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

Optimal. Leaf size=156 \[ \frac {2}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )-\frac {2}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )-\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^5+x^3}}{\sqrt {x^5+x^3}-x^2}\right )+\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^5+x^3}}{\sqrt {2}}}{x \sqrt [4]{x^5+x^3}}\right ) \]

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Rubi [C]  time = 0.74, antiderivative size = 320, normalized size of antiderivative = 2.05, number of steps used = 18, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2056, 6725, 959, 466, 510} \begin {gather*} -\frac {4 \left (1-\sqrt [3]{-1}\right ) \sqrt [4]{x^5+x^3} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^2,-\sqrt [3]{-1} x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {4 \left (1+(-1)^{2/3}\right ) \sqrt [4]{x^5+x^3} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^2,(-1)^{2/3} x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {8 \sqrt [4]{x^5+x^3} F_1\left (\frac {3}{8};1,-\frac {1}{4};\frac {11}{8};x^2,-x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {4 \left (1+(-1)^{2/3}\right ) x \sqrt [4]{x^5+x^3} F_1\left (\frac {7}{8};-\frac {1}{4},1;\frac {15}{8};-x^2,-\sqrt [3]{-1} x^2\right )}{21 \sqrt [4]{x^2+1}}-\frac {4 \left (1-\sqrt [3]{-1}\right ) x \sqrt [4]{x^5+x^3} F_1\left (\frac {7}{8};-\frac {1}{4},1;\frac {15}{8};-x^2,(-1)^{2/3} x^2\right )}{21 \sqrt [4]{x^2+1}}-\frac {8 x \sqrt [4]{x^5+x^3} F_1\left (\frac {7}{8};1,-\frac {1}{4};\frac {15}{8};x^2,-x^2\right )}{21 \sqrt [4]{x^2+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-4*(1 - (-1)^(1/3))*(x^3 + x^5)^(1/4)*AppellF1[3/8, -1/4, 1, 11/8, -x^2, -((-1)^(1/3)*x^2)])/(9*(1 + x^2)^(1/
4)) - (4*(1 + (-1)^(2/3))*(x^3 + x^5)^(1/4)*AppellF1[3/8, -1/4, 1, 11/8, -x^2, (-1)^(2/3)*x^2])/(9*(1 + x^2)^(
1/4)) - (8*(x^3 + x^5)^(1/4)*AppellF1[3/8, 1, -1/4, 11/8, x^2, -x^2])/(9*(1 + x^2)^(1/4)) - (4*(1 + (-1)^(2/3)
)*x*(x^3 + x^5)^(1/4)*AppellF1[7/8, -1/4, 1, 15/8, -x^2, -((-1)^(1/3)*x^2)])/(21*(1 + x^2)^(1/4)) - (4*(1 - (-
1)^(1/3))*x*(x^3 + x^5)^(1/4)*AppellF1[7/8, -1/4, 1, 15/8, -x^2, (-1)^(2/3)*x^2])/(21*(1 + x^2)^(1/4)) - (8*x*
(x^3 + x^5)^(1/4)*AppellF1[7/8, 1, -1/4, 15/8, x^2, -x^2])/(21*(1 + x^2)^(1/4))

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 959

Int[(((g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[(d*(g*x)^n)/x^n, In
t[(x^n*(a + c*x^2)^p)/(d^2 - e^2*x^2), x], x] - Dist[(e*(g*x)^n)/x^n, Int[(x^(n + 1)*(a + c*x^2)^p)/(d^2 - e^2
*x^2), x], x] /; FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] &&  !IntegersQ[n, 2
*p]

