3.9.99 \(\int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^5} \, dx\) [899]

Optimal. Leaf size=137 \[ -\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{1+x}}{192 x}-\frac {11}{64} \tan ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {11}{64} \tanh ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right ) \]

[Out]

-1/4*(1-x)^(3/4)*(1+x)^(1/4)/x^4-7/24*(1-x)^(3/4)*(1+x)^(1/4)/x^3-29/96*(1-x)^(3/4)*(1+x)^(1/4)/x^2-83/192*(1-
x)^(3/4)*(1+x)^(1/4)/x-11/64*arctan((1+x)^(1/4)/(1-x)^(1/4))-11/64*arctanh((1+x)^(1/4)/(1-x)^(1/4))

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Rubi [A]
time = 0.03, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {101, 156, 12, 95, 218, 212, 209} \begin {gather*} -\frac {11}{64} \text {ArcTan}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{x+1}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{x+1}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{x+1}}{192 x}-\frac {11}{64} \tanh ^{-1}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((1 - x)^(3/4)*(1 + x)^(1/4))/x^4 - (7*(1 - x)^(3/4)*(1 + x)^(1/4))/(24*x^3) - (29*(1 - x)^(3/4)*(1 + x)^
(1/4))/(96*x^2) - (83*(1 - x)^(3/4)*(1 + x)^(1/4))/(192*x) - (11*ArcTan[(1 + x)^(1/4)/(1 - x)^(1/4)])/64 - (11
*ArcTanh[(1 + x)^(1/4)/(1 - x)^(1/4)])/64

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 209

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

Rule 212

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

Rule 218

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] &&  !Gt
Q[a/b, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^5} \, dx &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}+\frac {1}{4} \int \frac {\frac {7}{2}+3 x}{\sqrt [4]{1-x} x^4 (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {1}{12} \int \frac {-\frac {29}{4}-7 x}{\sqrt [4]{1-x} x^3 (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}+\frac {1}{24} \int \frac {\frac {83}{8}+\frac {29 x}{4}}{\sqrt [4]{1-x} x^2 (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{1+x}}{192 x}-\frac {1}{24} \int -\frac {33}{16 \sqrt [4]{1-x} x (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{1+x}}{192 x}+\frac {11}{128} \int \frac {1}{\sqrt [4]{1-x} x (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{1+x}}{192 x}+\frac {11}{32} \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{1+x}}{192 x}-\frac {11}{64} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {11}{64} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{1+x}}{192 x}-\frac {11}{64} \tan ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {11}{64} \tanh ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 81, normalized size = 0.59 \begin {gather*} \frac {1}{192} \left (-\frac {(1-x)^{3/4} \sqrt [4]{1+x} \left (48+56 x+58 x^2+83 x^3\right )}{x^4}-33 \tan ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-33 \tanh ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(((1 - x)^(3/4)*(1 + x)^(1/4)*(48 + 56*x + 58*x^2 + 83*x^3))/x^4) - 33*ArcTan[(1 + x)^(1/4)/(1 - x)^(1/4)] -
33*ArcTanh[(1 + x)^(1/4)/(1 - x)^(1/4)])/192

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.52, size = 394, normalized size = 2.88

method	result	size
risch	\(\frac {\left (1+x \right )^{\frac {1}{4}} \left (-1+x \right ) \left (83 x^{3}+58 x^{2}+56 x +48\right ) \left (\left (1-x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}}}{192 x^{4} \left (-\left (-1+x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}} \left (1-x \right )^{\frac {1}{4}}}+\frac {\left (-\frac {11 \ln \left (\frac {\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}+\sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}+\sqrt {-x^{4}-2 x^{3}+2 x +1}+2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x +x^{2}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}+2 x +1}{x \left (1+x \right )^{2}}\right )}{128}-\frac {11 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}-2 x^{3}+2 x +1}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x -\RootOf \left (\textit {\_Z}^{2}+1\right )+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}}{x \left (1+x \right )^{2}}\right )}{128}\right ) \left (\left (1-x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}}}{\left (1+x \right )^{\frac {3}{4}} \left (1-x \right )^{\frac {1}{4}}}\)	\(394\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(1+x)^(1/4)*(-1+x)*(83*x^3+58*x^2+56*x+48)/x^4/(-(-1+x)*(1+x)^3)^(1/4)*((1-x)*(1+x)^3)^(1/4)/(1-x)^(1/4)
+(-11/128*ln(((-x^4-2*x^3+2*x+1)^(3/4)+(-x^4-2*x^3+2*x+1)^(1/2)*x+(-x^4-2*x^3+2*x+1)^(1/4)*x^2+(-x^4-2*x^3+2*x
+1)^(1/2)+2*(-x^4-2*x^3+2*x+1)^(1/4)*x+x^2+(-x^4-2*x^3+2*x+1)^(1/4)+2*x+1)/x/(1+x)^2)-11/128*RootOf(_Z^2+1)*ln
(-(RootOf(_Z^2+1)*(-x^4-2*x^3+2*x+1)^(1/2)*x+RootOf(_Z^2+1)*(-x^4-2*x^3+2*x+1)^(1/2)-RootOf(_Z^2+1)*x^2-(-x^4-
2*x^3+2*x+1)^(3/4)+(-x^4-2*x^3+2*x+1)^(1/4)*x^2-2*RootOf(_Z^2+1)*x+2*(-x^4-2*x^3+2*x+1)^(1/4)*x-RootOf(_Z^2+1)
+(-x^4-2*x^3+2*x+1)^(1/4))/x/(1+x)^2))/(1+x)^(3/4)*((1-x)*(1+x)^3)^(1/4)/(1-x)^(1/4)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.87, size = 117, normalized size = 0.85 \begin {gather*} \frac {66 \, x^{4} \arctan \left (\frac {{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) + 33 \, x^{4} \log \left (\frac {x + {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - 33 \, x^{4} \log \left (-\frac {x - {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - 2 \, {\left (83 \, x^{3} + 58 \, x^{2} + 56 \, x + 48\right )} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{384 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(66*x^4*arctan((x + 1)^(1/4)*(-x + 1)^(3/4)/(x - 1)) + 33*x^4*log((x + (x + 1)^(1/4)*(-x + 1)^(3/4) - 1)
/(x - 1)) - 33*x^4*log(-(x - (x + 1)^(1/4)*(-x + 1)^(3/4) - 1)/(x - 1)) - 2*(83*x^3 + 58*x^2 + 56*x + 48)*(x +
1)^(1/4)*(-x + 1)^(3/4))/x^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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