3.8.92 \(\int \frac {x^2}{(1+x^2) \sqrt [5]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx\) [792]

3.8.92.1 Optimal result

Integrand size = 65, antiderivative size = 60 \[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [5]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\frac {(-3+x) (-1+4 x) \left (\frac {\arctan (x)}{170}+\frac {9}{110} \log (-3+x)-\frac {1}{187} \log (-1+4 x)-\frac {13}{340} \log \left (1+x^2\right )\right )}{\sqrt [5]{\left (3-13 x+4 x^2\right )^5}} \]

output

(-3+x)*(-1+4*x)*(1/170*arctan(x)+9/110*ln(-3+x)-1/187*ln(-1+4*x)-13/340*ln 
(x^2+1))/((4*x^2-13*x+3)^5)^(1/5)
 

3.8.92.2 Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98 \[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [5]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\frac {\left (3-13 x+4 x^2\right ) \left (22 \arctan (x)-20 \log (1-4 x)+306 \log (3-x)-143 \log \left (1+x^2\right )\right )}{3740 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}} \]

input

Integrate[x^2/((1 + x^2)*(243 - 5265*x + 47250*x^2 - 225810*x^3 + 615255*x 
^4 - 954733*x^5 + 820340*x^6 - 401440*x^7 + 112000*x^8 - 16640*x^9 + 1024* 
x^10)^(1/5)),x]
 

output

((3 - 13*x + 4*x^2)*(22*ArcTan[x] - 20*Log[1 - 4*x] + 306*Log[3 - x] - 143 
*Log[1 + x^2]))/(3740*((3 - 13*x + 4*x^2)^5)^(1/5))
 

3.8.92.3 Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {7239, 7271, 2142, 25, 452, 216, 240, 1141, 2009}

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.

\(\displaystyle \int \frac {x^2}{\left (x^2+1\right ) \sqrt [5]{1024 x^{10}-16640 x^9+112000 x^8-401440 x^7+820340 x^6-954733 x^5+615255 x^4-225810 x^3+47250 x^2-5265 x+243}} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^2}{\left (x^2+1\right ) \sqrt [5]{\left (4 x^2-13 x+3\right )^5}}dx\)

\(\Big \downarrow \) 7271

\(\displaystyle \frac {\left (4 x^2-13 x+3\right ) \int \frac {x^2}{\left (x^2+1\right ) \left (4 x^2-13 x+3\right )}dx}{\sqrt [5]{\left (4 x^2-13 x+3\right )^5}}\)

\(\Big \downarrow \) 2142

\(\displaystyle \frac {\left (4 x^2-13 x+3\right ) \left (\frac {1}{170} \int \frac {1-13 x}{x^2+1}dx+\frac {1}{170} \int -\frac {3-52 x}{4 x^2-13 x+3}dx\right )}{\sqrt [5]{\left (4 x^2-13 x+3\right )^5}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (4 x^2-13 x+3\right ) \left (\frac {1}{170} \int \frac {1-13 x}{x^2+1}dx-\frac {1}{170} \int \frac {3-52 x}{4 x^2-13 x+3}dx\right )}{\sqrt [5]{\left (4 x^2-13 x+3\right )^5}}\)

\(\Big \downarrow \) 452

\(\displaystyle \frac {\left (4 x^2-13 x+3\right ) \left (\frac {1}{170} \left (\int \frac {1}{x^2+1}dx-13 \int \frac {x}{x^2+1}dx\right )-\frac {1}{170} \int \frac {3-52 x}{4 x^2-13 x+3}dx\right )}{\sqrt [5]{\left (4 x^2-13 x+3\right )^5}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\left (4 x^2-13 x+3\right ) \left (\frac {1}{170} \left (\arctan (x)-13 \int \frac {x}{x^2+1}dx\right )-\frac {1}{170} \int \frac {3-52 x}{4 x^2-13 x+3}dx\right )}{\sqrt [5]{\left (4 x^2-13 x+3\right )^5}}\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {\left (4 x^2-13 x+3\right ) \left (\frac {1}{170} \left (\arctan (x)-\frac {13}{2} \log \left (x^2+1\right )\right )-\frac {1}{170} \int \frac {3-52 x}{4 x^2-13 x+3}dx\right )}{\sqrt [5]{\left (4 x^2-13 x+3\right )^5}}\)

