3.4.49 \(\int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \sqrt {3-x+2 x^2}} \, dx\) [349]

3.4.49.1 Optimal result
3.4.49.2 Mathematica [A] (verified)
3.4.49.3 Rubi [A] (verified)
3.4.49.4 Maple [F(-1)]
3.4.49.5 Fricas [A] (verification not implemented)
3.4.49.6 Sympy [F]
3.4.49.7 Maxima [A] (verification not implemented)
3.4.49.8 Giac [B] (verification not implemented)
3.4.49.9 Mupad [F(-1)]

3.4.49.1 Optimal result

Integrand size = 40, antiderivative size = 135 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \sqrt {3-x+2 x^2}} \, dx=-\frac {3667 \sqrt {3-x+2 x^2}}{1728 (5+2 x)^3}+\frac {394907 \sqrt {3-x+2 x^2}}{248832 (5+2 x)^2}-\frac {3163415 \sqrt {3-x+2 x^2}}{5971968 (5+2 x)}-\frac {5 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{16 \sqrt {2}}+\frac {22389491 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{71663616 \sqrt {2}} \]

output
-5/32*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)+22389491/143327232*arctanh(1/ 
24*(17-22*x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)-3667/1728*(2*x^2-x+3)^(1/2 
)/(5+2*x)^3+394907/248832*(2*x^2-x+3)^(1/2)/(5+2*x)^2-3163415/5971968*(2*x 
^2-x+3)^(1/2)/(5+2*x)
 
3.4.49.2 Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.74 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \sqrt {3-x+2 x^2}} \, dx=\frac {-\frac {12 \sqrt {3-x+2 x^2} \left (44369687+44312764 x+12653660 x^2\right )}{(5+2 x)^3}-22389491 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )-11197440 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{71663616} \]

input
Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)^4*Sqrt[3 - x + 2*x^2]), 
x]
 
output
((-12*Sqrt[3 - x + 2*x^2]*(44369687 + 44312764*x + 12653660*x^2))/(5 + 2*x 
)^3 - 22389491*Sqrt[2]*ArcTanh[(5 + 2*x - Sqrt[6 - 2*x + 4*x^2])/6] - 1119 
7440*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/71663616
 
3.4.49.3 Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2181, 27, 2181, 2181, 27, 1269, 1090, 222, 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.

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

\(\Big \downarrow \) 2181

\(\displaystyle -\frac {1}{216} \int \frac {-8640 x^3+23328 x^2-34168 x+28687}{16 (2 x+5)^3 \sqrt {2 x^2-x+3}}dx-\frac {3667 \sqrt {2 x^2-x+3}}{1728 (2 x+5)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {-8640 x^3+23328 x^2-34168 x+28687}{(2 x+5)^3 \sqrt {2 x^2-x+3}}dx}{3456}-\frac {3667 \sqrt {2 x^2-x+3}}{1728 (2 x+5)^3}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {\frac {1}{144} \int \frac {622080 x^2-1655188 x+1464275}{(2 x+5)^2 \sqrt {2 x^2-x+3}}dx+\frac {394907 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}}{3456}-\frac {3667 \sqrt {2 x^2-x+3}}{1728 (2 x+5)^3}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {\frac {1}{144} \left (-\frac {1}{72} \int \frac {3 (3727091-7464960 x)}{(2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {3163415 \sqrt {2 x^2-x+3}}{12 (2 x+5)}\right )+\frac {394907 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}}{3456}-\frac {3667 \sqrt {2 x^2-x+3}}{1728 (2 x+5)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{144} \left (-\frac {1}{24} \int \frac {3727091-7464960 x}{(2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {3163415 \sqrt {2 x^2-x+3}}{12 (2 x+5)}\right )+\frac {394907 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}}{3456}-\frac {3667 \sqrt {2 x^2-x+3}}{1728 (2 x+5)^3}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{24} \left (3732480 \int \frac {1}{\sqrt {2 x^2-x+3}}dx-22389491 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx\right )-\frac {3163415 \sqrt {2 x^2-x+3}}{12 (2 x+5)}\right )+\frac {394907 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}}{3456}-\frac {3667 \sqrt {2 x^2-x+3}}{1728 (2 x+5)^3}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{24} \left (1866240 \sqrt {\frac {2}{23}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-22389491 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx\right )-\frac {3163415 \sqrt {2 x^2-x+3}}{12 (2 x+5)}\right )+\frac {394907 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}}{3456}-\frac {3667 \sqrt {2 x^2-x+3}}{1728 (2 x+5)^3}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{24} \left (1866240 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )-22389491 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx\right )-\frac {3163415 \sqrt {2 x^2-x+3}}{12 (2 x+5)}\right )+\frac {394907 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}}{3456}-\frac {3667 \sqrt {2 x^2-x+3}}{1728 (2 x+5)^3}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{24} \left (44778982 \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}+1866240 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )-\frac {3163415 \sqrt {2 x^2-x+3}}{12 (2 x+5)}\right )+\frac {394907 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}}{3456}-\frac {3667 \sqrt {2 x^2-x+3}}{1728 (2 x+5)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{144} \left (\frac {1}{24} \left (1866240 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )+\frac {22389491 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{6 \sqrt {2}}\right )-\frac {3163415 \sqrt {2 x^2-x+3}}{12 (2 x+5)}\right )+\frac {394907 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}}{3456}-\frac {3667 \sqrt {2 x^2-x+3}}{1728 (2 x+5)^3}\)

