3.15.27 \(\int \frac {\sqrt [4]{-x^3+x^4} (-1+x^8)}{x^4} \, dx\)

Optimal. Leaf size=101 \[ \frac {1463 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )}{32768}-\frac {1463 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )}{32768}+\frac {\sqrt [4]{x^4-x^3} \left (122880 x^8-6144 x^7-7296 x^6-9120 x^5-12540 x^4-21945 x^3-262144 x^2-65536 x+327680\right )}{737280 x^3} \]

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Rubi [B]  time = 0.48, antiderivative size = 242, normalized size of antiderivative = 2.40, number of steps used = 16, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2052, 2016, 2014, 2021, 2024, 2032, 63, 240, 212, 206, 203} \begin {gather*} -\frac {1}{120} \sqrt [4]{x^4-x^3} x^4-\frac {19 \sqrt [4]{x^4-x^3} x^3}{1920}-\frac {209 \sqrt [4]{x^4-x^3} x}{12288}-\frac {1463 \sqrt [4]{x^4-x^3}}{49152}-\frac {1463 (x-1)^{3/4} x^{9/4} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{32768 \left (x^4-x^3\right )^{3/4}}-\frac {1463 (x-1)^{3/4} x^{9/4} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{32768 \left (x^4-x^3\right )^{3/4}}-\frac {4 \left (x^4-x^3\right )^{5/4}}{9 x^6}+\frac {1}{6} \sqrt [4]{x^4-x^3} x^5-\frac {16 \left (x^4-x^3\right )^{5/4}}{45 x^5}-\frac {19 \sqrt [4]{x^4-x^3} x^2}{1536} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-x^3 + x^4)^(1/4)*(-1 + x^8))/x^4,x]

[Out]

(-1463*(-x^3 + x^4)^(1/4))/49152 - (209*x*(-x^3 + x^4)^(1/4))/12288 - (19*x^2*(-x^3 + x^4)^(1/4))/1536 - (19*x
^3*(-x^3 + x^4)^(1/4))/1920 - (x^4*(-x^3 + x^4)^(1/4))/120 + (x^5*(-x^3 + x^4)^(1/4))/6 - (4*(-x^3 + x^4)^(5/4
))/(9*x^6) - (16*(-x^3 + x^4)^(5/4))/(45*x^5) - (1463*(-1 + x)^(3/4)*x^(9/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(
32768*(-x^3 + x^4)^(3/4)) - (1463*(-1 + x)^(3/4)*x^(9/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/(32768*(-x^3 + x^4)^
(3/4))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, 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 206

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

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rule 2021

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b
*x^n)^p)/(c*(m + n*p + 1)), x] + Dist[(a*(n - j)*p)/(c^j*(m + n*p + 1)), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2024

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n +
1)*(a*x^j + b*x^n)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^(n - j)*(m + j*p - n + j + 1))/(b*(m + n*p + 1)
), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j
, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[m]*(a*x^j + b*x^n)^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rule 2052