\(\Big \downarrow \) 1141

\(\displaystyle \frac {\left (4 x^2-13 x+3\right ) \left (\frac {1}{170} \left (\arctan (x)-\frac {13}{2} \log \left (x^2+1\right )\right )-\frac {2}{85} \int \left (\frac {153}{44 (3-x)}-\frac {10}{11 (1-4 x)}\right )dx\right )}{\sqrt [5]{\left (4 x^2-13 x+3\right )^5}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (4 x^2-13 x+3\right ) \left (\frac {1}{170} \left (\arctan (x)-\frac {13}{2} \log \left (x^2+1\right )\right )-\frac {2}{85} \left (\frac {5}{22} \log (1-4 x)-\frac {153}{44} \log (3-x)\right )\right )}{\sqrt [5]{\left (4 x^2-13 x+3\right )^5}}\)

input

Int[x^2/((1 + x^2)*(243 - 5265*x + 47250*x^2 - 225810*x^3 + 615255*x^4 - 9 
54733*x^5 + 820340*x^6 - 401440*x^7 + 112000*x^8 - 16640*x^9 + 1024*x^10)^ 
(1/5)),x]
 

output

((3 - 13*x + 4*x^2)*((-2*((5*Log[1 - 4*x])/22 - (153*Log[3 - x])/44))/85 + 
 (ArcTan[x] - (13*Log[1 + x^2])/2)/170))/((3 - 13*x + 4*x^2)^5)^(1/5)
 

3.8.92.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 216

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

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 452

Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c   Int[1/ 
(a + b*x^2), x], x] + Simp[d   Int[x/(a + b*x^2), x], x] /; FreeQ[{a, b, c, 
 d}, x] && NeQ[b*c^2 + a*d^2, 0]
 

rule 1141

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ 
Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p   Int[ExpandIntegrand[ 
(d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 
1] ||  !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 
0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2142

Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)), x_Sym 
bol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2] 
, q = c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, Simp[1/q   Int[(A*c^2*d - a 
*c*C*d + A*b^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d + A*b*f - 
 a*B*f)*x)/(a + b*x + c*x^2), x], x] + Simp[1/q   Int[(c*C*d^2 + b*B*d*f - 
A*c*d*f - a*C*d*f + a*A*f^2 - f*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(d + f*x 
^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, f}, x] && PolyQ[Px, x, 2]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7271

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ 
FracPart[p]/v^(m*FracPart[p]))   Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, 
x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&  !(Eq 
Q[v, x] && EqQ[m, 1])
 

3.8.92.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.17

method	result	size
risch	\(-\frac {\left (4 x^{2}-13 x +3\right ) \ln \left (-1+4 x \right )}{187 {\left (\left (4 x^{2}-13 x +3\right )^{5}\right )}^{\frac {1}{5}}}+\frac {9 \left (4 x^{2}-13 x +3\right ) \ln \left (-3+x \right )}{110 {\left (\left (4 x^{2}-13 x +3\right )^{5}\right )}^{\frac {1}{5}}}+\frac {\left (-\frac {13}{340}+\frac {i}{340}\right ) \left (4 x^{2}-13 x +3\right ) \ln \left (i+x \right )}{{\left (\left (4 x^{2}-13 x +3\right )^{5}\right )}^{\frac {1}{5}}}+\frac {\left (-\frac {13}{340}-\frac {i}{340}\right ) \left (4 x^{2}-13 x +3\right ) \ln \left (-i+x \right )}{{\left (\left (4 x^{2}-13 x +3\right )^{5}\right )}^{\frac {1}{5}}}\)	\(130\)

input

int(x^2/(x^2+1)/(1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^6-9547 
33*x^5+615255*x^4-225810*x^3+47250*x^2-5265*x+243)^(1/5),x,method=_RETURNV 
ERBOSE)
 

output

-1/187/((4*x^2-13*x+3)^5)^(1/5)*(4*x^2-13*x+3)*ln(-1+4*x)+9/110/((4*x^2-13 
*x+3)^5)^(1/5)*(4*x^2-13*x+3)*ln(-3+x)+(-13/340+1/340*I)/((4*x^2-13*x+3)^5 
)^(1/5)*(4*x^2-13*x+3)*ln(I+x)-(13/340+1/340*I)/((4*x^2-13*x+3)^5)^(1/5)*( 
4*x^2-13*x+3)*ln(-I+x)
 

