\(\int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^5} \, dx\) [509]

Optimal result

Integrand size = 20, antiderivative size = 137 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^5} \, dx=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{1+x}}{192 x}-\frac {11}{64} \arctan \left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {11}{64} \text {arctanh}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right ) \] Output:

-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))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^5} \, dx=\frac {1}{192} \left (-\frac {(1-x)^{3/4} \sqrt [4]{1+x} \left (48+56 x+58 x^2+83 x^3\right )}{x^4}-33 \arctan \left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-33 \text {arctanh}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {110, 27, 168, 27, 168, 27, 168, 27, 104, 756, 216, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 110

\(\displaystyle \frac {1}{4} \int \frac {6 x+7}{2 \sqrt [4]{1-x} x^4 (x+1)^{3/4}}dx-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \int \frac {6 x+7}{\sqrt [4]{1-x} x^4 (x+1)^{3/4}}dx-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{8} \left (-\frac {1}{3} \int -\frac {28 x+29}{2 \sqrt [4]{1-x} x^3 (x+1)^{3/4}}dx-\frac {7 (1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \frac {28 x+29}{\sqrt [4]{1-x} x^3 (x+1)^{3/4}}dx-\frac {7 (1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (-\frac {1}{2} \int -\frac {58 x+83}{2 \sqrt [4]{1-x} x^2 (x+1)^{3/4}}dx-\frac {29 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {7 (1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {58 x+83}{\sqrt [4]{1-x} x^2 (x+1)^{3/4}}dx-\frac {29 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {7 (1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (-\int -\frac {33}{2 \sqrt [4]{1-x} x (x+1)^{3/4}}dx-\frac {83 (1-x)^{3/4} \sqrt [4]{x+1}}{x}\right )-\frac {29 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {7 (1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {33}{2} \int \frac {1}{\sqrt [4]{1-x} x (x+1)^{3/4}}dx-\frac {83 (1-x)^{3/4} \sqrt [4]{x+1}}{x}\right )-\frac {29 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {7 (1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (66 \int \frac {1}{\frac {x+1}{1-x}-1}d\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}-\frac {83 (1-x)^{3/4} \sqrt [4]{x+1}}{x}\right )-\frac {29 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {7 (1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (66 \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {x+1}}{\sqrt {1-x}}}d\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {x+1}}{\sqrt {1-x}}+1}d\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {83 (1-x)^{3/4} \sqrt [4]{x+1}}{x}\right )-\frac {29 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {7 (1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (66 \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {x+1}}{\sqrt {1-x}}}d\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-\frac {83 (1-x)^{3/4} \sqrt [4]{x+1}}{x}\right )-\frac {29 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {7 (1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (66 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-\frac {83 (1-x)^{3/4} \sqrt [4]{x+1}}{x}\right )-\frac {29 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {7 (1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}\)

Input:

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

Output:

-1/4*((1 - x)^(3/4)*(1 + x)^(1/4))/x^4 + ((-7*(1 - x)^(3/4)*(1 + x)^(1/4)) 
/(3*x^3) + ((-29*(1 - x)^(3/4)*(1 + x)^(1/4))/(2*x^2) + ((-83*(1 - x)^(3/4 
)*(1 + x)^(1/4))/x + 66*(-1/2*ArcTan[(1 + x)^(1/4)/(1 - x)^(1/4)] - ArcTan 
h[(1 + x)^(1/4)/(1 - x)^(1/4)]/2))/4)/6)/8
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[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] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 110

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

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S 
imp[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 216

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

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 756

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.69 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.88

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

Input:

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

Output:

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*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))/(1+x)^2/x)-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)/(1+x)^2/x))/(1+ 
x)^(3/4)*((1-x)*(1+x)^3)^(1/4)/(1-x)^(1/4)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^5} \, dx=\frac {66 \, x^{4} \arctan \left (\frac {{\left (-x + 1\right )}^{\frac {1}{4}}}{{\left (x + 1\right )}^{\frac {1}{4}}}\right ) + 33 \, x^{4} \log \left (\frac {x + {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - 33 \, x^{4} \log \left (-\frac {x - {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - 2 \, {\left (83 \, x^{3} + 58 \, x^{2} + 56 \, x + 48\right )} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{384 \, x^{4}} \] Input:

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

Output:

1/384*(66*x^4*arctan((-x + 1)^(1/4)/(x + 1)^(1/4)) + 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
 

Sympy [F]

\[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^5} \, dx=\int \frac {\sqrt [4]{x + 1}}{x^{5} \sqrt [4]{1 - x}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^5} \, dx=\int { \frac {{\left (x + 1\right )}^{\frac {1}{4}}}{x^{5} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^5} \, dx=\int { \frac {{\left (x + 1\right )}^{\frac {1}{4}}}{x^{5} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^5} \, dx=\int \frac {{\left (x+1\right )}^{1/4}}{x^5\,{\left (1-x\right )}^{1/4}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^5} \, dx=\int \frac {\left (x +1\right )^{\frac {1}{4}}}{\left (1-x \right )^{\frac {1}{4}} x^{5}}d x \] Input:

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

Output:

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