\(\int \frac {\sqrt [3]{a+b x^n}}{x} \, dx\) [553]

Optimal result

Integrand size = 15, antiderivative size = 106 \[ \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx=\frac {3 \sqrt [3]{a+b x^n}}{n}-\frac {\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^n}}{\sqrt {3} \sqrt [3]{a}}\right )}{n}-\frac {1}{2} \sqrt [3]{a} \log (x)+\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^n}\right )}{2 n} \] Output:

3*(a+b*x^n)^(1/3)/n-3^(1/2)*a^(1/3)*arctan(1/3*(a^(1/3)+2*(a+b*x^n)^(1/3)) 
*3^(1/2)/a^(1/3))/n-1/2*a^(1/3)*ln(x)+3/2*a^(1/3)*ln(a^(1/3)-(a+b*x^n)^(1/ 
3))/n
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx=\frac {6 \sqrt [3]{a+b x^n}-2 \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^n}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^n}\right )-\sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^n}+\left (a+b x^n\right )^{2/3}\right )}{2 n} \] Input:

Integrate[(a + b*x^n)^(1/3)/x,x]
 

Output:

(6*(a + b*x^n)^(1/3) - 2*Sqrt[3]*a^(1/3)*ArcTan[(1 + (2*(a + b*x^n)^(1/3)) 
/a^(1/3))/Sqrt[3]] + 2*a^(1/3)*Log[a^(1/3) - (a + b*x^n)^(1/3)] - a^(1/3)* 
Log[a^(2/3) + a^(1/3)*(a + b*x^n)^(1/3) + (a + b*x^n)^(2/3)])/(2*n)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {798, 60, 69, 16, 1082, 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.

\(\displaystyle \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {\int x^{-n} \sqrt [3]{b x^n+a}dx^n}{n}\)

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 69

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

Int[(a + b*x^n)^(1/3)/x,x]
 

Output:

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

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 69

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), 
 x] + (-Simp[3/(2*b*q)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] - Simp[3/(2*b*q^2)   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 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {3 \left (a +b \,x^{n}\right )^{\frac {1}{3}}+3 \left (\frac {\ln \left (\left (a +b \,x^{n}\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (a +b \,x^{n}\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +b \,x^{n}\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +b \,x^{n}\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\right ) a}{n}\)	\(104\)
default	\(\frac {3 \left (a +b \,x^{n}\right )^{\frac {1}{3}}+3 \left (\frac {\ln \left (\left (a +b \,x^{n}\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (a +b \,x^{n}\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +b \,x^{n}\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +b \,x^{n}\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\right ) a}{n}\)	\(104\)

Input:

int((a+b*x^n)^(1/3)/x,x,method=_RETURNVERBOSE)
 

Output:

1/n*(3*(a+b*x^n)^(1/3)+3*(1/3/a^(2/3)*ln((a+b*x^n)^(1/3)-a^(1/3))-1/6/a^(2 
/3)*ln((a+b*x^n)^(2/3)+a^(1/3)*(a+b*x^n)^(1/3)+a^(2/3))-1/3/a^(2/3)*3^(1/2 
)*arctan(1/3*3^(1/2)*(2/a^(1/3)*(a+b*x^n)^(1/3)+1)))*a)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx=-\frac {2 \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x^{n} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + a^{\frac {1}{3}} \log \left ({\left (b x^{n} + a\right )}^{\frac {2}{3}} + {\left (b x^{n} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 2 \, a^{\frac {1}{3}} \log \left ({\left (b x^{n} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 6 \, {\left (b x^{n} + a\right )}^{\frac {1}{3}}}{2 \, n} \] Input:

integrate((a+b*x^n)^(1/3)/x,x, algorithm="fricas")
 

Output:

-1/2*(2*sqrt(3)*a^(1/3)*arctan(1/3*(2*sqrt(3)*(b*x^n + a)^(1/3)*a^(2/3) + 
sqrt(3)*a)/a) + a^(1/3)*log((b*x^n + a)^(2/3) + (b*x^n + a)^(1/3)*a^(1/3) 
+ a^(2/3)) - 2*a^(1/3)*log((b*x^n + a)^(1/3) - a^(1/3)) - 6*(b*x^n + a)^(1 
/3))/n
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.79 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx=- \frac {\sqrt [3]{b} x^{\frac {n}{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {a x^{- n} e^{i \pi }}{b}} \right )}}{n \Gamma \left (\frac {2}{3}\right )} \] Input:

integrate((a+b*x**n)**(1/3)/x,x)
 

Output:

-b**(1/3)*x**(n/3)*gamma(-1/3)*hyper((-1/3, -1/3), (2/3,), a*exp_polar(I*p 
i)/(b*x**n))/(n*gamma(2/3))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx=-\frac {\sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{n} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{n} - \frac {a^{\frac {1}{3}} \log \left ({\left (b x^{n} + a\right )}^{\frac {2}{3}} + {\left (b x^{n} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{2 \, n} + \frac {a^{\frac {1}{3}} \log \left ({\left (b x^{n} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{n} + \frac {3 \, {\left (b x^{n} + a\right )}^{\frac {1}{3}}}{n} \] Input:

integrate((a+b*x^n)^(1/3)/x,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

-sqrt(3)*a^(1/3)*arctan(1/3*sqrt(3)*(2*(b*x^n + a)^(1/3) + a^(1/3))/a^(1/3 
))/n - 1/2*a^(1/3)*log((b*x^n + a)^(2/3) + (b*x^n + a)^(1/3)*a^(1/3) + a^( 
2/3))/n + a^(1/3)*log((b*x^n + a)^(1/3) - a^(1/3))/n + 3*(b*x^n + a)^(1/3) 
/n
 

Giac [F]

\[ \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{\frac {1}{3}}}{x} \,d x } \] Input:

integrate((a+b*x^n)^(1/3)/x,x, algorithm="giac")
 

Output:

integrate((b*x^n + a)^(1/3)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx=\int \frac {{\left (a+b\,x^n\right )}^{1/3}}{x} \,d x \] Input:

int((a + b*x^n)^(1/3)/x,x)
 

Output:

int((a + b*x^n)^(1/3)/x, x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx=\frac {3 \left (x^{n} b +a \right )^{\frac {1}{3}}+\left (\int \frac {\left (x^{n} b +a \right )^{\frac {1}{3}}}{x^{n} b x +a x}d x \right ) a n}{n} \] Input:

int((a+b*x^n)^(1/3)/x,x)
 

Output:

(3*(x**n*b + a)**(1/3) + int((x**n*b + a)**(1/3)/(x**n*b*x + a*x),x)*a*n)/ 
n