3.24.82 \(\int \frac {\sqrt [4]{x^3+x^5} (1+x^4+x^8)}{x^4 (-1+x^4)} \, dx\)

Optimal. Leaf size=190 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )}{2\ 2^{3/4}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )}{2\ 2^{3/4}}-\frac {3 \tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^5+x^3}}{\sqrt {2} x^2-\sqrt {x^5+x^3}}\right )}{4 \sqrt [4]{2}}-\frac {3 \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^5+x^3}}{2^{3/4}}}{x \sqrt [4]{x^5+x^3}}\right )}{4 \sqrt [4]{2}}+\frac {4 \sqrt [4]{x^5+x^3} \left (x^4+2 x^2+1\right )}{9 x^3} \]

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Rubi [C]  time = 0.61, antiderivative size = 211, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2056, 1586, 6725, 364, 466, 510} \begin {gather*} \frac {4 \sqrt [4]{x^5+x^3} F_1\left (-\frac {9}{8};1,\frac {3}{4};-\frac {1}{8};x^2,-x^2\right )}{3 \sqrt [4]{x^2+1} x^3}+\frac {4 \sqrt [4]{x^5+x^3} x^3 \, _2F_1\left (\frac {3}{4},\frac {15}{8};\frac {23}{8};-x^2\right )}{15 \sqrt [4]{x^2+1}}+\frac {4 \sqrt [4]{x^5+x^3} x \, _2F_1\left (\frac {3}{4},\frac {7}{8};\frac {15}{8};-x^2\right )}{7 \sqrt [4]{x^2+1}}-\frac {8 \sqrt [4]{x^5+x^3} \, _2F_1\left (-\frac {1}{8},\frac {3}{4};\frac {7}{8};-x^2\right )}{\sqrt [4]{x^2+1} x}-\frac {8 \sqrt [4]{x^5+x^3} \, _2F_1\left (-\frac {9}{8},\frac {3}{4};-\frac {1}{8};-x^2\right )}{9 \sqrt [4]{x^2+1} x^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((x^3 + x^5)^(1/4)*(1 + x^4 + x^8))/(x^4*(-1 + x^4)),x]

[Out]

(4*(x^3 + x^5)^(1/4)*AppellF1[-9/8, 1, 3/4, -1/8, x^2, -x^2])/(3*x^3*(1 + x^2)^(1/4)) - (8*(x^3 + x^5)^(1/4)*H
ypergeometric2F1[-9/8, 3/4, -1/8, -x^2])/(9*x^3*(1 + x^2)^(1/4)) - (8*(x^3 + x^5)^(1/4)*Hypergeometric2F1[-1/8
, 3/4, 7/8, -x^2])/(x*(1 + x^2)^(1/4)) + (4*x*(x^3 + x^5)^(1/4)*Hypergeometric2F1[3/4, 7/8, 15/8, -x^2])/(7*(1
+ x^2)^(1/4)) + (4*x^3*(x^3 + x^5)^(1/4)*Hypergeometric2F1[3/4, 15/8, 23/8, -x^2])/(15*(1 + x^2)^(1/4))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx &=\frac {\sqrt [4]{x^3+x^5} \int \frac {\sqrt [4]{1+x^2} \left (1+x^4+x^8\right )}{x^{13/4} \left (-1+x^4\right )} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^3+x^5} \int \frac {1+x^4+x^8}{x^{13/4} \left (-1+x^2\right ) \left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^3+x^5} \int \left (\frac {2}{x^{13/4} \left (1+x^2\right )^{3/4}}+\frac {2}{x^{5/4} \left (1+x^2\right )^{3/4}}+\frac {x^{3/4}}{\left (1+x^2\right )^{3/4}}+\frac {x^{11/4}}{\left (1+x^2\right )^{3/4}}+\frac {3}{x^{13/4} \left (-1+x^2\right ) \left (1+x^2\right )^{3/4}}\right ) \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^3+x^5} \int \frac {x^{3/4}}{\left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^3+x^5} \int \frac {x^{11/4}}{\left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {1}{x^{13/4} \left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {1}{x^{5/4} \left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (3 \sqrt [4]{x^3+x^5}\right ) \int \frac {1}{x^{13/4} \left (-1+x^2\right ) \left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=-\frac {8 \sqrt [4]{x^3+x^5} \, _2F_1\left (-\frac {9}{8},\frac {3}{4};-\frac {1}{8};-x^2\right )}{9 x^3 \sqrt [4]{1+x^2}}-\frac {8 \sqrt [4]{x^3+x^5} \, _2F_1\left (-\frac {1}{8},\frac {3}{4};\frac {7}{8};-x^2\right )}{x \sqrt [4]{1+x^2}}+\frac {4 x \sqrt [4]{x^3+x^5} \, _2F_1\left (\frac {3}{4},\frac {7}{8};\frac {15}{8};-x^2\right )}{7 \sqrt [4]{1+x^2}}+\frac {4 x^3 \sqrt [4]{x^3+x^5} \, _2F_1\left (\frac {3}{4},\frac {15}{8};\frac {23}{8};-x^2\right )}{15 \sqrt [4]{1+x^2}}+\frac {\left (12 \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{10} \left (-1+x^8\right ) \left (1+x^8\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=\frac {4 \sqrt [4]{x^3+x^5} F_1\left (-\frac {9}{8};1,\frac {3}{4};-\frac {1}{8};x^2,-x^2\right )}{3 x^3 \sqrt [4]{1+x^2}}-\frac {8 \sqrt [4]{x^3+x^5} \, _2F_1\left (-\frac {9}{8},\frac {3}{4};-\frac {1}{8};-x^2\right )}{9 x^3 \sqrt [4]{1+x^2}}-\frac {8 \sqrt [4]{x^3+x^5} \, _2F_1\left (-\frac {1}{8},\frac {3}{4};\frac {7}{8};-x^2\right )}{x \sqrt [4]{1+x^2}}+\frac {4 x \sqrt [4]{x^3+x^5} \, _2F_1\left (\frac {3}{4},\frac {7}{8};\frac {15}{8};-x^2\right )}{7 \sqrt [4]{1+x^2}}+\frac {4 x^3 \sqrt [4]{x^3+x^5} \, _2F_1\left (\frac {3}{4},\frac {15}{8};\frac {23}{8};-x^2\right )}{15 \sqrt [4]{1+x^2}}\\ \end {align*}