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 {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx &=\frac {\sqrt [4]{x^3+x^5} \int \frac {(1+x) \sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (-1+x^3\right )} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^3+x^5} \int \left (-\frac {2 \sqrt [4]{1+x^2}}{3 (1-x) \sqrt [4]{x}}-\frac {\left (1+(-1)^{2/3}\right ) \sqrt [4]{1+x^2}}{3 \sqrt [4]{x} \left (1+\sqrt [3]{-1} x\right )}-\frac {\left (1-\sqrt [3]{-1}\right ) \sqrt [4]{1+x^2}}{3 \sqrt [4]{x} \left (1-(-1)^{2/3} x\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=-\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{(1-x) \sqrt [4]{x}} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1-(-1)^{2/3} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}\\ &=-\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1-x^2\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x^2}}{1-x^2} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1+\sqrt [3]{-1} x^2\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left ((-1)^{2/3} \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x^2}}{1+\sqrt [3]{-1} x^2} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1-(-1)^{2/3} x^2\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (\sqrt [3]{-1} \left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x^2}}{1-(-1)^{2/3} x^2} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}\\ &=-\frac {\left (8 \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1-x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (8 \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [4]{1+x^8}}{1-x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (4 \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1+\sqrt [3]{-1} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (4 (-1)^{2/3} \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [4]{1+x^8}}{1+\sqrt [3]{-1} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (4 \left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1-(-1)^{2/3} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (4 \sqrt [3]{-1} \left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [4]{1+x^8}}{1-(-1)^{2/3} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}\\ &=-\frac {4 \left (1-\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^2,-\sqrt [3]{-1} x^2\right )}{9 \sqrt [4]{1+x^2}}-\frac {4 \left (1+(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^2,(-1)^{2/3} x^2\right )}{9 \sqrt [4]{1+x^2}}-\frac {8 \sqrt [4]{x^3+x^5} F_1\left (\frac {3}{8};1,-\frac {1}{4};\frac {11}{8};x^2,-x^2\right )}{9 \sqrt [4]{1+x^2}}-\frac {4 (-1)^{2/3} \left (1-\sqrt [3]{-1}\right ) x \sqrt [4]{x^3+x^5} F_1\left (\frac {7}{8};-\frac {1}{4},1;\frac {15}{8};-x^2,-\sqrt [3]{-1} x^2\right )}{21 \sqrt [4]{1+x^2}}-\frac {4 \left (1-\sqrt [3]{-1}\right ) x \sqrt [4]{x^3+x^5} F_1\left (\frac {7}{8};-\frac {1}{4},1;\frac {15}{8};-x^2,(-1)^{2/3} x^2\right )}{21 \sqrt [4]{1+x^2}}-\frac {8 x \sqrt [4]{x^3+x^5} F_1\left (\frac {7}{8};1,-\frac {1}{4};\frac {15}{8};x^2,-x^2\right )}{21 \sqrt [4]{1+x^2}}\\ \end {align*}

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Mathematica [F]  time = 1.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.53, size = 156, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )-\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3+x^5}}{-x^2+\sqrt {x^3+x^5}}\right )-\frac {2}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )+\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^3+x^5}}{\sqrt {2}}}{x \sqrt [4]{x^3+x^5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*2^(1/4)*ArcTan[(2^(1/4)*x)/(x^3 + x^5)^(1/4)])/3 - (Sqrt[2]*ArcTan[(Sqrt[2]*x*(x^3 + x^5)^(1/4))/(-x^2 + Sq
rt[x^3 + x^5])])/3 - (2*2^(1/4)*ArcTanh[(2^(1/4)*x)/(x^3 + x^5)^(1/4)])/3 + (Sqrt[2]*ArcTanh[(x^2/Sqrt[2] + Sq
rt[x^3 + x^5]/Sqrt[2])/(x*(x^3 + x^5)^(1/4))])/3

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fricas [B]  time = 5.06, size = 953, normalized size = 6.11

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(2)*arctan((x^6 + 2*x^5 + 3*x^4 + 2*x^3 + 2*sqrt(2)*(x^5 + x^3)^(3/4)*(x^2 - 3*x + 1) + x^2 + 2*sqrt(
2)*(x^5 + x^3)^(1/4)*(3*x^4 - x^3 + 3*x^2) + 4*sqrt(x^5 + x^3)*(x^3 + x^2 + x) + (2*sqrt(2)*sqrt(x^5 + x^3)*(x
^3 - 3*x^2 + x) + 16*(x^5 + x^3)^(3/4)*x + sqrt(2)*(x^6 - 8*x^5 + x^4 - 8*x^3 + x^2) + 4*(x^5 + x^3)^(1/4)*(x^
4 + x^3 + x^2))*sqrt((x^4 + x^3 + 2*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + x^2 + 4*sqrt(x^5 + x^3)*x + 2*sqrt(2)*(x^5
+ x^3)^(3/4))/(x^4 + x^3 + x^2)))/(x^6 - 14*x^5 + 3*x^4 - 14*x^3 + x^2)) + 1/3*sqrt(2)*arctan((x^6 + 2*x^5 +
3*x^4 + 2*x^3 - 2*sqrt(2)*(x^5 + x^3)^(3/4)*(x^2 - 3*x + 1) + x^2 - 2*sqrt(2)*(x^5 + x^3)^(1/4)*(3*x^4 - x^3 +
3*x^2) + 4*sqrt(x^5 + x^3)*(x^3 + x^2 + x) - (2*sqrt(2)*sqrt(x^5 + x^3)*(x^3 - 3*x^2 + x) - 16*(x^5 + x^3)^(3
/4)*x + sqrt(2)*(x^6 - 8*x^5 + x^4 - 8*x^3 + x^2) - 4*(x^5 + x^3)^(1/4)*(x^4 + x^3 + x^2))*sqrt((x^4 + x^3 - 2
*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + x^2 + 4*sqrt(x^5 + x^3)*x - 2*sqrt(2)*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)))/
(x^6 - 14*x^5 + 3*x^4 - 14*x^3 + x^2)) + 1/12*sqrt(2)*log(4*(x^4 + x^3 + 2*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + x^2
+ 4*sqrt(x^5 + x^3)*x + 2*sqrt(2)*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)) - 1/12*sqrt(2)*log(4*(x^4 + x^3 - 2*s
qrt(2)*(x^5 + x^3)^(1/4)*x^2 + x^2 + 4*sqrt(x^5 + x^3)*x - 2*sqrt(2)*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)) + 2
/3*2^(1/4)*arctan(1/2*(4*2^(3/4)*(x^5 + x^3)^(1/4)*x^2 + 2^(3/4)*(2*2^(3/4)*sqrt(x^5 + x^3)*x + 2^(1/4)*(x^4 +
2*x^3 + x^2)) + 4*2^(1/4)*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) - 1/6*2^(1/4)*log(-(4*sqrt(2)*(x^5 + x^3)^(
1/4)*x^2 + 2^(3/4)*(x^4 + 2*x^3 + x^2) + 4*2^(1/4)*sqrt(x^5 + x^3)*x + 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2
)) + 1/6*2^(1/4)*log(-(4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 - 2^(3/4)*(x^4 + 2*x^3 + x^2) - 4*2^(1/4)*sqrt(x^5 + x^
3)*x + 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 12.56, size = 732, normalized size = 4.69

