\(\int \frac {(1+x^5)^{2/3}}{x^6} \, dx\) [1242]

Optimal result

Integrand size = 13, antiderivative size = 90 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=-\frac {\left (1+x^5\right )^{2/3}}{5 x^5}+\frac {2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^5}}{\sqrt {3}}\right )}{5 \sqrt {3}}+\frac {2}{15} \log \left (-1+\sqrt [3]{1+x^5}\right )-\frac {1}{15} \log \left (1+\sqrt [3]{1+x^5}+\left (1+x^5\right )^{2/3}\right ) \] Output:

-1/5*(x^5+1)^(2/3)/x^5+2/15*3^(1/2)*arctan(1/3*3^(1/2)+2/3*(x^5+1)^(1/3)*3 
^(1/2))+2/15*ln(-1+(x^5+1)^(1/3))-1/15*ln(1+(x^5+1)^(1/3)+(x^5+1)^(2/3))
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=\frac {1}{15} \left (-\frac {3 \left (1+x^5\right )^{2/3}}{x^5}+2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^5}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1+x^5}\right )-\log \left (1+\sqrt [3]{1+x^5}+\left (1+x^5\right )^{2/3}\right )\right ) \] Input:

Integrate[(1 + x^5)^(2/3)/x^6,x]
 

Output:

((-3*(1 + x^5)^(2/3))/x^5 + 2*Sqrt[3]*ArcTan[(1 + 2*(1 + x^5)^(1/3))/Sqrt[ 
3]] + 2*Log[-1 + (1 + x^5)^(1/3)] - Log[1 + (1 + x^5)^(1/3) + (1 + x^5)^(2 
/3)])/15
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {798, 51, 67, 16, 1083, 217}

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.

\(\displaystyle \int \frac {\left (x^5+1\right )^{2/3}}{x^6} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {1}{5} \int \frac {\left (x^5+1\right )^{2/3}}{x^{10}}dx^5\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{5} \left (\frac {2}{3} \int \frac {1}{x^5 \sqrt [3]{x^5+1}}dx^5-\frac {\left (x^5+1\right )^{2/3}}{x^5}\right )\)

\(\Big \downarrow \) 67

\(\displaystyle \frac {1}{5} \left (\frac {2}{3} \left (-\frac {3}{2} \int \frac {1}{1-\sqrt [3]{x^5+1}}d\sqrt [3]{x^5+1}+\frac {3}{2} \int \frac {1}{x^{10}+\sqrt [3]{x^5+1}+1}d\sqrt [3]{x^5+1}-\frac {1}{2} \log \left (x^5\right )\right )-\frac {\left (x^5+1\right )^{2/3}}{x^5}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{5} \left (\frac {2}{3} \left (\frac {3}{2} \int \frac {1}{x^{10}+\sqrt [3]{x^5+1}+1}d\sqrt [3]{x^5+1}-\frac {1}{2} \log \left (x^5\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^5+1}\right )\right )-\frac {\left (x^5+1\right )^{2/3}}{x^5}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{5} \left (\frac {2}{3} \left (-3 \int \frac {1}{-x^{10}-3}d\left (2 \sqrt [3]{x^5+1}+1\right )-\frac {1}{2} \log \left (x^5\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^5+1}\right )\right )-\frac {\left (x^5+1\right )^{2/3}}{x^5}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{5} \left (\frac {2}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^5+1}+1}{\sqrt {3}}\right )-\frac {\log \left (x^5\right )}{2}+\frac {3}{2} \log \left (1-\sqrt [3]{x^5+1}\right )\right )-\frac {\left (x^5+1\right )^{2/3}}{x^5}\right )\)

Input:

Int[(1 + x^5)^(2/3)/x^6,x]
 

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 51

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

rule 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

rule 217

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

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3.

Time = 3.83 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84

method	result	size
meijerg	\(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {\pi \sqrt {3}\, x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], -x^{5}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+5 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{5}}\right )}{15 \pi }\)	\(76\)
risch	\(-\frac {\left (x^{5}+1\right )^{\frac {2}{3}}}{5 x^{5}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{5}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+5 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{15 \pi }\)	\(76\)
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{5}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right ) x^{5}-\ln \left (1+\left (x^{5}+1\right )^{\frac {1}{3}}+\left (x^{5}+1\right )^{\frac {2}{3}}\right ) x^{5}+2 \ln \left (-1+\left (x^{5}+1\right )^{\frac {1}{3}}\right ) x^{5}-3 \left (x^{5}+1\right )^{\frac {2}{3}}}{15 \left (1+\left (x^{5}+1\right )^{\frac {1}{3}}+\left (x^{5}+1\right )^{\frac {2}{3}}\right ) \left (-1+\left (x^{5}+1\right )^{\frac {1}{3}}\right )}\)	\(104\)
trager	\(-\frac {\left (x^{5}+1\right )^{\frac {2}{3}}}{5 x^{5}}+\frac {2 \ln \left (\frac {-115344 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}-5619 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}+55010 x^{5}+137820 \left (x^{5}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-78769 \left (x^{5}+1\right )^{\frac {2}{3}}-374127 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {1}{3}}+115344 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-45940 \left (x^{5}+1\right )^{\frac {1}{3}}+274755 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+137525}{x^{5}}\right )}{15}+\frac {2 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {-247545 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}-285993 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}-38448 x^{5}+137820 \left (x^{5}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+124709 \left (x^{5}+1\right )^{\frac {2}{3}}+236307 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {1}{3}}+247545 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-45940 \left (x^{5}+1\right )^{\frac {1}{3}}-291612 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-51264}{x^{5}}\right )}{5}\)	\(294\)

