Integrand size = 15, antiderivative size = 107 \[ \int \frac {\sqrt [3]{a+b x^2}}{x^3} \, dx=-\frac {\sqrt [3]{a+b x^2}}{2 x^2}-\frac {b \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^2}}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} a^{2/3}}-\frac {b \log (x)}{6 a^{2/3}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{2/3}} \]
-1/2*(b*x^2+a)^(1/3)/x^2-1/6*b*ln(x)/a^(2/3)+1/4*b*ln(a^(1/3)-(b*x^2+a)^(1 /3))/a^(2/3)-1/6*b*arctan(1/3*(a^(1/3)+2*(b*x^2+a)^(1/3))/a^(1/3)*3^(1/2)) /a^(2/3)*3^(1/2)
Time = 0.17 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt [3]{a+b x^2}}{x^3} \, dx=-\frac {6 a^{2/3} \sqrt [3]{a+b x^2}+2 \sqrt {3} b x^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b x^2 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^2}\right )+b x^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}\right )}{12 a^{2/3} x^2} \]
-1/12*(6*a^(2/3)*(a + b*x^2)^(1/3) + 2*Sqrt[3]*b*x^2*ArcTan[(1 + (2*(a + b *x^2)^(1/3))/a^(1/3))/Sqrt[3]] - 2*b*x^2*Log[-a^(1/3) + (a + b*x^2)^(1/3)] + b*x^2*Log[a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3)])/(a^ (2/3)*x^2)
Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {243, 51, 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.
\(\displaystyle \int \frac {\sqrt [3]{a+b x^2}}{x^3} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt [3]{b x^2+a}}{x^4}dx^2\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} b \int \frac {1}{x^2 \left (b x^2+a\right )^{2/3}}dx^2-\frac {\sqrt [3]{a+b x^2}}{x^2}\right )\) |
\(\Big \downarrow \) 69 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} b \left (-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}}{2 a^{2/3}}-\frac {3 \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 a^{2/3}}\right )-\frac {\sqrt [3]{a+b x^2}}{x^2}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} b \left (-\frac {3 \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 a^{2/3}}-\frac {\log \left (x^2\right )}{2 a^{2/3}}\right )-\frac {\sqrt [3]{a+b x^2}}{x^2}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} b \left (\frac {3 \int \frac {1}{-x^4-3}d\left (\frac {2 \sqrt [3]{b x^2+a}}{\sqrt [3]{a}}+1\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 a^{2/3}}-\frac {\log \left (x^2\right )}{2 a^{2/3}}\right )-\frac {\sqrt [3]{a+b x^2}}{x^2}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} b \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 a^{2/3}}-\frac {\log \left (x^2\right )}{2 a^{2/3}}\right )-\frac {\sqrt [3]{a+b x^2}}{x^2}\right )\) |
(-((a + b*x^2)^(1/3)/x^2) + (b*(-((Sqrt[3]*ArcTan[(1 + (2*(a + b*x^2)^(1/3 ))/a^(1/3))/Sqrt[3]])/a^(2/3)) - Log[x^2]/(2*a^(2/3)) + (3*Log[a^(1/3) - ( a + b*x^2)^(1/3)])/(2*a^(2/3))))/3)/2
3.7.70.3.1 Defintions of rubi rules used
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 + 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]
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_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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 = 2.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.04
method | result | size |
pseudoelliptic | \(\frac {-b \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{2}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\, x^{2}+b \ln \left (\left (b \,x^{2}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) x^{2}-\frac {b \ln \left (a^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{2}+a \right )^{\frac {1}{3}}+\left (b \,x^{2}+a \right )^{\frac {2}{3}}\right ) x^{2}}{2}-3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} a^{\frac {2}{3}}}{6 a^{\frac {2}{3}} x^{2}}\) | \(111\) |
1/6/a^(2/3)*(-b*arctan(1/3*(a^(1/3)+2*(b*x^2+a)^(1/3))/a^(1/3)*3^(1/2))*3^ (1/2)*x^2+b*ln((b*x^2+a)^(1/3)-a^(1/3))*x^2-1/2*b*ln(a^(2/3)+a^(1/3)*(b*x^ 2+a)^(1/3)+(b*x^2+a)^(2/3))*x^2-3*(b*x^2+a)^(1/3)*a^(2/3))/x^2
Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt [3]{a+b x^2}}{x^3} \, dx=-\frac {2 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{6}} a b x^{2} \arctan \left (\frac {{\left (a^{2}\right )}^{\frac {1}{6}} {\left (\sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} a + 2 \, \sqrt {3} {\left (b x^{2} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}\right )}}{3 \, a^{2}}\right ) + {\left (a^{2}\right )}^{\frac {2}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} a + {\left (a^{2}\right )}^{\frac {1}{3}} a + {\left (b x^{2} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}\right ) - 2 \, {\left (a^{2}\right )}^{\frac {2}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + 6 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{2}}{12 \, a^{2} x^{2}} \]
-1/12*(2*sqrt(3)*(a^2)^(1/6)*a*b*x^2*arctan(1/3*(a^2)^(1/6)*(sqrt(3)*(a^2) ^(1/3)*a + 2*sqrt(3)*(b*x^2 + a)^(1/3)*(a^2)^(2/3))/a^2) + (a^2)^(2/3)*b*x ^2*log((b*x^2 + a)^(2/3)*a + (a^2)^(1/3)*a + (b*x^2 + a)^(1/3)*(a^2)^(2/3) ) - 2*(a^2)^(2/3)*b*x^2*log((b*x^2 + a)^(1/3)*a - (a^2)^(2/3)) + 6*(b*x^2 + a)^(1/3)*a^2)/(a^2*x^2)
Result contains complex when optimal does not.
Time = 0.72 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt [3]{a+b x^2}}{x^3} \, dx=- \frac {\sqrt [3]{b} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{\frac {4}{3}} \Gamma \left (\frac {5}{3}\right )} \]
-b**(1/3)*gamma(2/3)*hyper((-1/3, 2/3), (5/3,), a*exp_polar(I*pi)/(b*x**2) )/(2*x**(4/3)*gamma(5/3))
Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{a+b x^2}}{x^3} \, dx=-\frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{6 \, a^{\frac {2}{3}}} - \frac {b \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{12 \, a^{\frac {2}{3}}} + \frac {b \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{6 \, a^{\frac {2}{3}}} - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}}}{2 \, x^{2}} \]
-1/6*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3)) /a^(2/3) - 1/12*b*log((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2 /3))/a^(2/3) + 1/6*b*log((b*x^2 + a)^(1/3) - a^(1/3))/a^(2/3) - 1/2*(b*x^2 + a)^(1/3)/x^2
Time = 0.49 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt [3]{a+b x^2}}{x^3} \, dx=-\frac {\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {2}{3}}} + \frac {b^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {2}{3}}} - \frac {2 \, b^{2} \log \left ({\left | {\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {2}{3}}} + \frac {6 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} b}{x^{2}}}{12 \, b} \]
-1/12*(2*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^ (1/3))/a^(2/3) + b^2*log((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a ^(2/3))/a^(2/3) - 2*b^2*log(abs((b*x^2 + a)^(1/3) - a^(1/3)))/a^(2/3) + 6* (b*x^2 + a)^(1/3)*b/x^2)/b
Time = 5.01 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt [3]{a+b x^2}}{x^3} \, dx=\frac {b\,\ln \left (\frac {3\,b\,{\left (b\,x^2+a\right )}^{1/3}}{2}-\frac {3\,a^{1/3}\,b}{2}\right )}{6\,a^{2/3}}-\frac {{\left (b\,x^2+a\right )}^{1/3}}{2\,x^2}-\frac {\ln \left (\frac {3\,a^{1/3}\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{4}+\frac {3\,b\,{\left (b\,x^2+a\right )}^{1/3}}{2}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{12\,a^{2/3}}-\frac {\ln \left (\frac {3\,a^{1/3}\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{4}+\frac {3\,b\,{\left (b\,x^2+a\right )}^{1/3}}{2}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{12\,a^{2/3}} \]
(b*log((3*b*(a + b*x^2)^(1/3))/2 - (3*a^(1/3)*b)/2))/(6*a^(2/3)) - (a + b* x^2)^(1/3)/(2*x^2) - (log((3*a^(1/3)*(b - 3^(1/2)*b*1i))/4 + (3*b*(a + b*x ^2)^(1/3))/2)*(b - 3^(1/2)*b*1i))/(12*a^(2/3)) - (log((3*a^(1/3)*(b + 3^(1 /2)*b*1i))/4 + (3*b*(a + b*x^2)^(1/3))/2)*(b + 3^(1/2)*b*1i))/(12*a^(2/3))