Integrand size = 15, antiderivative size = 104 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x^3} \, dx=-\frac {\left (a+b x^2\right )^{2/3}}{2 x^2}+\frac {b \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^2}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a}}-\frac {b \log (x)}{3 \sqrt [3]{a}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}} \] Output:
-1/2*(b*x^2+a)^(2/3)/x^2+1/3*b*arctan(1/3*(a^(1/3)+2*(b*x^2+a)^(1/3))*3^(1 /2)/a^(1/3))*3^(1/2)/a^(1/3)-1/3*b*ln(x)/a^(1/3)+1/2*b*ln(a^(1/3)-(b*x^2+a )^(1/3))/a^(1/3)
Time = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x^3} \, dx=-\frac {\left (a+b x^2\right )^{2/3}}{2 x^2}+\frac {b \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a}}+\frac {b \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^2}\right )}{3 \sqrt [3]{a}}-\frac {b \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}\right )}{6 \sqrt [3]{a}} \] Input:
Integrate[(a + b*x^2)^(2/3)/x^3,x]
Output:
-1/2*(a + b*x^2)^(2/3)/x^2 + (b*ArcTan[1/Sqrt[3] + (2*(a + b*x^2)^(1/3))/( Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)) + (b*Log[-a^(1/3) + (a + b*x^2)^(1/3) ])/(3*a^(1/3)) - (b*Log[a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^ (2/3)])/(6*a^(1/3))
Time = 0.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {243, 51, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{2/3}}{x^3} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^{2/3}}{x^4}dx^2\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {2}{3} b \int \frac {1}{x^2 \sqrt [3]{b x^2+a}}dx^2-\frac {\left (a+b x^2\right )^{2/3}}{x^2}\right )\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle \frac {1}{2} \left (\frac {2}{3} b \left (\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}\right )-\frac {\left (a+b x^2\right )^{2/3}}{x^2}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (\frac {2}{3} b \left (\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}\right )-\frac {\left (a+b x^2\right )^{2/3}}{x^2}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\frac {2}{3} b \left (-\frac {3 \int \frac {1}{-x^4-3}d\left (\frac {2 \sqrt [3]{b x^2+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}\right )-\frac {\left (a+b x^2\right )^{2/3}}{x^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {2}{3} b \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}\right )-\frac {\left (a+b x^2\right )^{2/3}}{x^2}\right )\) |
Input:
Int[(a + b*x^2)^(2/3)/x^3,x]
Output:
(-((a + b*x^2)^(2/3)/x^2) + (2*b*((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))))/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 + 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_))^(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 = 0.54 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.08
| method | result | size |
| pseudoelliptic | \(\frac {2 b \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (b \,x^{2}+a \right )^{\frac {1}{3}}}{3 a^{\frac {1}{3}}}+\frac {\sqrt {3}}{3}\right ) x^{2}+2 b \ln \left (\left (b \,x^{2}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) x^{2}-b \ln \left (a^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{2}+a \right )^{\frac {1}{3}}+\left (b \,x^{2}+a \right )^{\frac {2}{3}}\right ) x^{2}-3 \left (b \,x^{2}+a \right )^{\frac {2}{3}} a^{\frac {1}{3}}}{6 x^{2} a^{\frac {1}{3}}}\) | \(112\) |
Input:
int((b*x^2+a)^(2/3)/x^3,x,method=_RETURNVERBOSE)
Output:
1/6*(2*b*3^(1/2)*arctan(2/3*3^(1/2)/a^(1/3)*(b*x^2+a)^(1/3)+1/3*3^(1/2))*x ^2+2*b*ln((b*x^2+a)^(1/3)-a^(1/3))*x^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)^(2/3)*a^(1/3))/x^2/a^(1/3)
Time = 0.08 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.79 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x^3} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b x^{2} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x^{2} + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x^{2}}\right ) - a^{\frac {2}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, a^{\frac {2}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 3 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a}{6 \, a x^{2}}, \frac {6 \, \sqrt {\frac {1}{3}} a^{\frac {2}{3}} b x^{2} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{a^{\frac {1}{3}}}\right ) - a^{\frac {2}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, a^{\frac {2}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 3 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a}{6 \, a x^{2}}\right ] \] Input:
integrate((b*x^2+a)^(2/3)/x^3,x, algorithm="fricas")
Output:
[1/6*(3*sqrt(1/3)*a*b*x^2*sqrt(-1/a^(2/3))*log((2*b*x^2 + 3*sqrt(1/3)*(2*( b*x^2 + a)^(2/3)*a^(2/3) - (b*x^2 + a)^(1/3)*a - a^(4/3))*sqrt(-1/a^(2/3)) - 3*(b*x^2 + a)^(1/3)*a^(2/3) + 3*a)/x^2) - a^(2/3)*b*x^2*log((b*x^2 + a) ^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3)) + 2*a^(2/3)*b*x^2*log((b*x^2 + a)^(1/3) - a^(1/3)) - 3*(b*x^2 + a)^(2/3)*a)/(a*x^2), 1/6*(6*sqrt(1/3)* a^(2/3)*b*x^2*arctan(sqrt(1/3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3)) - a^(2/3)*b*x^2*log((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3)) + 2*a^(2/3)*b*x^2*log((b*x^2 + a)^(1/3) - a^(1/3)) - 3*(b*x^2 + a)^(2/3)* a)/(a*x^2)]
Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.40 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x^3} \, dx=- \frac {b^{\frac {2}{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} \] Input:
integrate((b*x**2+a)**(2/3)/x**3,x)
Output:
-b**(2/3)*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), a*exp_polar(I*pi)/(b*x**2) )/(2*x**(2/3)*gamma(4/3))
Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x^3} \, dx=\frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {1}{3}}} - \frac {b \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{6 \, a^{\frac {1}{3}}} + \frac {b \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{3 \, a^{\frac {1}{3}}} - \frac {{\left (b x^{2} + a\right )}^{\frac {2}{3}}}{2 \, x^{2}} \] Input:
integrate((b*x^2+a)^(2/3)/x^3,x, algorithm="maxima")
Output:
1/3*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))/ a^(1/3) - 1/6*b*log((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3 ))/a^(1/3) + 1/3*b*log((b*x^2 + a)^(1/3) - a^(1/3))/a^(1/3) - 1/2*(b*x^2 + a)^(2/3)/x^2
Time = 0.44 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x^3} \, dx=\frac {1}{6} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - \frac {\log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} + \frac {2 \, \log \left ({\left | {\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {1}{3}}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}}}{b x^{2}}\right )} b \] Input:
integrate((b*x^2+a)^(2/3)/x^3,x, algorithm="giac")
Output:
1/6*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3)) /a^(1/3) - log((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3))/a^ (1/3) + 2*log(abs((b*x^2 + a)^(1/3) - a^(1/3)))/a^(1/3) - 3*(b*x^2 + a)^(2 /3)/(b*x^2))*b
Time = 0.40 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a+b x^2\right )^{2/3}}{x^3} \, dx=\frac {b\,\ln \left (a^{1/3}\,b^2-b^2\,{\left (b\,x^2+a\right )}^{1/3}\right )}{3\,a^{1/3}}-\frac {{\left (b\,x^2+a\right )}^{2/3}}{2\,x^2}-\frac {\ln \left (\frac {a^{1/3}\,{\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{4}-b^2\,{\left (b\,x^2+a\right )}^{1/3}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{6\,a^{1/3}}-\frac {\ln \left (\frac {a^{1/3}\,{\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{4}-b^2\,{\left (b\,x^2+a\right )}^{1/3}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{6\,a^{1/3}} \] Input:
int((a + b*x^2)^(2/3)/x^3,x)
Output:
(b*log(a^(1/3)*b^2 - b^2*(a + b*x^2)^(1/3)))/(3*a^(1/3)) - (a + b*x^2)^(2/ 3)/(2*x^2) - (log((a^(1/3)*(b - 3^(1/2)*b*1i)^2)/4 - b^2*(a + b*x^2)^(1/3) )*(b - 3^(1/2)*b*1i))/(6*a^(1/3)) - (log((a^(1/3)*(b + 3^(1/2)*b*1i)^2)/4 - b^2*(a + b*x^2)^(1/3))*(b + 3^(1/2)*b*1i))/(6*a^(1/3))
\[ \int \frac {\left (a+b x^2\right )^{2/3}}{x^3} \, dx=\frac {-3 \left (b \,x^{2}+a \right )^{\frac {2}{3}}+4 \left (\int \frac {\left (b \,x^{2}+a \right )^{\frac {2}{3}}}{b \,x^{3}+a x}d x \right ) b \,x^{2}}{6 x^{2}} \] Input:
int((b*x^2+a)^(2/3)/x^3,x)
Output:
( - 3*(a + b*x**2)**(2/3) + 4*int((a + b*x**2)**(2/3)/(a*x + b*x**3),x)*b* x**2)/(6*x**2)