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Mathematica [F]  time = 0.55, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((x^3 + x^5)^(1/4)*(1 + x^4 + x^8))/(x^4*(-1 + x^4)),x]

[Out]

Integrate[((x^3 + x^5)^(1/4)*(1 + x^4 + x^8))/(x^4*(-1 + x^4)), x]

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IntegrateAlgebraic [A]  time = 0.66, size = 190, normalized size = 1.00 \begin {gather*} \frac {4 \left (1+2 x^2+x^4\right ) \sqrt [4]{x^3+x^5}}{9 x^3}+\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{2\ 2^{3/4}}-\frac {3 \tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^3+x^5}}{\sqrt {2} x^2-\sqrt {x^3+x^5}}\right )}{4 \sqrt [4]{2}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{2\ 2^{3/4}}-\frac {3 \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^3+x^5}}{2^{3/4}}}{x \sqrt [4]{x^3+x^5}}\right )}{4 \sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((x^3 + x^5)^(1/4)*(1 + x^4 + x^8))/(x^4*(-1 + x^4)),x]

[Out]

(4*(1 + 2*x^2 + x^4)*(x^3 + x^5)^(1/4))/(9*x^3) + (3*ArcTan[(2^(1/4)*x)/(x^3 + x^5)^(1/4)])/(2*2^(3/4)) - (3*A
rcTan[(2^(3/4)*x*(x^3 + x^5)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^3 + x^5])])/(4*2^(1/4)) - (3*ArcTanh[(2^(1/4)*x)/(x^
3 + x^5)^(1/4)])/(2*2^(3/4)) - (3*ArcTanh[(x^2/2^(1/4) + Sqrt[x^3 + x^5]/2^(3/4))/(x*(x^3 + x^5)^(1/4))])/(4*2
^(1/4))

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fricas [B]  time = 6.71, size = 1133, normalized size = 5.96

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+x^3)^(1/4)*(x^8+x^4+1)/x^4/(x^4-1),x, algorithm="fricas")

[Out]

