Integrand size = 17, antiderivative size = 85 \[ \int \frac {1}{x \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^{3/2}}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{a^{2/3}} \]
-1/2*ln(x)/a^(2/3)+ln(a^(1/3)-(a+b*x^(3/2))^(1/3))/a^(2/3)-2/3*arctan(1/3* (a^(1/3)+2*(a+b*x^(3/2))^(1/3))/a^(1/3)*3^(1/2))/a^(2/3)*3^(1/2)
Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^{3/2}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^{3/2}}\right )+\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^{3/2}}+\left (a+b x^{3/2}\right )^{2/3}\right )}{3 a^{2/3}} \]
-1/3*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^(3/2))^(1/3))/a^(1/3))/Sqrt[3]] - 2*Log[-a^(1/3) + (a + b*x^(3/2))^(1/3)] + Log[a^(2/3) + a^(1/3)*(a + b*x^( 3/2))^(1/3) + (a + b*x^(3/2))^(2/3)])/a^(2/3)
Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {798, 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.
\(\displaystyle \int \frac {1}{x \left (a+b x^{3/2}\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {2}{3} \int \frac {1}{x^{3/2} \left (b x^{3/2}+a\right )^{2/3}}dx^{3/2}\) |
\(\Big \downarrow \) 69 |
\(\displaystyle \frac {2}{3} \left (-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^{3/2}+a}}d\sqrt [3]{b x^{3/2}+a}}{2 a^{2/3}}-\frac {3 \int \frac {1}{x^3+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^{3/2}+a}}d\sqrt [3]{b x^{3/2}+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^{3/2}\right )}{2 a^{2/3}}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {2}{3} \left (-\frac {3 \int \frac {1}{x^3+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^{3/2}+a}}d\sqrt [3]{b x^{3/2}+a}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{2 a^{2/3}}-\frac {\log \left (x^{3/2}\right )}{2 a^{2/3}}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {2}{3} \left (\frac {3 \int \frac {1}{-x^3-3}d\left (\frac {2 \sqrt [3]{b x^{3/2}+a}}{\sqrt [3]{a}}+1\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{2 a^{2/3}}-\frac {\log \left (x^{3/2}\right )}{2 a^{2/3}}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {2}{3} \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^{3/2}}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{2 a^{2/3}}-\frac {\log \left (x^{3/2}\right )}{2 a^{2/3}}\right )\) |
(2*(-((Sqrt[3]*ArcTan[(1 + (2*(a + b*x^(3/2))^(1/3))/a^(1/3))/Sqrt[3]])/a^ (2/3)) - Log[x^(3/2)]/(2*a^(2/3)) + (3*Log[a^(1/3) - (a + b*x^(3/2))^(1/3) ])/(2*a^(2/3))))/3
3.23.79.3.1 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[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 = 5.74 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {2 \ln \left (\left (a +b \,x^{\frac {3}{2}}\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (a +b \,x^{\frac {3}{2}}\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +b \,x^{\frac {3}{2}}\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +b \,x^{\frac {3}{2}}\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\) | \(85\) |
default | \(\frac {2 \ln \left (\left (a +b \,x^{\frac {3}{2}}\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (a +b \,x^{\frac {3}{2}}\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +b \,x^{\frac {3}{2}}\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +b \,x^{\frac {3}{2}}\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\) | \(85\) |
2/3/a^(2/3)*ln((a+b*x^(3/2))^(1/3)-a^(1/3))-1/3/a^(2/3)*ln((a+b*x^(3/2))^( 2/3)+a^(1/3)*(a+b*x^(3/2))^(1/3)+a^(2/3))-2/3/a^(2/3)*3^(1/2)*arctan(1/3*3 ^(1/2)*(2/a^(1/3)*(a+b*x^(3/2))^(1/3)+1))
Timed out. \[ \int \frac {1}{x \left (a+b x^{3/2}\right )^{2/3}} \, dx=\text {Timed out} \]
Result contains complex when optimal does not.
Time = 0.67 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x \left (a+b x^{3/2}\right )^{2/3}} \, dx=- \frac {2 \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{\frac {3}{2}}}} \right )}}{3 b^{\frac {2}{3}} x \Gamma \left (\frac {5}{3}\right )} \]
-2*gamma(2/3)*hyper((2/3, 2/3), (5/3,), a*exp_polar(I*pi)/(b*x**(3/2)))/(3 *b**(2/3)*x*gamma(5/3))
Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {2}{3}}} - \frac {\log \left ({\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}} + {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{3 \, a^{\frac {2}{3}}} + \frac {2 \, \log \left ({\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{3 \, a^{\frac {2}{3}}} \]
-2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^(3/2) + a)^(1/3) + a^(1/3))/a^(1/3 ))/a^(2/3) - 1/3*log((b*x^(3/2) + a)^(2/3) + (b*x^(3/2) + a)^(1/3)*a^(1/3) + a^(2/3))/a^(2/3) + 2/3*log((b*x^(3/2) + a)^(1/3) - a^(1/3))/a^(2/3)
Time = 1.60 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {2}{3}}} - \frac {\log \left ({\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}} + {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{3 \, a^{\frac {2}{3}}} + \frac {2 \, \log \left ({\left | {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {2}{3}}} \]
-2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^(3/2) + a)^(1/3) + a^(1/3))/a^(1/3 ))/a^(2/3) - 1/3*log((b*x^(3/2) + a)^(2/3) + (b*x^(3/2) + a)^(1/3)*a^(1/3) + a^(2/3))/a^(2/3) + 2/3*log(abs((b*x^(3/2) + a)^(1/3) - a^(1/3)))/a^(2/3 )
Time = 6.01 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x \left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {2\,\ln \left (6\,{\left (a+b\,x^{3/2}\right )}^{1/3}-6\,a^{1/3}\right )}{3\,a^{2/3}}+\frac {\ln \left (3\,a^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )-6\,{\left (a+b\,x^{3/2}\right )}^{1/3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{3\,a^{2/3}}-\frac {\ln \left (3\,a^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )+6\,{\left (a+b\,x^{3/2}\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{3\,a^{2/3}} \]