Integrand size = 15, antiderivative size = 59 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x} \, dx=-\frac {2}{3} a \sqrt {a+\frac {b}{x^3}}-\frac {2}{9} \left (a+\frac {b}{x^3}\right )^{3/2}+\frac {2}{3} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right ) \] Output:
-2/3*a*(a+b/x^3)^(1/2)-2/9*(a+b/x^3)^(3/2)+2/3*a^(3/2)*arctanh((a+b/x^3)^( 1/2)/a^(1/2))
Time = 0.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x} \, dx=\frac {2 \sqrt {a+\frac {b}{x^3}} \left (-\sqrt {b+a x^3} \left (b+4 a x^3\right )+3 a^{3/2} x^{9/2} \log \left (\sqrt {a} x^{3/2}+\sqrt {b+a x^3}\right )\right )}{9 x^3 \sqrt {b+a x^3}} \] Input:
Integrate[(a + b/x^3)^(3/2)/x,x]
Output:
(2*Sqrt[a + b/x^3]*(-(Sqrt[b + a*x^3]*(b + 4*a*x^3)) + 3*a^(3/2)*x^(9/2)*L og[Sqrt[a]*x^(3/2) + Sqrt[b + a*x^3]]))/(9*x^3*Sqrt[b + a*x^3])
Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 60, 60, 73, 221}
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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{3} \int \left (a+\frac {b}{x^3}\right )^{3/2} x^3d\frac {1}{x^3}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{3} \left (-a \int \sqrt {a+\frac {b}{x^3}} x^3d\frac {1}{x^3}-\frac {2}{3} \left (a+\frac {b}{x^3}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{3} \left (-a \left (a \int \frac {x^3}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x^3}+2 \sqrt {a+\frac {b}{x^3}}\right )-\frac {2}{3} \left (a+\frac {b}{x^3}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (-a \left (\frac {2 a \int \frac {1}{\frac {1}{b x^6}-\frac {a}{b}}d\sqrt {a+\frac {b}{x^3}}}{b}+2 \sqrt {a+\frac {b}{x^3}}\right )-\frac {2}{3} \left (a+\frac {b}{x^3}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (-a \left (2 \sqrt {a+\frac {b}{x^3}}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )\right )-\frac {2}{3} \left (a+\frac {b}{x^3}\right )^{3/2}\right )\) |
Input:
Int[(a + b/x^3)^(3/2)/x,x]
Output:
((-2*(a + b/x^3)^(3/2))/3 - a*(2*Sqrt[a + b/x^3] - 2*Sqrt[a]*ArcTanh[Sqrt[ a + b/x^3]/Sqrt[a]]))/3
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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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]]
Time = 0.45 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.46
method | result | size |
risch | \(-\frac {2 \left (4 a \,x^{3}+b \right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{9 x^{3}}+\frac {2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {x \left (a \,x^{3}+b \right )}}{x^{2} \sqrt {a}}\right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, x \sqrt {x \left (a \,x^{3}+b \right )}}{3 \left (a \,x^{3}+b \right )}\) | \(86\) |
default | \(\frac {2 \left (\frac {a \,x^{3}+b}{x^{3}}\right )^{\frac {3}{2}} \left (3 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {x \left (a \,x^{3}+b \right )}}{x^{2} \sqrt {a}}\right ) x^{5}-a \sqrt {a \,x^{4}+b x}\, x^{3}-3 a \,x^{3} \sqrt {x \left (a \,x^{3}+b \right )}-\sqrt {a \,x^{4}+b x}\, b \right )}{9 \left (a \,x^{3}+b \right ) \sqrt {x \left (a \,x^{3}+b \right )}}\) | \(112\) |
Input:
int((a+b/x^3)^(3/2)/x,x,method=_RETURNVERBOSE)
Output:
-2/9*(4*a*x^3+b)/x^3*((a*x^3+b)/x^3)^(1/2)+2/3*a^(3/2)*arctanh((x*(a*x^3+b ))^(1/2)/x^2/a^(1/2))*((a*x^3+b)/x^3)^(1/2)*x*(x*(a*x^3+b))^(1/2)/(a*x^3+b )