1/1152*(108*8^(3/4)*sqrt(2)*x^3*arctan(1/8*(8*x^6 + 32*x^5 + 48*x^4 + 4*8^(3/4)*sqrt(2)*(x^5 + x^3)^(3/4)*(x^2
- 6*x + 1) + 32*x^3 + 16*8^(1/4)*sqrt(2)*(x^5 + x^3)^(1/4)*(3*x^4 - 2*x^3 + 3*x^2) + 32*sqrt(2)*sqrt(x^5 + x^
3)*(x^3 + 2*x^2 + x) + 8*x^2 + sqrt(2)*(8^(3/4)*sqrt(2)*(x^6 - 16*x^5 - 2*x^4 - 16*x^3 + x^2) + 8*8^(1/4)*sqrt
(2)*sqrt(x^5 + x^3)*(x^3 - 6*x^2 + x) + 128*sqrt(2)*(x^5 + x^3)^(3/4)*x + 32*(x^5 + x^3)^(1/4)*(x^4 + 2*x^3 +
x^2))*sqrt((8^(3/4)*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + 2*8^(1/4)*sqrt(2)*(x^5 + x^3)^(3/4) + sqrt(2)*(x^4 + 2*x^3
+ x^2) + 8*sqrt(x^5 + x^3)*x)/(x^4 + 2*x^3 + x^2)))/(x^6 - 28*x^5 + 6*x^4 - 28*x^3 + x^2)) - 108*8^(3/4)*sqrt
(2)*x^3*arctan(1/8*(8*x^6 + 32*x^5 + 48*x^4 - 4*8^(3/4)*sqrt(2)*(x^5 + x^3)^(3/4)*(x^2 - 6*x + 1) + 32*x^3 - 1
6*8^(1/4)*sqrt(2)*(x^5 + x^3)^(1/4)*(3*x^4 - 2*x^3 + 3*x^2) + 32*sqrt(2)*sqrt(x^5 + x^3)*(x^3 + 2*x^2 + x) + 8
*x^2 - sqrt(2)*(8^(3/4)*sqrt(2)*(x^6 - 16*x^5 - 2*x^4 - 16*x^3 + x^2) + 8*8^(1/4)*sqrt(2)*sqrt(x^5 + x^3)*(x^3
- 6*x^2 + x) - 128*sqrt(2)*(x^5 + x^3)^(3/4)*x - 32*(x^5 + x^3)^(1/4)*(x^4 + 2*x^3 + x^2))*sqrt(-(8^(3/4)*sqr
t(2)*(x^5 + x^3)^(1/4)*x^2 + 2*8^(1/4)*sqrt(2)*(x^5 + x^3)^(3/4) - sqrt(2)*(x^4 + 2*x^3 + x^2) - 8*sqrt(x^5 +
x^3)*x)/(x^4 + 2*x^3 + x^2)))/(x^6 - 28*x^5 + 6*x^4 - 28*x^3 + x^2)) - 27*8^(3/4)*sqrt(2)*x^3*log(8*(8^(3/4)*s
qrt(2)*(x^5 + x^3)^(1/4)*x^2 + 2*8^(1/4)*sqrt(2)*(x^5 + x^3)^(3/4) + sqrt(2)*(x^4 + 2*x^3 + x^2) + 8*sqrt(x^5
+ x^3)*x)/(x^4 + 2*x^3 + x^2)) + 27*8^(3/4)*sqrt(2)*x^3*log(-8*(8^(3/4)*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + 2*8^(1
/4)*sqrt(2)*(x^5 + x^3)^(3/4) - sqrt(2)*(x^4 + 2*x^3 + x^2) - 8*sqrt(x^5 + x^3)*x)/(x^4 + 2*x^3 + x^2)) - 216*
8^(3/4)*x^3*arctan(-1/8*(16*8^(1/4)*(x^5 + x^3)^(1/4)*x^2 - 2^(3/4)*(8^(3/4)*(x^4 + 2*x^3 + x^2) + 8*8^(1/4)*s
qrt(x^5 + x^3)*x) + 4*8^(3/4)*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) - 54*8^(3/4)*x^3*log((4*sqrt(2)*(x^5 + x
^3)^(1/4)*x^2 + 8^(3/4)*sqrt(x^5 + x^3)*x + 8^(1/4)*(x^4 + 2*x^3 + x^2) + 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 +
x^2)) + 54*8^(3/4)*x^3*log((4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 - 8^(3/4)*sqrt(x^5 + x^3)*x - 8^(1/4)*(x^4 + 2*x^3
+ x^2) + 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) + 512*(x^5 + x^3)^(1/4)*(x^4 + 2*x^2 + 1))/x^3

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} + x^{4} + 1\right )} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}}{{\left (x^{4} - 1\right )} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+x^3)^(1/4)*(x^8+x^4+1)/x^4/(x^4-1),x, algorithm="giac")

[Out]

integrate((x^8 + x^4 + 1)*(x^5 + x^3)^(1/4)/((x^4 - 1)*x^4), x)

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maple [C]  time = 29.58, size = 742, normalized size = 3.91

method	result	size
trager	\(\frac {4 \left (x^{4}+2 x^{2}+1\right ) \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}+\frac {3 \RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{4}-2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{2}-4 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}+2\right ) x +4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{\left (1+x \right )^{2} x^{2}}\right )}{8}-\frac {3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{3}-4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}+4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \sqrt {x^{5}+x^{3}}\, x +4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{\left (1+x \right )^{2} x^{2}}\right )}{8}-\frac {3 \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \sqrt {x^{5}+x^{3}}\, x -4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}+2\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}+2\right ) x^{2}}{x^{2} \left (-1+x \right )^{2}}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{3}}{16}-\frac {3 \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \sqrt {x^{5}+x^{3}}\, x -4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}+2\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}+2\right ) x^{2}}{x^{2} \left (-1+x \right )^{2}}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )}{16}+\frac {3 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {x^{5}+x^{3}}\, x +2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \sqrt {x^{5}+x^{3}}\, x +4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{4}-\RootOf \left (\textit {\_Z}^{4}+2\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{3}-2 \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}-\RootOf \left (\textit {\_Z}^{4}+2\right ) x^{2}}{x^{2} \left (-1+x \right )^{2}}\right )}{8}\)	\(742\)
risch	\(\text {Expression too large to display}\)	\(1784\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5+x^3)^(1/4)*(x^8+x^4+1)/x^4/(x^4-1),x,method=_RETURNVERBOSE)

