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

Optimal. Leaf size=185 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )}{2 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )}{2 \sqrt [4]{2}}+\frac {2 \left (x^5+x^3\right )^{3/4}}{x^2 \left (x^2+1\right )}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^5+x^3}}{\sqrt {2} x^2-\sqrt {x^5+x^3}}\right )}{2\ 2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^5+x^3}}{2^{3/4}}}{x \sqrt [4]{x^5+x^3}}\right )}{2\ 2^{3/4}} \]

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Rubi [C]  time = 0.58, antiderivative size = 81, normalized size of antiderivative = 0.44, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2056, 6715, 6725, 245, 1404, 429} \begin {gather*} \frac {8 x \sqrt [4]{x^2+1} F_1\left (\frac {1}{8};1,\frac {5}{4};\frac {9}{8};x^2,-x^2\right )}{\sqrt [4]{x^5+x^3}}-\frac {4 x \sqrt [4]{x^2+1} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^2\right )}{\sqrt [4]{x^5+x^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(8*x*(1 + x^2)^(1/4)*AppellF1[1/8, 1, 5/4, 9/8, x^2, -x^2])/(x^3 + x^5)^(1/4) - (4*x*(1 + x^2)^(1/4)*Hypergeom
etric2F1[1/8, 1/4, 9/8, -x^2])/(x^3 + x^5)^(1/4)

Rule 245

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

Rule 429

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

Rule 1404

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(d + e*x^n)^(p + q)*(a/d
+ (c*x^n)/e)^p, x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[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 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{1+x^2}\right ) \int \frac {1+x^4}{x^{3/4} \sqrt [4]{1+x^2} \left (1-x^4\right )} \, dx}{\sqrt [4]{x^3+x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1+x^{16}}{\sqrt [4]{1+x^8} \left (1-x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt [4]{1+x^8}}+\frac {2}{\sqrt [4]{1+x^8} \left (1-x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}\\ &=-\frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}+\frac {\left (8 x^{3/4} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8} \left (1-x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}\\ &=-\frac {4 x \sqrt [4]{1+x^2} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^2\right )}{\sqrt [4]{x^3+x^5}}+\frac {\left (8 x^{3/4} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^8\right ) \left (1+x^8\right )^{5/4}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}\\ &=\frac {8 x \sqrt [4]{1+x^2} F_1\left (\frac {1}{8};1,\frac {5}{4};\frac {9}{8};x^2,-x^2\right )}{\sqrt [4]{x^3+x^5}}-\frac {4 x \sqrt [4]{1+x^2} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^2\right )}{\sqrt [4]{x^3+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.58, size = 185, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^3+x^5\right )^{3/4}}{x^2 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{2 \sqrt [4]{2}}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^3+x^5}}{\sqrt {2} x^2-\sqrt {x^3+x^5}}\right )}{2\ 2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{2 \sqrt [4]{2}}+\frac {\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 )}{2\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 14.92, size = 1100, normalized size = 5.95

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(4*2^(3/4)*(x^4 + x^2)*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)) - 2^(3/4)*(x^4 + x^2)*lo
g((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)) + 2^(3/4)*(x^4 + x^2)*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)) - 4*2^(1/4)*(x^4 + x^2)*arct
an(1/2*(2*x^6 + 8*x^5 + 12*x^4 + 8*x^3 + 4*2^(3/4)*(x^5 + x^3)^(3/4)*(x^2 - 6*x + 1) + 8*sqrt(2)*sqrt(x^5 + x^
3)*(x^3 + 2*x^2 + x) + 2*x^2 + sqrt(2)*(32*sqrt(2)*(x^5 + x^3)^(3/4)*x + 2^(3/4)*(x^6 - 16*x^5 - 2*x^4 - 16*x^
3 + x^2) + 4*2^(1/4)*sqrt(x^5 + x^3)*(x^3 - 6*x^2 + x) + 8*(x^5 + x^3)^(1/4)*(x^4 + 2*x^3 + x^2))*sqrt((4*2^(3
/4)*(x^5 + x^3)^(1/4)*x^2 + sqrt(2)*(x^4 + 2*x^3 + x^2) + 8*sqrt(x^5 + x^3)*x + 4*2^(1/4)*(x^5 + x^3)^(3/4))/(
x^4 + 2*x^3 + x^2)) + 8*2^(1/4)*(x^5 + x^3)^(1/4)*(3*x^4 - 2*x^3 + 3*x^2))/(x^6 - 28*x^5 + 6*x^4 - 28*x^3 + x^
2)) + 4*2^(1/4)*(x^4 + x^2)*arctan(1/2*(2*x^6 + 8*x^5 + 12*x^4 + 8*x^3 - 4*2^(3/4)*(x^5 + x^3)^(3/4)*(x^2 - 6*
x + 1) + 8*sqrt(2)*sqrt(x^5 + x^3)*(x^3 + 2*x^2 + x) + 2*x^2 + sqrt(2)*(32*sqrt(2)*(x^5 + x^3)^(3/4)*x - 2^(3/
4)*(x^6 - 16*x^5 - 2*x^4 - 16*x^3 + x^2) - 4*2^(1/4)*sqrt(x^5 + x^3)*(x^3 - 6*x^2 + x) + 8*(x^5 + x^3)^(1/4)*(
x^4 + 2*x^3 + x^2))*sqrt(-(4*2^(3/4)*(x^5 + x^3)^(1/4)*x^2 - sqrt(2)*(x^4 + 2*x^3 + x^2) - 8*sqrt(x^5 + x^3)*x
+ 4*2^(1/4)*(x^5 + x^3)^(3/4))/(x^4 + 2*x^3 + x^2)) - 8*2^(1/4)*(x^5 + x^3)^(1/4)*(3*x^4 - 2*x^3 + 3*x^2))/(x
^6 - 28*x^5 + 6*x^4 - 28*x^3 + x^2)) - 2^(1/4)*(x^4 + x^2)*log(8*(4*2^(3/4)*(x^5 + x^3)^(1/4)*x^2 + sqrt(2)*(x
^4 + 2*x^3 + x^2) + 8*sqrt(x^5 + x^3)*x + 4*2^(1/4)*(x^5 + x^3)^(3/4))/(x^4 + 2*x^3 + x^2)) + 2^(1/4)*(x^4 + x
^2)*log(-8*(4*2^(3/4)*(x^5 + x^3)^(1/4)*x^2 - sqrt(2)*(x^4 + 2*x^3 + x^2) - 8*sqrt(x^5 + x^3)*x + 4*2^(1/4)*(x
^5 + x^3)^(3/4))/(x^4 + 2*x^3 + x^2)) - 32*(x^5 + x^3)^(3/4))/(x^4 + x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 23.98, size = 737, normalized size = 3.98