3.8.92.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.45 \[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [5]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\frac {1}{170} \, \arctan \left (x\right ) - \frac {13}{340} \, \log \left (x^{2} + 1\right ) - \frac {1}{187} \, \log \left (4 \, x - 1\right ) + \frac {9}{110} \, \log \left (x - 3\right ) \]

input

integrate(x^2/(x^2+1)/(1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^ 
6-954733*x^5+615255*x^4-225810*x^3+47250*x^2-5265*x+243)^(1/5),x, algorith 
m="fricas")
 

output

1/170*arctan(x) - 13/340*log(x^2 + 1) - 1/187*log(4*x - 1) + 9/110*log(x - 
 3)
 

3.8.92.6 Sympy [F]

\[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [5]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\int \frac {x^{2}}{\sqrt [5]{\left (x - 3\right )^{5} \left (4 x - 1\right )^{5}} \left (x^{2} + 1\right )}\, dx \]

input

integrate(x**2/(x**2+1)/(1024*x**10-16640*x**9+112000*x**8-401440*x**7+820 
340*x**6-954733*x**5+615255*x**4-225810*x**3+47250*x**2-5265*x+243)**(1/5) 
,x)
 

output

Integral(x**2/(((x - 3)**5*(4*x - 1)**5)**(1/5)*(x**2 + 1)), x)
 

3.8.92.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.45 \[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [5]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\frac {1}{170} \, \arctan \left (x\right ) - \frac {13}{340} \, \log \left (x^{2} + 1\right ) - \frac {1}{187} \, \log \left (4 \, x - 1\right ) + \frac {9}{110} \, \log \left (x - 3\right ) \]

input

integrate(x^2/(x^2+1)/(1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^ 
6-954733*x^5+615255*x^4-225810*x^3+47250*x^2-5265*x+243)^(1/5),x, algorith 
m="maxima")
 

output

1/170*arctan(x) - 13/340*log(x^2 + 1) - 1/187*log(4*x - 1) + 9/110*log(x - 
 3)
 

3.8.92.8 Giac [F]

\[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [5]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\int { \frac {x^{2}}{{\left (1024 \, x^{10} - 16640 \, x^{9} + 112000 \, x^{8} - 401440 \, x^{7} + 820340 \, x^{6} - 954733 \, x^{5} + 615255 \, x^{4} - 225810 \, x^{3} + 47250 \, x^{2} - 5265 \, x + 243\right )}^{\frac {1}{5}} {\left (x^{2} + 1\right )}} \,d x } \]

input

integrate(x^2/(x^2+1)/(1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^ 
6-954733*x^5+615255*x^4-225810*x^3+47250*x^2-5265*x+243)^(1/5),x, algorith 
m="giac")
 

output

integrate(x^2/((1024*x^10 - 16640*x^9 + 112000*x^8 - 401440*x^7 + 820340*x 
^6 - 954733*x^5 + 615255*x^4 - 225810*x^3 + 47250*x^2 - 5265*x + 243)^(1/5 
)*(x^2 + 1)), x)
 

3.8.92.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [5]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\int \frac {x^2}{\left (x^2+1\right )\,{\left (1024\,x^{10}-16640\,x^9+112000\,x^8-401440\,x^7+820340\,x^6-954733\,x^5+615255\,x^4-225810\,x^3+47250\,x^2-5265\,x+243\right )}^{1/5}} \,d x \]

input

int(x^2/((x^2 + 1)*(47250*x^2 - 5265*x - 225810*x^3 + 615255*x^4 - 954733* 
x^5 + 820340*x^6 - 401440*x^7 + 112000*x^8 - 16640*x^9 + 1024*x^10 + 243)^ 
(1/5)),x)
 

output

int(x^2/((x^2 + 1)*(47250*x^2 - 5265*x - 225810*x^3 + 615255*x^4 - 954733* 
x^5 + 820340*x^6 - 401440*x^7 + 112000*x^8 - 16640*x^9 + 1024*x^10 + 243)^ 
(1/5)), x)