Time = 0.17 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.92 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x} \, dx=\left [\frac {3 \, a^{\frac {3}{2}} x^{3} \log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \, {\left (2 \, a x^{6} + b x^{3}\right )} \sqrt {a} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right ) - 4 \, {\left (4 \, a x^{3} + b\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{18 \, x^{3}}, -\frac {3 \, \sqrt {-a} a x^{3} \arctan \left (\frac {{\left (2 \, a x^{3} + b\right )} \sqrt {-a} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{2 \, {\left (a^{2} x^{3} + a b\right )}}\right ) + 2 \, {\left (4 \, a x^{3} + b\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{9 \, x^{3}}\right ] \] Input:
integrate((a+b/x^3)^(3/2)/x,x, algorithm="fricas")
Output:
[1/18*(3*a^(3/2)*x^3*log(-8*a^2*x^6 - 8*a*b*x^3 - b^2 - 4*(2*a*x^6 + b*x^3 )*sqrt(a)*sqrt((a*x^3 + b)/x^3)) - 4*(4*a*x^3 + b)*sqrt((a*x^3 + b)/x^3))/ x^3, -1/9*(3*sqrt(-a)*a*x^3*arctan(1/2*(2*a*x^3 + b)*sqrt(-a)*sqrt((a*x^3 + b)/x^3)/(a^2*x^3 + a*b)) + 2*(4*a*x^3 + b)*sqrt((a*x^3 + b)/x^3))/x^3]
Time = 1.49 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x} \, dx=- \frac {8 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}}{9} - \frac {a^{\frac {3}{2}} \log {\left (\frac {b}{a x^{3}} \right )}}{3} + \frac {2 a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {b}{a x^{3}}} + 1 \right )}}{3} - \frac {2 \sqrt {a} b \sqrt {1 + \frac {b}{a x^{3}}}}{9 x^{3}} \] Input:
integrate((a+b/x**3)**(3/2)/x,x)
Output:
-8*a**(3/2)*sqrt(1 + b/(a*x**3))/9 - a**(3/2)*log(b/(a*x**3))/3 + 2*a**(3/ 2)*log(sqrt(1 + b/(a*x**3)) + 1)/3 - 2*sqrt(a)*b*sqrt(1 + b/(a*x**3))/(9*x **3)
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x} \, dx=-\frac {1}{3} \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x^{3}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{3}}} + \sqrt {a}}\right ) - \frac {2}{9} \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {a + \frac {b}{x^{3}}} a \] Input:
integrate((a+b/x^3)^(3/2)/x,x, algorithm="maxima")
Output:
-1/3*a^(3/2)*log((sqrt(a + b/x^3) - sqrt(a))/(sqrt(a + b/x^3) + sqrt(a))) - 2/9*(a + b/x^3)^(3/2) - 2/3*sqrt(a + b/x^3)*a
Time = 0.18 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x} \, dx=-\frac {2 \, a^{2} \arctan \left (\frac {\sqrt {a + \frac {b}{x^{3}}}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} - \frac {2}{9} \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {a + \frac {b}{x^{3}}} a \] Input:
integrate((a+b/x^3)^(3/2)/x,x, algorithm="giac")
Output:
-2/3*a^2*arctan(sqrt(a + b/x^3)/sqrt(-a))/sqrt(-a) - 2/9*(a + b/x^3)^(3/2) - 2/3*sqrt(a + b/x^3)*a
Time = 0.90 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x} \, dx=\frac {2\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3}-\frac {2\,a\,\sqrt {a+\frac {b}{x^3}}}{3}-\frac {2\,{\left (a+\frac {b}{x^3}\right )}^{3/2}}{9} \] Input:
int((a + b/x^3)^(3/2)/x,x)
Output:
(2*a^(3/2)*atanh((a + b/x^3)^(1/2)/a^(1/2)))/3 - (2*a*(a + b/x^3)^(1/2))/3 - (2*(a + b/x^3)^(3/2))/9
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x} \, dx=\frac {-8 \sqrt {a \,x^{3}+b}\, a \,x^{3}-2 \sqrt {a \,x^{3}+b}\, b -3 \sqrt {x}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a \,x^{3}+b}-\sqrt {x}\, \sqrt {a}\, x \right ) a \,x^{4}+3 \sqrt {x}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a \,x^{3}+b}+\sqrt {x}\, \sqrt {a}\, x \right ) a \,x^{4}}{9 \sqrt {x}\, x^{4}} \] Input:
int((a+b/x^3)^(3/2)/x,x)
Output:
( - 8*sqrt(a*x**3 + b)*a*x**3 - 2*sqrt(a*x**3 + b)*b - 3*sqrt(x)*sqrt(a)*l og(sqrt(a*x**3 + b) - sqrt(x)*sqrt(a)*x)*a*x**4 + 3*sqrt(x)*sqrt(a)*log(sq rt(a*x**3 + b) + sqrt(x)*sqrt(a)*x)*a*x**4)/(9*sqrt(x)*x**4)