\(\int \frac {\sqrt [3]{a+b x^n}}{x} \, dx\) [2517]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 52, 59, 631, 210, 31} \[ \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx=-\frac {\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {2 \sqrt [3]{a+b x^n}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{n}+\frac {3 \sqrt [3]{a+b x^n}}{n}+\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^n}\right )}{2 n}-\frac {1}{2} \sqrt [3]{a} \log (x) \]

[In]

Int[(a + b*x^n)^(1/3)/x,x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 59

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[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 210

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{x} \, dx,x,x^n\right )}{n} \\ & = \frac {3 \sqrt [3]{a+b x^n}}{n}+\frac {a \text {Subst}\left (\int \frac {1}{x (a+b x)^{2/3}} \, dx,x,x^n\right )}{n} \\ & = \frac {3 \sqrt [3]{a+b x^n}}{n}-\frac {1}{2} \sqrt [3]{a} \log (x)-\frac {\left (3 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^n}\right )}{2 n}-\frac {\left (3 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^n}\right )}{2 n} \\ & = \frac {3 \sqrt [3]{a+b x^n}}{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}+\frac {\left (3 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^n}}{\sqrt [3]{a}}\right )}{n} \\ & = \frac {3 \sqrt [3]{a+b x^n}}{n}-\frac {\sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^n}}{\sqrt [3]{a}}}{\sqrt {3}}\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx=\frac {6 \sqrt [3]{a+b x^n}-2 \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^n}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^n}\right )-\sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^n}+\left (a+b x^n\right )^{2/3}\right )}{2 n} \]

[In]

Integrate[(a + b*x^n)^(1/3)/x,x]

[Out]

(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)

Maple [A] (verified)

Time = 3.71 (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	\(\frac {3 \left (a +b \,x^{n}\right )^{\frac {1}{3}}+3 \left (\frac {\ln \left (\left (a +b \,x^{n}\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (a +b \,x^{n}\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +b \,x^{n}\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +b \,x^{n}\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\right ) a}{n}\)	\(104\)

[In]

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

[Out]

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)

none

Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx=-\frac {2 \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x^{n} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + a^{\frac {1}{3}} \log \left ({\left (b x^{n} + a\right )}^{\frac {2}{3}} + {\left (b x^{n} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 2 \, a^{\frac {1}{3}} \log \left ({\left (b x^{n} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 6 \, {\left (b x^{n} + a\right )}^{\frac {1}{3}}}{2 \, n} \]

[In]

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

[Out]

-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 = 0.74 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx=- \frac {\sqrt [3]{b} x^{\frac {n}{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {a x^{- n} e^{i \pi }}{b}} \right )}}{n \Gamma \left (\frac {2}{3}\right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt [3]{a+b x^n}}{x} \, dx=-\frac {\sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{n} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{n} - \frac {a^{\frac {1}{3}} \log \left ({\left (b x^{n} + a\right )}^{\frac {2}{3}} + {\left (b x^{n} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{2 \, n} + \frac {a^{\frac {1}{3}} \log \left ({\left (b x^{n} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{n} + \frac {3 \, {\left (b x^{n} + a\right )}^{\frac {1}{3}}}{n} \]

[In]

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

[Out]

-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]

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

[In]

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

[Out]

integrate((b*x^n + a)^(1/3)/x, x)

Mupad [F(-1)]

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

[In]

int((a + b*x^n)^(1/3)/x,x)

[Out]

int((a + b*x^n)^(1/3)/x, x)