Input:

int((x^5+1)^(2/3)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/15/Pi*3^(1/2)*GAMMA(2/3)*(1/9*Pi*3^(1/2)/GAMMA(2/3)*x^5*hypergeom([1,1, 
4/3],[2,3],-x^5)-2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)-1+5*ln(x))*Pi*3^(1/2)/GAMM 
A(2/3)+Pi*3^(1/2)/GAMMA(2/3)/x^5)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=\frac {2 \, \sqrt {3} x^{5} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{5} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - x^{5} \log \left ({\left (x^{5} + 1\right )}^{\frac {2}{3}} + {\left (x^{5} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, x^{5} \log \left ({\left (x^{5} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{15 \, x^{5}} \] Input:

integrate((x^5+1)^(2/3)/x^6,x, algorithm="fricas")
 

Output:

1/15*(2*sqrt(3)*x^5*arctan(2/3*sqrt(3)*(x^5 + 1)^(1/3) + 1/3*sqrt(3)) - x^ 
5*log((x^5 + 1)^(2/3) + (x^5 + 1)^(1/3) + 1) + 2*x^5*log((x^5 + 1)^(1/3) - 
 1) - 3*(x^5 + 1)^(2/3))/x^5
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.68 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.38 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{5}}} \right )}}{5 x^{\frac {5}{3}} \Gamma \left (\frac {4}{3}\right )} \] Input:

integrate((x**5+1)**(2/3)/x**6,x)
 

Output:

-gamma(1/3)*hyper((-2/3, 1/3), (4/3,), exp_polar(I*pi)/x**5)/(5*x**(5/3)*g 
amma(4/3))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=\frac {2}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{5} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{5} + 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} - \frac {1}{15} \, \log \left ({\left (x^{5} + 1\right )}^{\frac {2}{3}} + {\left (x^{5} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{15} \, \log \left ({\left (x^{5} + 1\right )}^{\frac {1}{3}} - 1\right ) \] Input:

integrate((x^5+1)^(2/3)/x^6,x, algorithm="maxima")
 

Output:

2/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^5 + 1)^(1/3) + 1)) - 1/5*(x^5 + 1)^( 
2/3)/x^5 - 1/15*log((x^5 + 1)^(2/3) + (x^5 + 1)^(1/3) + 1) + 2/15*log((x^5 
 + 1)^(1/3) - 1)
 

Giac [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=\frac {2}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{5} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{5} + 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} - \frac {1}{15} \, \log \left ({\left (x^{5} + 1\right )}^{\frac {2}{3}} + {\left (x^{5} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{15} \, \log \left ({\left | {\left (x^{5} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \] Input:

integrate((x^5+1)^(2/3)/x^6,x, algorithm="giac")
 

Output:

2/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^5 + 1)^(1/3) + 1)) - 1/5*(x^5 + 1)^( 
2/3)/x^5 - 1/15*log((x^5 + 1)^(2/3) + (x^5 + 1)^(1/3) + 1) + 2/15*log(abs( 
(x^5 + 1)^(1/3) - 1))
 

Mupad [B] (verification not implemented)

Time = 8.96 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=\frac {2\,\ln \left (\frac {4\,{\left (x^5+1\right )}^{1/3}}{25}-\frac {4}{25}\right )}{15}+\ln \left (\frac {4\,{\left (x^5+1\right )}^{1/3}}{25}-9\,{\left (-\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )}^2\right )\,\left (-\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )-\ln \left (\frac {4\,{\left (x^5+1\right )}^{1/3}}{25}-9\,{\left (\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )}^2\right )\,\left (\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )-\frac {{\left (x^5+1\right )}^{2/3}}{5\,x^5} \] Input:

int((x^5 + 1)^(2/3)/x^6,x)
                                                                                    
                                                                                    
 

Output:

(2*log((4*(x^5 + 1)^(1/3))/25 - 4/25))/15 + log((4*(x^5 + 1)^(1/3))/25 - 9 
*((3^(1/2)*1i)/15 - 1/15)^2)*((3^(1/2)*1i)/15 - 1/15) - log((4*(x^5 + 1)^( 
1/3))/25 - 9*((3^(1/2)*1i)/15 + 1/15)^2)*((3^(1/2)*1i)/15 + 1/15) - (x^5 + 
 1)^(2/3)/(5*x^5)
 

Reduce [F]

\[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=\frac {-3 \left (x^{5}+1\right )^{\frac {2}{3}}+10 \left (\int \frac {\left (x^{5}+1\right )^{\frac {2}{3}}}{x^{6}+x}d x \right ) x^{5}}{15 x^{5}} \] Input:

int((x^5+1)^(2/3)/x^6,x)
 

Output:

( - 3*(x**5 + 1)**(2/3) + 10*int((x**5 + 1)**(2/3)/(x**6 + x),x)*x**5)/(15 
*x**5)