Integrand size = 15, antiderivative size = 59 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x} \, dx=-\frac {1}{2} a \sqrt {a+\frac {b}{x^4}}-\frac {1}{6} \left (a+\frac {b}{x^4}\right )^{3/2}+\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right ) \] Output:
-1/2*a*(a+b/x^4)^(1/2)-1/6*(a+b/x^4)^(3/2)+1/2*a^(3/2)*arctanh((a+b/x^4)^( 1/2)/a^(1/2))
Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x} \, dx=\frac {\sqrt {a+\frac {b}{x^4}} \left (-\sqrt {b+a x^4} \left (b+4 a x^4\right )+3 a^{3/2} x^6 \log \left (\sqrt {a} x^2+\sqrt {b+a x^4}\right )\right )}{6 x^4 \sqrt {b+a x^4}} \] Input:
Integrate[(a + b/x^4)^(3/2)/x,x]
Output:
(Sqrt[a + b/x^4]*(-(Sqrt[b + a*x^4]*(b + 4*a*x^4)) + 3*a^(3/2)*x^6*Log[Sqr t[a]*x^2 + Sqrt[b + a*x^4]]))/(6*x^4*Sqrt[b + a*x^4])
Time = 0.31 (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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{4} \int \left (a+\frac {b}{x^4}\right )^{3/2} x^4d\frac {1}{x^4}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (-a \int \sqrt {a+\frac {b}{x^4}} x^4d\frac {1}{x^4}-\frac {2}{3} \left (a+\frac {b}{x^4}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (-a \left (a \int \frac {x^4}{\sqrt {a+\frac {b}{x^4}}}d\frac {1}{x^4}+2 \sqrt {a+\frac {b}{x^4}}\right )-\frac {2}{3} \left (a+\frac {b}{x^4}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (-a \left (\frac {2 a \int \frac {1}{\frac {1}{b x^8}-\frac {a}{b}}d\sqrt {a+\frac {b}{x^4}}}{b}+2 \sqrt {a+\frac {b}{x^4}}\right )-\frac {2}{3} \left (a+\frac {b}{x^4}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (-a \left (2 \sqrt {a+\frac {b}{x^4}}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )\right )-\frac {2}{3} \left (a+\frac {b}{x^4}\right )^{3/2}\right )\) |
Input:
Int[(a + b/x^4)^(3/2)/x,x]
Output:
((-2*(a + b/x^4)^(3/2))/3 - a*(2*Sqrt[a + b/x^4] - 2*Sqrt[a]*ArcTanh[Sqrt[ a + b/x^4]/Sqrt[a]]))/4
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 = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.29
method | result | size |
risch | \(-\frac {\left (4 a \,x^{4}+b \right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}}{6 x^{4}}+\frac {a^{\frac {3}{2}} \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}{2 \sqrt {a \,x^{4}+b}}\) | \(76\) |
default | \(-\frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} \left (-3 a^{\frac {3}{2}} \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right ) x^{6}+4 a \,x^{4} \sqrt {a \,x^{4}+b}+b \sqrt {a \,x^{4}+b}\right )}{6 \left (a \,x^{4}+b \right )^{\frac {3}{2}}}\) | \(78\) |
Input:
int((a+b/x^4)^(3/2)/x,x,method=_RETURNVERBOSE)
Output:
-1/6*(4*a*x^4+b)/x^4*((a*x^4+b)/x^4)^(1/2)+1/2*a^(3/2)*ln(a^(1/2)*x^2+(a*x ^4+b)^(1/2))*((a*x^4+b)/x^4)^(1/2)*x^2/(a*x^4+b)^(1/2)
Time = 0.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x} \, dx=\left [\frac {3 \, a^{\frac {3}{2}} x^{4} \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right ) - 2 \, {\left (4 \, a x^{4} + b\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{12 \, x^{4}}, -\frac {3 \, \sqrt {-a} a x^{4} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a}\right ) + {\left (4 \, a x^{4} + b\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{6 \, x^{4}}\right ] \] Input:
integrate((a+b/x^4)^(3/2)/x,x, algorithm="fricas")
Output:
[1/12*(3*a^(3/2)*x^4*log(-2*a*x^4 - 2*sqrt(a)*x^4*sqrt((a*x^4 + b)/x^4) - b) - 2*(4*a*x^4 + b)*sqrt((a*x^4 + b)/x^4))/x^4, -1/6*(3*sqrt(-a)*a*x^4*ar ctan(sqrt(-a)*sqrt((a*x^4 + b)/x^4)/a) + (4*a*x^4 + b)*sqrt((a*x^4 + b)/x^ 4))/x^4]
Time = 1.55 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x} \, dx=- \frac {2 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{4}}}}{3} - \frac {a^{\frac {3}{2}} \log {\left (\frac {b}{a x^{4}} \right )}}{4} + \frac {a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {b}{a x^{4}}} + 1 \right )}}{2} - \frac {\sqrt {a} b \sqrt {1 + \frac {b}{a x^{4}}}}{6 x^{4}} \] Input:
integrate((a+b/x**4)**(3/2)/x,x)
Output:
-2*a**(3/2)*sqrt(1 + b/(a*x**4))/3 - a**(3/2)*log(b/(a*x**4))/4 + a**(3/2) *log(sqrt(1 + b/(a*x**4)) + 1)/2 - sqrt(a)*b*sqrt(1 + b/(a*x**4))/(6*x**4)
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x} \, dx=-\frac {1}{4} \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right ) - \frac {1}{6} \, {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {a + \frac {b}{x^{4}}} a \] Input:
integrate((a+b/x^4)^(3/2)/x,x, algorithm="maxima")
Output:
-1/4*a^(3/2)*log((sqrt(a + b/x^4) - sqrt(a))/(sqrt(a + b/x^4) + sqrt(a))) - 1/6*(a + b/x^4)^(3/2) - 1/2*sqrt(a + b/x^4)*a
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (43) = 86\).
Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.07 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x} \, dx=-\frac {1}{4} \, a^{\frac {3}{2}} \log \left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2}\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{4} a^{\frac {3}{2}} b - 3 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} a^{\frac {3}{2}} b^{2} + 2 \, a^{\frac {3}{2}} b^{3}\right )}}{3 \, {\left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} - b\right )}^{3}} \] Input:
integrate((a+b/x^4)^(3/2)/x,x, algorithm="giac")
Output:
-1/4*a^(3/2)*log((sqrt(a)*x^2 - sqrt(a*x^4 + b))^2) + 2/3*(3*(sqrt(a)*x^2 - sqrt(a*x^4 + b))^4*a^(3/2)*b - 3*(sqrt(a)*x^2 - sqrt(a*x^4 + b))^2*a^(3/ 2)*b^2 + 2*a^(3/2)*b^3)/((sqrt(a)*x^2 - sqrt(a*x^4 + b))^2 - b)^3
Time = 0.77 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x} \, dx=\frac {a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2}-\frac {a\,\sqrt {a+\frac {b}{x^4}}}{2}-\frac {{\left (a+\frac {b}{x^4}\right )}^{3/2}}{6} \] Input:
int((a + b/x^4)^(3/2)/x,x)
Output:
(a^(3/2)*atanh((a + b/x^4)^(1/2)/a^(1/2)))/2 - (a*(a + b/x^4)^(1/2))/2 - ( a + b/x^4)^(3/2)/6
Time = 0.22 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.56 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x} \, dx=\frac {12 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a^{2} x^{10}+3 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a b \,x^{6}-16 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, a^{2} x^{10}-16 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, a b \,x^{6}-3 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, b^{2} x^{2}+12 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a^{3} x^{12}+9 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a^{2} b \,x^{8}-16 a^{3} x^{12}-24 a^{2} b \,x^{8}-9 a \,b^{2} x^{4}-b^{3}}{6 x^{6} \left (4 \sqrt {a \,x^{4}+b}\, a \,x^{4}+\sqrt {a \,x^{4}+b}\, b +4 \sqrt {a}\, a \,x^{6}+3 \sqrt {a}\, b \,x^{2}\right )} \] Input:
int((a+b/x^4)^(3/2)/x,x)
Output:
(12*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x**4 + b) + sqrt(a)*x**2)/sqrt(b) )*a**2*x**10 + 3*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x**4 + b) + sqrt(a)* x**2)/sqrt(b))*a*b*x**6 - 16*sqrt(a)*sqrt(a*x**4 + b)*a**2*x**10 - 16*sqrt (a)*sqrt(a*x**4 + b)*a*b*x**6 - 3*sqrt(a)*sqrt(a*x**4 + b)*b**2*x**2 + 12* log((sqrt(a*x**4 + b) + sqrt(a)*x**2)/sqrt(b))*a**3*x**12 + 9*log((sqrt(a* x**4 + b) + sqrt(a)*x**2)/sqrt(b))*a**2*b*x**8 - 16*a**3*x**12 - 24*a**2*b *x**8 - 9*a*b**2*x**4 - b**3)/(6*x**6*(4*sqrt(a*x**4 + b)*a*x**4 + sqrt(a* x**4 + b)*b + 4*sqrt(a)*a*x**6 + 3*sqrt(a)*b*x**2))