input
Int[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)^4*Sqrt[3 - x + 2*x^2]),x]
 
output
(-3667*Sqrt[3 - x + 2*x^2])/(1728*(5 + 2*x)^3) + ((394907*Sqrt[3 - x + 2*x 
^2])/(72*(5 + 2*x)^2) + ((-3163415*Sqrt[3 - x + 2*x^2])/(12*(5 + 2*x)) + ( 
1866240*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]] + (22389491*ArcTanh[(17 - 22* 
x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(6*Sqrt[2]))/24)/144)/3456
 

3.4.49.3.1 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 222
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 1090
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* 
(c/(b^2 - 4*a*c)))^p)   Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, 
b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 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 1269
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[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]
 
3.4.49.4 Maple [F(-1)]

Timed out.

hanged

input
int((5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4/(2*x^2-x+3)^(1/2),x)
 
output
int((5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4/(2*x^2-x+3)^(1/2),x)
 
3.4.49.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.21 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \sqrt {3-x+2 x^2}} \, dx=\frac {22394880 \, \sqrt {2} {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 22389491 \, \sqrt {2} {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) - 48 \, {\left (12653660 \, x^{2} + 44312764 \, x + 44369687\right )} \sqrt {2 \, x^{2} - x + 3}}{286654464 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} \]

input
integrate((5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4/(2*x^2-x+3)^(1/2),x, algorithm=" 
fricas")
 
output
1/286654464*(22394880*sqrt(2)*(8*x^3 + 60*x^2 + 150*x + 125)*log(-4*sqrt(2 
)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 22389491*sqrt(2)*( 
8*x^3 + 60*x^2 + 150*x + 125)*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*(12653660*x^2 + 
44312764*x + 44369687)*sqrt(2*x^2 - x + 3))/(8*x^3 + 60*x^2 + 150*x + 125)
 
3.4.49.6 Sympy [F]

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

input
integrate((5*x**4-x**3+3*x**2+x+2)/(5+2*x)**4/(2*x**2-x+3)**(1/2),x)
 
output
Integral((5*x**4 - x**3 + 3*x**2 + x + 2)/((2*x + 5)**4*sqrt(2*x**2 - x + 
3)), x)
 
3.4.49.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.97 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \sqrt {3-x+2 x^2}} \, dx=\frac {5}{32} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) - \frac {22389491}{143327232} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {3667 \, \sqrt {2 \, x^{2} - x + 3}}{1728 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} + \frac {394907 \, \sqrt {2 \, x^{2} - x + 3}}{248832 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac {3163415 \, \sqrt {2 \, x^{2} - x + 3}}{5971968 \, {\left (2 \, x + 5\right )}} \]

input
integrate((5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4/(2*x^2-x+3)^(1/2),x, algorithm=" 
maxima")
 
output
5/32*sqrt(2)*arcsinh(4/23*sqrt(23)*x - 1/23*sqrt(23)) - 22389491/143327232 
*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 
5)) - 3667/1728*sqrt(2*x^2 - x + 3)/(8*x^3 + 60*x^2 + 150*x + 125) + 39490 
7/248832*sqrt(2*x^2 - x + 3)/(4*x^2 + 20*x + 25) - 3163415/5971968*sqrt(2* 
x^2 - x + 3)/(2*x + 5)
 
3.4.49.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (108) = 216\).

Time = 0.28 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.11 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \sqrt {3-x+2 x^2}} \, dx=-\frac {5}{32} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {22389491}{143327232} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {22389491}{143327232} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {\sqrt {2} {\left (215012404 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} + 3010410772 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} + 2740802468 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} - 21459328844 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 14434519361 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 5957650879\right )}}{11943936 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{3}} \]

input
integrate((5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4/(2*x^2-x+3)^(1/2),x, algorithm=" 
giac")
 
output
-5/32*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 2238 
9491/143327232*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 
 3))) - 22389491/143327232*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*s 
qrt(2*x^2 - x + 3))) - 1/11943936*sqrt(2)*(215012404*sqrt(2)*(sqrt(2)*x - 
sqrt(2*x^2 - x + 3))^5 + 3010410772*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^4 + 
2740802468*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 - 21459328844*(sqrt 
(2)*x - sqrt(2*x^2 - x + 3))^2 + 14434519361*sqrt(2)*(sqrt(2)*x - sqrt(2*x 
^2 - x + 3)) - 5957650879)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqr 
t(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^3
 
3.4.49.9 Mupad [F(-1)]

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

input
int((x + 3*x^2 - x^3 + 5*x^4 + 2)/((2*x + 5)^4*(2*x^2 - x + 3)^(1/2)),x)
 
output
int((x + 3*x^2 - x^3 + 5*x^4 + 2)/((2*x + 5)^4*(2*x^2 - x + 3)^(1/2)), x)