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(c*x)
^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !In
tegerQ[p] && NeQ[n, j]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx &=\int \left (-\frac {\sqrt [4]{-x^3+x^4}}{x^4}+x^4 \sqrt [4]{-x^3+x^4}\right ) \, dx\\ &=-\int \frac {\sqrt [4]{-x^3+x^4}}{x^4} \, dx+\int x^4 \sqrt [4]{-x^3+x^4} \, dx\\ &=\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {1}{24} \int \frac {x^7}{\left (-x^3+x^4\right )^{3/4}} \, dx-\frac {4}{9} \int \frac {\sqrt [4]{-x^3+x^4}}{x^3} \, dx\\ &=-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {19}{480} \int \frac {x^6}{\left (-x^3+x^4\right )^{3/4}} \, dx\\ &=-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {19}{512} \int \frac {x^5}{\left (-x^3+x^4\right )^{3/4}} \, dx\\ &=-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {209 \int \frac {x^4}{\left (-x^3+x^4\right )^{3/4}} \, dx}{6144}\\ &=-\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {1463 \int \frac {x^3}{\left (-x^3+x^4\right )^{3/4}} \, dx}{49152}\\ &=-\frac {1463 \sqrt [4]{-x^3+x^4}}{49152}-\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {1463 \int \frac {x^2}{\left (-x^3+x^4\right )^{3/4}} \, dx}{65536}\\ &=-\frac {1463 \sqrt [4]{-x^3+x^4}}{49152}-\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {\left (1463 (-1+x)^{3/4} x^{9/4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{65536 \left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {1463 \sqrt [4]{-x^3+x^4}}{49152}-\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {\left (1463 (-1+x)^{3/4} x^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{16384 \left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {1463 \sqrt [4]{-x^3+x^4}}{49152}-\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {\left (1463 (-1+x)^{3/4} x^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{16384 \left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {1463 \sqrt [4]{-x^3+x^4}}{49152}-\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {\left (1463 (-1+x)^{3/4} x^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32768 \left (-x^3+x^4\right )^{3/4}}-\frac {\left (1463 (-1+x)^{3/4} x^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32768 \left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {1463 \sqrt [4]{-x^3+x^4}}{49152}-\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {1463 (-1+x)^{3/4} x^{9/4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32768 \left (-x^3+x^4\right )^{3/4}}-\frac {1463 (-1+x)^{3/4} x^{9/4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32768 \left (-x^3+x^4\right )^{3/4}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 47, normalized size = 0.47 \begin {gather*} \frac {4 (x-1)^2 \left (9 x^{9/4} \, _2F_1\left (-\frac {19}{4},\frac {5}{4};\frac {9}{4};1-x\right )-4 x-5\right )}{45 \left ((x-1) x^3\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-x^3 + x^4)^(1/4)*(-1 + x^8))/x^4,x]

[Out]

(4*(-1 + x)^2*(-5 - 4*x + 9*x^(9/4)*Hypergeometric2F1[-19/4, 5/4, 9/4, 1 - x]))/(45*((-1 + x)*x^3)^(3/4))

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IntegrateAlgebraic [A]  time = 0.58, size = 101, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-x^3+x^4} \left (327680-65536 x-262144 x^2-21945 x^3-12540 x^4-9120 x^5-7296 x^6-6144 x^7+122880 x^8\right )}{737280 x^3}+\frac {1463 \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{32768}-\frac {1463 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{32768} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-x^3 + x^4)^(1/4)*(-1 + x^8))/x^4,x]

[Out]

((-x^3 + x^4)^(1/4)*(327680 - 65536*x - 262144*x^2 - 21945*x^3 - 12540*x^4 - 9120*x^5 - 7296*x^6 - 6144*x^7 +
122880*x^8))/(737280*x^3) + (1463*ArcTan[x/(-x^3 + x^4)^(1/4)])/32768 - (1463*ArcTanh[x/(-x^3 + x^4)^(1/4)])/3
2768

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fricas [A]  time = 0.45, size = 129, normalized size = 1.28 \begin {gather*} -\frac {131670 \, x^{3} \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 65835 \, x^{3} \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 65835 \, x^{3} \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (122880 \, x^{8} - 6144 \, x^{7} - 7296 \, x^{6} - 9120 \, x^{5} - 12540 \, x^{4} - 21945 \, x^{3} - 262144 \, x^{2} - 65536 \, x + 327680\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{2949120 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^3)^(1/4)*(x^8-1)/x^4,x, algorithm="fricas")

[Out]

-1/2949120*(131670*x^3*arctan((x^4 - x^3)^(1/4)/x) + 65835*x^3*log((x + (x^4 - x^3)^(1/4))/x) - 65835*x^3*log(
-(x - (x^4 - x^3)^(1/4))/x) - 4*(122880*x^8 - 6144*x^7 - 7296*x^6 - 9120*x^5 - 12540*x^4 - 21945*x^3 - 262144*
x^2 - 65536*x + 327680)*(x^4 - x^3)^(1/4))/x^3

