Integrand size = 15, antiderivative size = 101 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x} \, dx=\frac {3}{4} \left (a+b x^2\right )^{2/3}+\frac {1}{2} \sqrt {3} a^{2/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^2}}{\sqrt {3} \sqrt [3]{a}}\right )-\frac {1}{2} a^{2/3} \log (x)+\frac {3}{4} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \] Output:
3/4*(b*x^2+a)^(2/3)+1/2*3^(1/2)*a^(2/3)*arctan(1/3*(a^(1/3)+2*(b*x^2+a)^(1 /3))*3^(1/2)/a^(1/3))-1/2*a^(2/3)*ln(x)+3/4*a^(2/3)*ln(a^(1/3)-(b*x^2+a)^( 1/3))
Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x} \, dx=\frac {1}{4} \left (3 \left (a+b x^2\right )^{2/3}+2 \sqrt {3} a^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 a^{2/3} \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^2}\right )-a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}\right )\right ) \] Input:
Integrate[(a + b*x^2)^(2/3)/x,x]
Output:
(3*(a + b*x^2)^(2/3) + 2*Sqrt[3]*a^(2/3)*ArcTan[(1 + (2*(a + b*x^2)^(1/3)) /a^(1/3))/Sqrt[3]] + 2*a^(2/3)*Log[-a^(1/3) + (a + b*x^2)^(1/3)] - a^(2/3) *Log[a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3)])/4
Time = 0.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {243, 60, 67, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^{2/3}}{x} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^{2/3}}{x^2}dx^2\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (a \int \frac {1}{x^2 \sqrt [3]{b x^2+a}}dx^2+\frac {3}{2} \left (a+b x^2\right )^{2/3}\right )\) |
\(\Big \downarrow \) 67 |
\(\displaystyle \frac {1}{2} \left (a \left (\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}\right )+\frac {3}{2} \left (a+b x^2\right )^{2/3}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{2} \left (a \left (\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}\right )+\frac {3}{2} \left (a+b x^2\right )^{2/3}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {1}{2} \left (a \left (-\frac {3 \int \frac {1}{-x^4-3}d\left (\frac {2 \sqrt [3]{b x^2+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}\right )+\frac {3}{2} \left (a+b x^2\right )^{2/3}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (a \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}\right )+\frac {3}{2} \left (a+b x^2\right )^{2/3}\right )\) |
Input:
Int[(a + b*x^2)^(2/3)/x,x]
Output:
((3*(a + b*x^2)^(2/3))/2 + a*((Sqrt[3]*ArcTan[(1 + (2*(a + b*x^2)^(1/3))/a ^(1/3))/Sqrt[3]])/a^(1/3) - Log[x^2]/(2*a^(1/3)) + (3*Log[a^(1/3) - (a + b *x^2)^(1/3)])/(2*a^(1/3))))/2
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_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) 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 = 0.49 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(\frac {3 \left (b \,x^{2}+a \right )^{\frac {2}{3}}}{4}+\frac {a^{\frac {2}{3}} \ln \left (\left (b \,x^{2}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{2}-\frac {a^{\frac {2}{3}} \ln \left (a^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{2}+a \right )^{\frac {1}{3}}+\left (b \,x^{2}+a \right )^{\frac {2}{3}}\right )}{4}+\frac {a^{\frac {2}{3}} \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (b \,x^{2}+a \right )^{\frac {1}{3}}}{3 a^{\frac {1}{3}}}+\frac {\sqrt {3}}{3}\right )}{2}\) | \(98\) |
Input:
int((b*x^2+a)^(2/3)/x,x,method=_RETURNVERBOSE)
Output:
3/4*(b*x^2+a)^(2/3)+1/2*a^(2/3)*ln((b*x^2+a)^(1/3)-a^(1/3))-1/4*a^(2/3)*ln (a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b*x^2+a)^(2/3))+1/2*a^(2/3)*3^(1/2)*arct an(2/3*3^(1/2)/a^(1/3)*(b*x^2+a)^(1/3)+1/3*3^(1/2))
Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x} \, dx=\frac {1}{2} \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} a + 2 \, \sqrt {3} {\left (b x^{2} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {1}{3}}}{3 \, a}\right ) - \frac {1}{4} \, {\left (a^{2}\right )}^{\frac {1}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} a + {\left (a^{2}\right )}^{\frac {1}{3}} a + {\left (b x^{2} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + \frac {1}{2} \, {\left (a^{2}\right )}^{\frac {1}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + \frac {3}{4} \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} \] Input:
integrate((b*x^2+a)^(2/3)/x,x, algorithm="fricas")
Output:
1/2*sqrt(3)*(a^2)^(1/3)*arctan(1/3*(sqrt(3)*a + 2*sqrt(3)*(b*x^2 + a)^(1/3 )*(a^2)^(1/3))/a) - 1/4*(a^2)^(1/3)*log((b*x^2 + a)^(2/3)*a + (a^2)^(1/3)* a + (b*x^2 + a)^(1/3)*(a^2)^(2/3)) + 1/2*(a^2)^(1/3)*log((b*x^2 + a)^(1/3) *a - (a^2)^(2/3)) + 3/4*(b*x^2 + a)^(2/3)
Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.46 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x} \, dx=- \frac {b^{\frac {2}{3}} x^{\frac {4}{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 \Gamma \left (\frac {1}{3}\right )} \] Input:
integrate((b*x**2+a)**(2/3)/x,x)
Output:
-b**(2/3)*x**(4/3)*gamma(-2/3)*hyper((-2/3, -2/3), (1/3,), a*exp_polar(I*p i)/(b*x**2))/(2*gamma(1/3))
Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x} \, dx=\frac {1}{2} \, \sqrt {3} a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{4} \, a^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + \frac {1}{2} \, a^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {3}{4} \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} \] Input:
integrate((b*x^2+a)^(2/3)/x,x, algorithm="maxima")
Output:
1/2*sqrt(3)*a^(2/3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^( 1/3)) - 1/4*a^(2/3)*log((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^ (2/3)) + 1/2*a^(2/3)*log((b*x^2 + a)^(1/3) - a^(1/3)) + 3/4*(b*x^2 + a)^(2 /3)
Time = 0.43 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x} \, dx=\frac {1}{2} \, \sqrt {3} a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{4} \, a^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + \frac {1}{2} \, a^{\frac {2}{3}} \log \left ({\left | {\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) + \frac {3}{4} \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} \] Input:
integrate((b*x^2+a)^(2/3)/x,x, algorithm="giac")
Output:
1/2*sqrt(3)*a^(2/3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^( 1/3)) - 1/4*a^(2/3)*log((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^ (2/3)) + 1/2*a^(2/3)*log(abs((b*x^2 + a)^(1/3) - a^(1/3))) + 3/4*(b*x^2 + a)^(2/3)
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x} \, dx=\frac {3\,{\left (b\,x^2+a\right )}^{2/3}}{4}+\frac {a^{2/3}\,\ln \left (\frac {9\,a^2\,{\left (b\,x^2+a\right )}^{1/3}}{4}-\frac {9\,a^{7/3}}{4}\right )}{2}-\frac {a^{2/3}\,\ln \left (\frac {9\,a^2\,{\left (b\,x^2+a\right )}^{1/3}}{4}-\frac {9\,a^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{4}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2}+a^{2/3}\,\ln \left (\frac {9\,a^2\,{\left (b\,x^2+a\right )}^{1/3}}{4}-9\,a^{7/3}\,{\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}^2\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right ) \] Input:
int((a + b*x^2)^(2/3)/x,x)
Output:
(3*(a + b*x^2)^(2/3))/4 + (a^(2/3)*log((9*a^2*(a + b*x^2)^(1/3))/4 - (9*a^ (7/3))/4))/2 - (a^(2/3)*log((9*a^2*(a + b*x^2)^(1/3))/4 - (9*a^(7/3)*((3^( 1/2)*1i)/2 + 1/2)^2)/4)*((3^(1/2)*1i)/2 + 1/2))/2 + a^(2/3)*log((9*a^2*(a + b*x^2)^(1/3))/4 - 9*a^(7/3)*((3^(1/2)*1i)/4 - 1/4)^2)*((3^(1/2)*1i)/4 - 1/4)
\[ \int \frac {\left (a+b x^2\right )^{2/3}}{x} \, dx=\frac {3 \left (b \,x^{2}+a \right )^{\frac {2}{3}}}{4}+\left (\int \frac {\left (b \,x^{2}+a \right )^{\frac {2}{3}}}{b \,x^{3}+a x}d x \right ) a \] Input:
int((b*x^2+a)^(2/3)/x,x)
Output:
(3*(a + b*x**2)**(2/3) + 4*int((a + b*x**2)**(2/3)/(a*x + b*x**3),x)*a)/4