)^(3/2)) + ((-2*(27754539 - 3 
1190998*x))/(69*Sqrt[3 - x + 2*x^2]) - (92*((198018*Sqrt[3 - x + 2*x^2])/( 
5 + 2*x)^3 + (267411*Sqrt[3 - x + 2*x^2])/(8*(5 + 2*x)^2) + ((-475357*Sqrt 
[3 - x + 2*x^2])/(4*(5 + 2*x)) - (4778789*ArcTanh[(17 - 22*x)/(12*Sqrt[2]* 
Sqrt[3 - x + 2*x^2])])/(16*Sqrt[2]))/16))/3)/927148032
 

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 = 0.14 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.52

method	result	size
risch	\(\frac {6664404208 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}-6702882569 x^{2}[phone] x -95241881529}{341190475776 \left (5+2 x \right )^{3} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {4778789 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{15479341056}\)	\(83\)
trager	\(\frac {6664404208 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}-6702882569 x^{2}[phone] x -95241881529}{341190475776 \left (5+2 x \right )^{3} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {4778789 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {22 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x -17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-24 \sqrt {2 x^{2}-x +3}}{5+2 x}\right )}{15479341056}\)	\(103\)
default	\(\frac {\frac {5 x}{138}-\frac {5}{552}}{\left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {\frac {40 x}{1587}-\frac {10}{1587}}{\sqrt {2 x^{2}-x +3}}-\frac {3667}{13824 \left (x +\frac {5}{2}\right )^{3} \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}+\frac {25951}{110592 \left (x +\frac {5}{2}\right )^{2} \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}-\frac {34861}{3981312 \left (x +\frac {5}{2}\right ) \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}-\frac {4778789}{429981696 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}-\frac {72646615 \left (4 x -1\right )}{9889579008 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}-\frac {8183108657 \left (4 x -1\right )}{1364761903104 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}-\frac {4778789}{2579890176 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}+\frac {4778789 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{15479341056}\)	\(207\)

Input:

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

Output:

1/341190475776*(6664404208*x^6+34872810880*x^5+46210466520*x^4+27484986184 
*x^3-6702882569*x^2+73621973154*x-95241881529)/(5+2*x)^3/(2*x^2-x+3)^(3/2) 
+4778789/15479341056*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 = 170, normalized size of antiderivative = 1.06 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {2527979381 \, \sqrt {2} {\left (32 \, x^{7} + 208 \, x^{6} + 464 \, x^{5} + 632 \, x^{4} + 1162 \, x^{3} + 1265 \, x^{2} + 600 \, x + 1125\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (6664404208 \, x^{6} + 34872810880 \, x^{5} + 46210466520 \, x^{4} + 27484986184 \, x^{3} - 6702882569 \, x^{2} + 73621973154 \, x - 95241881529\right )} \sqrt {2 \, x^{2} - x + 3}}{16377142837248 \, {\left (32 \, x^{7} + 208 \, x^{6} + 464 \, x^{5} + 632 \, x^{4} + 1162 \, x^{3} + 1265 \, x^{2} + 600 \, x + 1125\right )}} \] Input:

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

Output:

1/16377142837248*(2527979381*sqrt(2)*(32*x^7 + 208*x^6 + 464*x^5 + 632*x^4 
 + 1162*x^3 + 1265*x^2 + 600*x + 1125)*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*(666440 
4208*x^6 + 34872810880*x^5 + 46210466520*x^4 + 27484986184*x^3 - 670288256 
9*x^2 + 73621973154*x - 95241881529)*sqrt(2*x^2 - x + 3))/(32*x^7 + 208*x^ 
6 + 464*x^5 + 632*x^4 + 1162*x^3 + 1265*x^2 + 600*x + 1125)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.54 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}} \, dx=-\frac {4778789}{15479341056} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {416525263 \, x}{341190475776 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {245375387}{113730158592 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {16932905 \, x}{2472394752 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {3667}{1728 \, {\left (8 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 60 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 150 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 125 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {25951}{27648 \, {\left (4 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 20 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 25 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {34861}{1990656 \, {\left (2 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 5 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {10570421}{824131584 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:

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

Output:

-4778789/15479341056*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23 
*sqrt(23)/abs(2*x + 5)) + 416525263/341190475776*x/sqrt(2*x^2 - x + 3) - 2 
45375387/113730158592/sqrt(2*x^2 - x + 3) + 16932905/2472394752*x/(2*x^2 - 
 x + 3)^(3/2) - 3667/1728/(8*(2*x^2 - x + 3)^(3/2)*x^3 + 60*(2*x^2 - x + 3 
)^(3/2)*x^2 + 150*(2*x^2 - x + 3)^(3/2)*x + 125*(2*x^2 - x + 3)^(3/2)) + 2 
5951/27648/(4*(2*x^2 - x + 3)^(3/2)*x^2 + 20*(2*x^2 - x + 3)^(3/2)*x + 25* 
(2*x^2 - x + 3)^(3/2)) - 34861/1990656/(2*(2*x^2 - x + 3)^(3/2)*x + 5*(2*x 
^2 - x + 3)^(3/2)) - 10570421/824131584/(2*x^2 - x + 3)^(3/2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (130) = 260\).