[Out]

4/9*(x^4+2*x^2+1)*(x^5+x^3)^(1/4)/x^3+3/8*RootOf(_Z^4+2)*ln(-(RootOf(_Z^4+2)^3*x^4-2*RootOf(_Z^4+2)^3*x^3+4*(x
^5+x^3)^(1/4)*RootOf(_Z^4+2)^2*x^2+RootOf(_Z^4+2)^3*x^2-4*(x^5+x^3)^(1/2)*RootOf(_Z^4+2)*x+4*(x^5+x^3)^(3/4))/
(1+x)^2/x^2)-3/8*RootOf(_Z^2+RootOf(_Z^4+2)^2)*ln(-(RootOf(_Z^4+2)^2*RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^4-2*RootO
f(_Z^2+RootOf(_Z^4+2)^2)*RootOf(_Z^4+2)^2*x^3-4*(x^5+x^3)^(1/4)*RootOf(_Z^4+2)^2*x^2+RootOf(_Z^4+2)^2*RootOf(_
Z^2+RootOf(_Z^4+2)^2)*x^2+4*RootOf(_Z^2+RootOf(_Z^4+2)^2)*(x^5+x^3)^(1/2)*x+4*(x^5+x^3)^(3/4))/(1+x)^2/x^2)-3/
16*ln(-(4*RootOf(_Z^4+2)^3*(x^5+x^3)^(1/2)*x-4*(x^5+x^3)^(1/4)*RootOf(_Z^4+2)^2*x^2-RootOf(_Z^4+2)*x^4+2*RootO
f(_Z^4+2)*x^3+4*(x^5+x^3)^(3/4)-RootOf(_Z^4+2)*x^2)/x^2/(-1+x)^2)*RootOf(_Z^4+2)^3-3/16*ln(-(4*RootOf(_Z^4+2)^
3*(x^5+x^3)^(1/2)*x-4*(x^5+x^3)^(1/4)*RootOf(_Z^4+2)^2*x^2-RootOf(_Z^4+2)*x^4+2*RootOf(_Z^4+2)*x^3+4*(x^5+x^3)
^(3/4)-RootOf(_Z^4+2)*x^2)/x^2/(-1+x)^2)*RootOf(_Z^4+2)^2*RootOf(_Z^2+RootOf(_Z^4+2)^2)+3/8*RootOf(_Z^4+2)^2*R
ootOf(_Z^2+RootOf(_Z^4+2)^2)*ln((-2*RootOf(_Z^2+RootOf(_Z^4+2)^2)*RootOf(_Z^4+2)^2*(x^5+x^3)^(1/2)*x+2*RootOf(
_Z^4+2)^3*(x^5+x^3)^(1/2)*x+4*(x^5+x^3)^(1/4)*RootOf(_Z^2+RootOf(_Z^4+2)^2)*RootOf(_Z^4+2)*x^2-RootOf(_Z^2+Roo
tOf(_Z^4+2)^2)*x^4-RootOf(_Z^4+2)*x^4-2*RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^3-2*RootOf(_Z^4+2)*x^3+4*(x^5+x^3)^(3/
4)-RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^2-RootOf(_Z^4+2)*x^2)/x^2/(-1+x)^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} + x^{4} + 1\right )} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}}{{\left (x^{4} - 1\right )} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+x^3)^(1/4)*(x^8+x^4+1)/x^4/(x^4-1),x, algorithm="maxima")

[Out]

integrate((x^8 + x^4 + 1)*(x^5 + x^3)^(1/4)/((x^4 - 1)*x^4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^5+x^3\right )}^{1/4}\,\left (x^8+x^4+1\right )}{x^4\,\left (x^4-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 + x^5)^(1/4)*(x^4 + x^8 + 1))/(x^4*(x^4 - 1)),x)

[Out]

int(((x^3 + x^5)^(1/4)*(x^4 + x^8 + 1))/(x^4*(x^4 - 1)), 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^{2} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}{x^{4} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5+x**3)**(1/4)*(x**8+x**4+1)/x**4/(x**4-1),x)

[Out]

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

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