method	result	size
trager	\(-\frac {\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}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{5}+x^{3}\right )^{\frac {1}{4}} 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}}}{x^{2} \left (-1+x \right )^{2}}\right )}{3}+\frac {\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}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{5}+x^{3}\right )^{\frac {1}{4}} 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}}}{x^{2} \left (-1+x \right )^{2}}\right )}{3}+\frac {\ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \sqrt {x^{5}+x^{3}}\, x +\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} x^{2}}{x^{2} \left (x^{2}+x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}}{6}+\frac {\ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \sqrt {x^{5}+x^{3}}\, x +\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} x^{2}}{x^{2} \left (x^{2}+x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{6}-\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \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 )^{3} \left (x^{5}+x^{3}\right )^{\frac {1}{4}} x^{2}-\RootOf \left (\textit {\_Z}^{4}-2\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{4}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}-2\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {x^{5}+x^{3}}\, x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{3}-2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \sqrt {x^{5}+x^{3}}\, x +\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-\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}^{4}-2\right )^{2} x^{2}-4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (x^{2}+x +1\right )}\right )}{3}\)	\(732\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*RootOf(_Z^4-2)*ln((RootOf(_Z^4-2)^3*x^4+2*RootOf(_Z^4-2)^3*x^3+RootOf(_Z^4-2)^3*x^2+4*RootOf(_Z^4-2)^2*(x
^5+x^3)^(1/4)*x^2+4*(x^5+x^3)^(1/2)*RootOf(_Z^4-2)*x+4*(x^5+x^3)^(3/4))/x^2/(-1+x)^2)+1/3*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*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2
)^2*x^3+RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^2*x^2-4*RootOf(_Z^4-2)^2*(x^5+x^3)^(1/4)*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))/x^2/(-1+x)^2)+1/6*ln((RootOf(_Z^4-2)^2*x^4+4*RootOf(_
Z^4-2)^2*(x^5+x^3)^(1/2)*x+RootOf(_Z^4-2)^2*x^3+RootOf(_Z^4-2)^2*x^2+4*(x^5+x^3)^(3/4)+4*(x^5+x^3)^(1/4)*x^2)/
x^2/(x^2+x+1))*RootOf(_Z^4-2)^2+1/6*ln((RootOf(_Z^4-2)^2*x^4+4*RootOf(_Z^4-2)^2*(x^5+x^3)^(1/2)*x+RootOf(_Z^4-
2)^2*x^3+RootOf(_Z^4-2)^2*x^2+4*(x^5+x^3)^(3/4)+4*(x^5+x^3)^(1/4)*x^2)/x^2/(x^2+x+1))*RootOf(_Z^4-2)*RootOf(_Z
^2+RootOf(_Z^4-2)^2)-1/3*RootOf(_Z^4-2)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*ln(-(2*RootOf(_Z^2+RootOf(_Z^4-2)^2)*Roo
tOf(_Z^4-2)^3*(x^5+x^3)^(1/4)*x^2-RootOf(_Z^4-2)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*x^4-RootOf(_Z^4-2)^2*x^4+2*Root
Of(_Z^4-2)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*(x^5+x^3)^(1/2)*x+RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)*x^3-2*
RootOf(_Z^4-2)^2*(x^5+x^3)^(1/2)*x+RootOf(_Z^4-2)^2*x^3-RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)*x^2-RootO
f(_Z^4-2)^2*x^2-4*(x^5+x^3)^(3/4))/x^2/(x^2+x+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^5 + x^3)^(1/4)*(x + 1)/((x^3 - 1)*x), 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+1\right )}{x\,\left (x^3-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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