Time = 0.20 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.74 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {4778789}{15479341056} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {4778789}{15479341056} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {{\left ({\left (15595499 \, x - 21675019\right )} x + 27298005\right )} x - 14440149}{7996651776 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {\sqrt {2} {\left (38030012 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} + 734231900 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} + 122834956 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} - 2154595396 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 1659431083 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 760577429\right )}}{3869835264 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{3}} \] Input:

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

Output:

4778789/15479341056*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 
- x + 3))) - 4778789/15479341056*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) 
 + 2*sqrt(2*x^2 - x + 3))) + 1/7996651776*(((15595499*x - 21675019)*x + 27 
298005)*x - 14440149)/(2*x^2 - x + 3)^(3/2) + 1/3869835264*sqrt(2)*(380300 
12*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 + 734231900*(sqrt(2)*x - sq 
rt(2*x^2 - x + 3))^4 + 122834956*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) 
^3 - 2154595396*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 1659431083*sqrt(2)*( 
sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 760577429)/(2*(sqrt(2)*x - sqrt(2*x^2 - 
 x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.90 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {159945700992 \sqrt {2 x^{2}-x +3}\, x^{6}[phone] \sqrt {2 x^{2}-x +3}\, x^{5}[phone] \sqrt {2 x^{2}-x +3}\, x^{4}[phone] \sqrt {2 x^{2}-x +3}\, x^{3}-160869181656 \sqrt {2 x^{2}-x +3}\, x^{2}[phone] \sqrt {2 x^{2}-x +3}\, x -2285805156696 \sqrt {2 x^{2}-x +3}[phone] \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{7}[phone] \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{6}[phone] \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{5}[phone] \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{4}[phone] \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{3}[phone] \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{2}[phone] \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x [phone] \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right )-80895340192 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{7}-525819711248 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{6}-1172982432784 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{5}-1597682968792 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{4}-2937512040722 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{3}-3197893916965 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{2}-1516787628600 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x -2843976803625 \sqrt {2}\, \mathrm {log}\left (2 x +5\right )}{262034285395968 x^{7}+1703222855073792 x^{6}+3799497138241536 x^{5}+5175177136570368 x^{4}+9515119988441088 x^{3}+10358542844559360 x^{2}+4913142851174400 x +9212142845952000} \] Input:

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

Output:

(159945700992*sqrt(2*x**2 - x + 3)*x**6 + 836947461120*sqrt(2*x**2 - x + 3 
)*x**5 + 1109051196480*sqrt(2*x**2 - x + 3)*x**4 + 659639668416*sqrt(2*x** 
2 - x + 3)*x**3 - 160869181656*sqrt(2*x**2 - x + 3)*x**2 + 1766927355696*s 
qrt(2*x**2 - x + 3)*x - 2285805156696*sqrt(2*x**2 - x + 3) + 80895340192*s 
qrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sqrt(2) + 22*x - 17)*x**7 + 52581971 
1248*sqrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sqrt(2) + 22*x - 17)*x**6 + 11 
72982432784*sqrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sqrt(2) + 22*x - 17)*x* 
*5 + 1597682968792*sqrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sqrt(2) + 22*x - 
 17)*x**4 + 2937512040722*sqrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sqrt(2) + 
 22*x - 17)*x**3 + 3197893916965*sqrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sq 
rt(2) + 22*x - 17)*x**2 + 1516787628600*sqrt(2)*log( - 12*sqrt(2*x**2 - x 
+ 3)*sqrt(2) + 22*x - 17)*x + 2843976803625*sqrt(2)*log( - 12*sqrt(2*x**2 
- x + 3)*sqrt(2) + 22*x - 17) - 80895340192*sqrt(2)*log(2*x + 5)*x**7 - 52 
5819711248*sqrt(2)*log(2*x + 5)*x**6 - 1172982432784*sqrt(2)*log(2*x + 5)* 
x**5 - 1597682968792*sqrt(2)*log(2*x + 5)*x**4 - 2937512040722*sqrt(2)*log 
(2*x + 5)*x**3 - 3197893916965*sqrt(2)*log(2*x + 5)*x**2 - 1516787628600*s 
qrt(2)*log(2*x + 5)*x - 2843976803625*sqrt(2)*log(2*x + 5))/(8188571418624 
*(32*x**7 + 208*x**6 + 464*x**5 + 632*x**4 + 1162*x**3 + 1265*x**2 + 600*x 
 + 1125))