method	result	size
risch	\(\frac {2 x}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x +2 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}}{\left (1+x \right )^{2} x^{2}}\right )}{8}+\frac {\RootOf \left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} \sqrt {x^{5}+x^{3}}\, x -2 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{4}+8\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}+8\right ) x^{2}}{\left (1+x \right )^{2} x^{2}}\right )}{8}-\frac {\ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{3}+8 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{2}+16 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}+8\right ) x +16 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (-1+x \right )^{2}}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )}{32}-\frac {\ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{3}+8 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{2}+16 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}+8\right ) x +16 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (-1+x \right )^{2}}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{3}}{32}+\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{4}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{4}-2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}+8 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}+8 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}+8\right ) x +8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \sqrt {x^{5}+x^{3}}\, x +16 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (-1+x \right )^{2}}\right )}{16}\)	\(737\)
trager	\(\frac {2 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (x^{2}+1\right )}+\frac {\RootOf \left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} \sqrt {x^{5}+x^{3}}\, x -2 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{4}+8\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}+8\right ) x^{2}}{\left (1+x \right )^{2} x^{2}}\right )}{8}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x +2 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}}{\left (1+x \right )^{2} x^{2}}\right )}{8}-\frac {\ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{3}+8 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{2}+16 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}+8\right ) x +16 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (-1+x \right )^{2}}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{3}}{32}-\frac {\ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{3}+8 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{2}+16 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}+8\right ) x +16 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (-1+x \right )^{2}}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )}{32}+\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{4}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{4}-2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}+8 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}+8 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}+8\right ) x +8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \sqrt {x^{5}+x^{3}}\, x +16 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (-1+x \right )^{2}}\right )}{16}\)	\(744\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x/(x^3*(x^2+1))^(1/4)+1/8*RootOf(_Z^2+RootOf(_Z^4+8)^2)*ln(-((x^5+x^3)^(1/2)*RootOf(_Z^4+8)^2*RootOf(_Z^2+Ro
otOf(_Z^4+8)^2)*x+2*(x^5+x^3)^(1/4)*RootOf(_Z^4+8)^2*x^2+RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^4-2*RootOf(_Z^2+RootO
f(_Z^4+8)^2)*x^3+4*(x^5+x^3)^(3/4)+RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^2)/(1+x)^2/x^2)+1/8*RootOf(_Z^4+8)*ln(-(-Ro
otOf(_Z^4+8)^3*(x^5+x^3)^(1/2)*x-2*(x^5+x^3)^(1/4)*RootOf(_Z^4+8)^2*x^2+RootOf(_Z^4+8)*x^4-2*RootOf(_Z^4+8)*x^
3+4*(x^5+x^3)^(3/4)+RootOf(_Z^4+8)*x^2)/(1+x)^2/x^2)-1/32*ln(-(-RootOf(_Z^4+8)^3*x^4+2*RootOf(_Z^4+8)^3*x^3+8*
(x^5+x^3)^(1/4)*RootOf(_Z^4+8)^2*x^2-RootOf(_Z^4+8)^3*x^2+16*(x^5+x^3)^(1/2)*RootOf(_Z^4+8)*x+16*(x^5+x^3)^(3/
4))/x^2/(-1+x)^2)*RootOf(_Z^4+8)^2*RootOf(_Z^2+RootOf(_Z^4+8)^2)-1/32*ln(-(-RootOf(_Z^4+8)^3*x^4+2*RootOf(_Z^4
+8)^3*x^3+8*(x^5+x^3)^(1/4)*RootOf(_Z^4+8)^2*x^2-RootOf(_Z^4+8)^3*x^2+16*(x^5+x^3)^(1/2)*RootOf(_Z^4+8)*x+16*(
x^5+x^3)^(3/4))/x^2/(-1+x)^2)*RootOf(_Z^4+8)^3+1/16*RootOf(_Z^4+8)^2*RootOf(_Z^2+RootOf(_Z^4+8)^2)*ln((-RootOf
(_Z^4+8)^3*x^4+RootOf(_Z^2+RootOf(_Z^4+8)^2)*RootOf(_Z^4+8)^2*x^4-2*RootOf(_Z^4+8)^3*x^3+2*RootOf(_Z^2+RootOf(
_Z^4+8)^2)*RootOf(_Z^4+8)^2*x^3+8*(x^5+x^3)^(1/4)*RootOf(_Z^4+8)*RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^2-RootOf(_Z^4
+8)^3*x^2+RootOf(_Z^4+8)^2*RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^2+8*(x^5+x^3)^(1/2)*RootOf(_Z^4+8)*x+8*RootOf(_Z^2+
RootOf(_Z^4+8)^2)*(x^5+x^3)^(1/2)*x+16*(x^5+x^3)^(3/4))/x^2/(-1+x)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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