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

Optimal. Leaf size=60 \[ \frac {(x-3) (4 x-1) \left (-\frac {13}{340} \log \left (x^2+1\right )+\frac {9}{110} \log (x-3)-\frac {1}{187} \log (4 x-1)+\frac {1}{170} \tan ^{-1}(x)\right )}{\sqrt [5]{\left (4 x^2-13 x+3\right )^5}} \]

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Rubi [B]  time = 0.95, antiderivative size = 141, normalized size of antiderivative = 2.35, number of steps used = 9, number of rules used = 8, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.123, Rules used = {6688, 6720, 1075, 632, 31, 635, 203, 260} \begin {gather*} -\frac {\left (4 x^2-13 x+3\right ) \log (1-4 x)}{187 \sqrt [5]{\left (4 x^2-13 x+3\right )^5}}+\frac {9 \left (4 x^2-13 x+3\right ) \log (3-x)}{110 \sqrt [5]{\left (4 x^2-13 x+3\right )^5}}-\frac {13 \left (4 x^2-13 x+3\right ) \log \left (x^2+1\right )}{340 \sqrt [5]{\left (4 x^2-13 x+3\right )^5}}+\frac {\left (4 x^2-13 x+3\right ) \tan ^{-1}(x)}{170 \sqrt [5]{\left (4 x^2-13 x+3\right )^5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[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]

[Out]

((3 - 13*x + 4*x^2)*ArcTan[x])/(170*((3 - 13*x + 4*x^2)^5)^(1/5)) - ((3 - 13*x + 4*x^2)*Log[1 - 4*x])/(187*((3
- 13*x + 4*x^2)^5)^(1/5)) + (9*(3 - 13*x + 4*x^2)*Log[3 - x])/(110*((3 - 13*x + 4*x^2)^5)^(1/5)) - (13*(3 - 1
3*x + 4*x^2)*Log[1 + x^2])/(340*((3 - 13*x + 4*x^2)^5)^(1/5))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 203

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 1075

Int[((A_.) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)), x_Symbol] :> With[{q =
c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, Dist[1/q, Int[(A*c^2*d - a*c*C*d + A*b^2*f - a*A*c*f + a^2*C*f + c*(
-(b*C*d) + A*b*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*C*d^2 - A*c*d*f - a*C*d*f + a*A*f^2 - f*(-(b
*C*d) + A*b*f)*x)/(d + f*x^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 6688

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

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(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]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rubi steps

\begin {align*} \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 (1+x^2\right ) \sqrt [5]{\left (3-13 x+4 x^2\right )^5}} \, dx\\ &=\frac {\left (3-13 x+4 x^2\right ) \int \frac {x^2}{\left (1+x^2\right ) \left (3-13 x+4 x^2\right )} \, dx}{\sqrt [5]{\left (3-13 x+4 x^2\right )^5}}\\ &=\frac {\left (3-13 x+4 x^2\right ) \int \frac {1-13 x}{1+x^2} \, dx}{170 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}+\frac {\left (3-13 x+4 x^2\right ) \int \frac {-3+52 x}{3-13 x+4 x^2} \, dx}{170 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}\\ &=\frac {\left (3-13 x+4 x^2\right ) \int \frac {1}{1+x^2} \, dx}{170 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}-\frac {\left (4 \left (3-13 x+4 x^2\right )\right ) \int \frac {1}{-1+4 x} \, dx}{187 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}-\frac {\left (13 \left (3-13 x+4 x^2\right )\right ) \int \frac {x}{1+x^2} \, dx}{170 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}+\frac {\left (18 \left (3-13 x+4 x^2\right )\right ) \int \frac {1}{-12+4 x} \, dx}{55 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}\\ &=\frac {\left (3-13 x+4 x^2\right ) \tan ^{-1}(x)}{170 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}-\frac {\left (3-13 x+4 x^2\right ) \log (1-4 x)}{187 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}+\frac {9 \left (3-13 x+4 x^2\right ) \log (3-x)}{110 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}-\frac {13 \left (3-13 x+4 x^2\right ) \log \left (1+x^2\right )}{340 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 59, normalized size = 0.98 \begin {gather*} \frac {\left (4 x^2-13 x+3\right ) \left (-143 \log \left (x^2+1\right )-20 \log (1-4 x)+306 \log (3-x)+22 \tan ^{-1}(x)\right )}{3740 \sqrt [5]{\left (4 x^2-13 x+3\right )^5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/((1 + x^2)*(243 - 5265*x + 47250*x^2 - 225810*x^3 + 615255*x^4 - 954733*x^5 + 820340*x^6 - 40144
0*x^7 + 112000*x^8 - 16640*x^9 + 1024*x^10)^(1/5)),x]

[Out]

((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))

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IntegrateAlgebraic [A]  time = 19.27, size = 60, normalized size = 1.00 \begin {gather*} \frac {(-3+x) (-1+4 x) \left (\frac {1}{170} \tan ^{-1}(x)+\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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]

[Out]

((-3 + x)*(-1 + 4*x)*(ArcTan[x]/170 + (9*Log[-3 + x])/110 - Log[-1 + 4*x]/187 - (13*Log[1 + x^2])/340))/((3 -
13*x + 4*x^2)^5)^(1/5)

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fricas [A]  time = 0.46, size = 27, normalized size = 0.45 \begin {gather*} \frac {1}{170} \, \arctan \relax (x) - \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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+4
7250*x^2-5265*x+243)^(1/5),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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+4
7250*x^2-5265*x+243)^(1/5),x, algorithm="giac")

[Out]

integrate(x^2/((1024*x^10 - 16640*x^9 + 112000*x^8 - 401440*x^7 + 820340*x^6 - 954733*x^5 + 615255*x^4 - 22581
0*x^3 + 47250*x^2 - 5265*x + 243)^(1/5)*(x^2 + 1)), x)

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maple [C]  time = 0.19, size = 130, normalized size = 2.17

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(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,method=_RETURNVERBOSE)

[Out]

(-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)-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)

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maxima [A]  time = 0.41, size = 27, normalized size = 0.45 \begin {gather*} \frac {1}{170} \, \arctan \relax (x) - \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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+4
7250*x^2-5265*x+243)^(1/5),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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 + 1120
00*x^8 - 16640*x^9 + 1024*x^10 + 243)^(1/5)),x)

[Out]

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 + 1120
00*x^8 - 16640*x^9 + 1024*x^10 + 243)^(1/5)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt [5]{\left (x - 3\right )^{5} \left (4 x - 1\right )^{5}} \left (x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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**4-225
810*x**3+47250*x**2-5265*x+243)**(1/5),x)

[Out]

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

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