Integrand size = 15, antiderivative size = 95 \[ \int \frac {\sqrt [3]{a+b x^3}}{x} \, dx=\sqrt [3]{a+b x^3}-\frac {\sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3}}-\frac {1}{2} \sqrt [3]{a} \log (x)+\frac {1}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right ) \] Output:
(b*x^3+a)^(1/3)-1/3*a^(1/3)*arctan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/3))*3^(1/2) /a^(1/3))*3^(1/2)-1/2*a^(1/3)*ln(x)+1/2*a^(1/3)*ln(a^(1/3)-(b*x^3+a)^(1/3) )
Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt [3]{a+b x^3}}{x} \, dx=\sqrt [3]{a+b x^3}-\frac {\sqrt [3]{a} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{a} \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )-\frac {1}{6} \sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right ) \] Input:
Integrate[(a + b*x^3)^(1/3)/x,x]
Output:
(a + b*x^3)^(1/3) - (a^(1/3)*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sq rt[3]])/Sqrt[3] + (a^(1/3)*Log[-a^(1/3) + (a + b*x^3)^(1/3)])/3 - (a^(1/3) *Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/6
Time = 0.38 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.07, 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.
\(\displaystyle \int \frac {\sqrt [3]{a+b x^3}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\sqrt [3]{b x^3+a}}{x^3}dx^3\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{3} \left (a \int \frac {1}{x^3 \left (b x^3+a\right )^{2/3}}dx^3+3 \sqrt [3]{a+b x^3}\right )\) |
\(\Big \downarrow \) 69 |
\(\displaystyle \frac {1}{3} \left (a \left (-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 a^{2/3}}-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{3} \left (a \left (-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {1}{3} \left (a \left (\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2 \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{3} \left (a \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )\) |
Input:
Int[(a + b*x^3)^(1/3)/x,x]
Output:
(3*(a + b*x^3)^(1/3) + a*(-((Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^( 1/3))/Sqrt[3]])/a^(2/3)) - Log[x^3]/(2*a^(2/3)) + (3*Log[a^(1/3) - (a + b* x^3)^(1/3)])/(2*a^(2/3))))/3
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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]
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]
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])
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]]
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]
Time = 1.85 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01
method | result | size |
pseudoelliptic | \(\left (b \,x^{3}+a \right )^{\frac {1}{3}}+\frac {a^{\frac {1}{3}} \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3}-\frac {a^{\frac {1}{3}} \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6}-\frac {a^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{3 a^{\frac {1}{3}}}+\frac {\sqrt {3}}{3}\right )}{3}\) | \(96\) |
Input:
int((b*x^3+a)^(1/3)/x,x,method=_RETURNVERBOSE)
Output:
(b*x^3+a)^(1/3)+1/3*a^(1/3)*ln((b*x^3+a)^(1/3)-a^(1/3))-1/6*a^(1/3)*ln((b* x^3+a)^(2/3)+a^(1/3)*(b*x^3+a)^(1/3)+a^(2/3))-1/3*a^(1/3)*3^(1/2)*arctan(2 /3*3^(1/2)/a^(1/3)*(b*x^3+a)^(1/3)+1/3*3^(1/2))
Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt [3]{a+b x^3}}{x} \, dx=-\frac {1}{3} \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) - \frac {1}{6} \, a^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + \frac {1}{3} \, a^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \] Input:
integrate((b*x^3+a)^(1/3)/x,x, algorithm="fricas")
Output:
-1/3*sqrt(3)*a^(1/3)*arctan(1/3*(2*sqrt(3)*(b*x^3 + a)^(1/3)*a^(2/3) + sqr t(3)*a)/a) - 1/6*a^(1/3)*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3)) + 1/3*a^(1/3)*log((b*x^3 + a)^(1/3) - a^(1/3)) + (b*x^3 + a)^( 1/3)
Result contains complex when optimal does not.
Time = 0.67 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt [3]{a+b x^3}}{x} \, dx=- \frac {\sqrt [3]{b} x \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{3}}} \right )}}{3 \Gamma \left (\frac {2}{3}\right )} \] Input:
integrate((b*x**3+a)**(1/3)/x,x)
Output:
-b**(1/3)*x*gamma(-1/3)*hyper((-1/3, -1/3), (2/3,), a*exp_polar(I*pi)/(b*x **3))/(3*gamma(2/3))
Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{a+b x^3}}{x} \, dx=-\frac {1}{3} \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{6} \, a^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + \frac {1}{3} \, a^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \] Input:
integrate((b*x^3+a)^(1/3)/x,x, algorithm="maxima")
Output:
-1/3*sqrt(3)*a^(1/3)*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^ (1/3)) - 1/6*a^(1/3)*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a ^(2/3)) + 1/3*a^(1/3)*log((b*x^3 + a)^(1/3) - a^(1/3)) + (b*x^3 + a)^(1/3)
Time = 0.43 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt [3]{a+b x^3}}{x} \, dx=-\frac {1}{3} \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{6} \, a^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + \frac {1}{3} \, a^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \] Input:
integrate((b*x^3+a)^(1/3)/x,x, algorithm="giac")
Output:
-1/3*sqrt(3)*a^(1/3)*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^ (1/3)) - 1/6*a^(1/3)*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a ^(2/3)) + 1/3*a^(1/3)*log(abs((b*x^3 + a)^(1/3) - a^(1/3))) + (b*x^3 + a)^ (1/3)
Time = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt [3]{a+b x^3}}{x} \, dx=\frac {a^{1/3}\,\ln \left (a\,{\left (b\,x^3+a\right )}^{1/3}-a^{4/3}\right )}{3}+{\left (b\,x^3+a\right )}^{1/3}-\frac {a^{1/3}\,\ln \left (3\,a^{4/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+3\,a\,{\left (b\,x^3+a\right )}^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3}+a^{1/3}\,\ln \left (3\,a\,{\left (b\,x^3+a\right )}^{1/3}-9\,a^{4/3}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \] Input:
int((a + b*x^3)^(1/3)/x,x)
Output:
(a^(1/3)*log(a*(a + b*x^3)^(1/3) - a^(4/3)))/3 + (a + b*x^3)^(1/3) - (a^(1 /3)*log(3*a^(4/3)*((3^(1/2)*1i)/2 + 1/2) + 3*a*(a + b*x^3)^(1/3))*((3^(1/2 )*1i)/2 + 1/2))/3 + a^(1/3)*log(3*a*(a + b*x^3)^(1/3) - 9*a^(4/3)*((3^(1/2 )*1i)/6 - 1/6))*((3^(1/2)*1i)/6 - 1/6)
\[ \int \frac {\sqrt [3]{a+b x^3}}{x} \, dx=\left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b \,x^{4}+a x}d x \right ) a \] Input:
int((b*x^3+a)^(1/3)/x,x)
Output:
(a + b*x**3)**(1/3) + int((a + b*x**3)**(1/3)/(a*x + b*x**4),x)*a