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giac [A]  time = 0.27, size = 171, normalized size = 1.69 \begin {gather*} \frac {1}{245760} \, {\left (7315 \, {\left (\frac {1}{x} - 1\right )}^{5} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 40755 \, {\left (\frac {1}{x} - 1\right )}^{4} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 92910 \, {\left (\frac {1}{x} - 1\right )}^{3} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 109782 \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 69327 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - 21945 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{6} - \frac {4}{9} \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \frac {4}{5} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} + \frac {1463}{32768} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1463}{65536} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1463}{65536} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^3)^(1/4)*(x^8-1)/x^4,x, algorithm="giac")

[Out]

1/245760*(7315*(1/x - 1)^5*(-1/x + 1)^(1/4) + 40755*(1/x - 1)^4*(-1/x + 1)^(1/4) + 92910*(1/x - 1)^3*(-1/x + 1
)^(1/4) + 109782*(1/x - 1)^2*(-1/x + 1)^(1/4) - 69327*(-1/x + 1)^(5/4) - 21945*(-1/x + 1)^(1/4))*x^6 - 4/9*(1/
x - 1)^2*(-1/x + 1)^(1/4) + 4/5*(-1/x + 1)^(5/4) + 1463/32768*arctan((-1/x + 1)^(1/4)) + 1463/65536*log((-1/x
+ 1)^(1/4) + 1) - 1463/65536*log(abs((-1/x + 1)^(1/4) - 1))

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maple [C]  time = 0.80, size = 64, normalized size = 0.63

method	result	size
meijerg	\(\frac {4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} x^{\frac {23}{4}} \hypergeom \left (\left [-\frac {1}{4}, \frac {23}{4}\right ], \left [\frac {27}{4}\right ], x\right )}{23 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}}}+\frac {4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} \left (-\frac {4}{5} x^{2}-\frac {1}{5} x +1\right ) \left (1-x \right )^{\frac {1}{4}}}{9 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {9}{4}}}\)	\(64\)
trager	\(\frac {\left (x^{4}-x^{3}\right )^{\frac {1}{4}} \left (122880 x^{8}-6144 x^{7}-7296 x^{6}-9120 x^{5}-12540 x^{4}-21945 x^{3}-262144 x^{2}-65536 x +327680\right )}{737280 x^{3}}-\frac {1463 \ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+2 x^{3}-x^{2}}{x^{2}}\right )}{65536}-\frac {1463 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )}{65536}\)	\(203\)
risch	\(\frac {\left (122880 x^{9}-129024 x^{8}-1152 x^{7}-1824 x^{6}-3420 x^{5}-9405 x^{4}-240199 x^{3}+196608 x^{2}+393216 x -327680\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{737280 x^{3} \left (-1+x \right )}+\frac {\left (-\frac {1463 \ln \left (\frac {2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}+2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, x +2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+2 x^{3}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -5 x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+4 x -1}{\left (-1+x \right )^{2}}\right )}{65536}+\frac {1463 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right )^{2}}\right )}{65536}\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x \left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (-1+x \right )}\)	\(445\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-x^3)^(1/4)*(x^8-1)/x^4,x,method=_RETURNVERBOSE)

[Out]

4/23*signum(-1+x)^(1/4)/(-signum(-1+x))^(1/4)*x^(23/4)*hypergeom([-1/4,23/4],[27/4],x)+4/9*signum(-1+x)^(1/4)/
(-signum(-1+x))^(1/4)/x^(9/4)*(-4/5*x^2-1/5*x+1)*(1-x)^(1/4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} - 1\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^3)^(1/4)*(x^8-1)/x^4,x, algorithm="maxima")

[Out]

integrate((x^8 - 1)*(x^4 - x^3)^(1/4)/x^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^8-1\right )\,{\left (x^4-x^3\right )}^{1/4}}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^8 - 1)*(x^4 - x^3)^(1/4))/x^4,x)

[Out]

int(((x^8 - 1)*(x^4 - x^3)^(1/4))/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-x**3)**(1/4)*(x**8-1)/x**4,x)

[Out]

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

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